Gravitationally induced decoherence vs space-time diffusion: testing the quantum nature of gravity

We consider two interacting systems when one is treated classically while the other system remains quantum. Consistent dynamics of this coupling has been shown to exist, and explored in the context of treating space-time classically. Here, we prove that any such hybrid dynamics necessarily results in decoherence of the quantum system, and a breakdown in predictability in the classical phase space. We further prove that a trade-off between the rate of this decoherence and the degree of diffusion induced in the classical system is a general feature of all classical quantum dynamics; long coherence times require strong diffusion in phase-space relative to the strength of the coupling. Applying the trade-off relation to gravity, we find a relationship between the strength of gravitationally-induced decoherence versus diffusion of the metric and its conjugate momenta. This provides an experimental signature of theories in which gravity is fundamentally classical. Bounds on decoherence rates arising from current interferometry experiments, combined with precision measurements of mass, place significant restrictions on theories where Einstein’s classical theory of gravity interacts with quantum matter. We find that part of the parameter space of such theories are already squeezed out, and provide figures of merit which can be used in future mass measurements and interference experiments.

We consider two interacting systems when one is treated classically while the other system remains quantum. Consistent dynamics of this coupling has been shown to exist, and explored in the context of treating space-time classically. Here, we prove that such hybrid dynamics necessarily results in decoherence of the quantum system, and a breakdown in predictability in the classical phase space. We further prove that a trade-off between the rate of this decoherence and the degree of diffusion induced in the classical system is a general feature of all classical quantum dynamics; long coherence times require strong diffusion in phase-space relative to the strength of the coupling. Applying the trade-off relation to gravity, we find a relationship between the strength of gravitationally-induced decoherence versus diffusion of the metric and its conjugate momenta. This provides an experimental signature of theories in which gravity is fundamentally classical. Bounds on decoherence rates arising from current interferometry experiments, combined with precision measurements of mass, place significant restrictions on theories where Einstein's classical theory of gravity interacts with quantum matter. We find that part of the parameter space of such theories are already squeezed out, and provide figures of merit which can be used in future mass measurements and interference experiments.

I. INTRODUCTION
When considering the dynamics of composite quantum systems, there are many regimes where one system can be taken to be classical and the other quantum-mechanical. For example, in quantum thermodynamics we often have a quantum system interacting with a large thermal reservoir that can be treated classically, whilst in atomic physics it is common to consider the behaviour of quantum atoms in the presence of classical electromagnetic fields. Things become more complicated when one considers classical-quantum (CQ) dynamics where the quantum system backreacts on the classical system. This is particularly relevant in gravity, because we would like to study the back-reaction of thermal radiation being emitted from black holes on space-time, and while the matter fields can be described by quantum field theory, we only know how to treat space-time classically. Likewise in cosmology, vacuum fluctuations are a quantum effect which we believe seeds galaxy formation, while the expanding space-time they live on can only be treated classically. In addition to the need for an effective theory which treats space-time in the classical limit, there has long been a debate about whether one should quantise gravity [1][2][3][4][5][6][7][8][9][10][11][12].
There has even been much discussion on whether quantum-classical coupling can even be consistent. Many proposals for such dynamics [13,14] are not completely positive (CP) 1 , meaning they are at best an approximation and fail outside a regime of validity [15,16]. The semi-classical Einstein's equation [17,18], which replaces the quantum operator corresponding to the stress-energy tensor by its expectation value, is another attempt to treat the classical limit from an effective point of view, but it is non-linear in the state, leading to pathological behavior if quantum fluctuations are of comparable magnitude to the stress-energy tensor [19]. This is often the precise regime we would like to understand.
However, dynamics introduced in [20,21] and studied in [11,[22][23][24] do not suffer from such problems, and lead to consistent dynamics. In particular, the master-equation shown in Equation (4), is linear, preserves the division of classical degrees of freedom and quantum ones, and is completely positive (CP) and preserves normalisation. This ensures that probabilities of measurement outcomes remain positive and always add to 1. The dynamics is related to the GKSL or Lindblad equation [25,26], which for bounded generators of the dynamics, is the most general Markovian dynamics for an open quantum system. Likewise, Equation (4) is the most general Markovian classicalquantum dynamics with bounded generators [11]. Sub-classes of this master equation along with meeasurement and feedback approaches have been discussed in the context of Newtonian models of gravity [23,[27][28][29][30][31][32], and further developed into a spatially covariant framework so that Einstein gravity in the ADM formalism [33] emerges as a limiting case [11,34].
In this work, we move away from specific realisations of CQ dynamics, in order to discuss their common features and the experimental signatures that follow from this. An early precursor to the discussion here, is the insight of Diósi [21] who added classical noise and quantum decoherence to the master equation of [13], and found the noise and decoherence trade-off required for the dynamics to become completely positive. Here we prove that the phenomena found in [11,21,24] are generic features of all CQ dynamics; the classical-quantum interaction necessarily induces decoherence on the quantum system, and there is a generic trade-off between the rate of decoherence and the amount of diffusion in the classical phase space. The stronger the interaction between the quantum system and the classical one, the greater the trade-off. One cannot have quantum systems with long-coherence times without inducing a lot of diffusion in the classical system. One can also generalise this result to a trade-off between the rate of diffusion and the strength of more general couplings to Lindblad operators, with decoherence being a special case. This is expressed as Equations (25) and (23), which bounds the product of diffusion coefficients and Lindblad coupling constants in terms of the strength of the CQ-interaction. It is precisely this trade-off which allows the theories considered here, to evade the no-go arguments of Feynmann [1,2], Aharonov [3], Eppley and Hannah [4] and others [1, 7-9, 15, 16, 35-43]. The essence of arguments against quantum-classical interactions is that they would prohibit superpositions of quantum systems which source a classical field. Since different classical fields are perfectly distinguishable in principle, if the classical field is in a distinct state for each quantum state in the superposition, the classical field could always be used to determine the state of the quantum system, causing it to decohere instantly. By satisfying the trade-off, the quantum system preserves coherence because diffusion of the classical degrees of freedom mean that the state of the classical field does not determine the state of the quantum system [11,23]. Equation (25) and other variants we derive, quantify the amount of diffusion required to preserve any amount of coherence. If space-time curvature is treated classically, then complete positivity of the dynamics means its interaction with quantum fields necessarily results in unpredictability and gravitationally induced decoherence.
This trade-off between the decoherence rate and diffusion provides an experimental signature, not only of models of hybrid Newtonian dynamics such as [27] or post-quantum theories of General Relativity such as [11] but of any theory which treats gravity as being fundamentally classical. The metric and their conjugate momenta necessarily diffuse away from what Einstein's General Relativity predicts. This experimental signature squeezes classical-quantum theories of gravity from both sides: if one has shorter decoherence times for superpositions of different mass distributions, one necessarily has more diffusion of the metric and conjugate momenta. In Appendix F we show that the latter effect causes imprecision in measurements of mass such as those undertaken in the Cavendish experiment [44][45][46] or in measurements of Newton's constant "Big G" [47][48][49]. The precision at which a mass can be measured in a short time, thus provides an upper bound on the amount of gravitational diffusion, as quantified by Equation (42), while decoherence experiments place a lower bound on the diffusion. Our estimates suggest that experimental lower bounds on the coherence time of large molecules [50][51][52][53][54][55], combined with gravitational experiments measuring the acceleration of small masses [56][57][58], already place strong restrictions on theories where space time isn't quantised. In Section V we show that several realisations of CQ-gravity are already ruled out, while other realisations produce enough diffusion away from General Relativity to be detectable by future table-top experiments. Although the absence of such deviations from General Relativity would not be as direct a confirmation of the quantum nature of gravity, such as experiments proposed in [59][60][61][62][63][64][65][66][67] to exhibit entanglement generated by gravitons, it would effectively rule out any sensible theory which treats space-time classically. While confirmation of gravitational diffusion would suggest that space-time is fundamentally classical.
The outline of this paper is as follows. In Sec. II we review the general form of the CQ master equation of classicalquantum systems. The CQ-map can be represented in a manner akin to the Kraus-representation [68] for quantum maps, with conditions for it to be Completely Positive and Trace Preserving (CPTP). We can perform a short time moment expansion of the CQ-map taking states at some initial time, to states at a later time. This gives us the CQ version of the Kramers-Moyal expansion [69,70]. The physical meaning of the moments is discussed in Subsection II B. In Sec. III we show that there is a general trade-off between decoherence of the quantum system and diffusion in the classical system. We generalize the trade-off to the case of fields in Section IV and in Subsection V, we apply the inequality in the gravitational setting. The positivity constraints mean that the considerations do not depend on the specifics of the theory, only that it treats gravity classically, and be Markovian. This allows us to discuss some of the observational implications of this result and we comment on the relevant figures of merit required in interference and precision mass measurements in order to constrain theories of gravity, as they are not always readily available in published reports. In addition to table-top constraints, we consider those due to cosmological observations. We then conclude with a discussion of our results in Sec. VI. The Appendix collects or previews a number of technical results.

II. CLASSICAL-QUANTUM DYNAMICS
Let us first review the general map and master equation governing classical-quantum dynamics. The classical degrees of freedom are described by a differential manifold M and we shall generically denote elements of the classical space by z. For example, we could take the classical degrees of freedom to be position and momenta in which case M = R 2 and z = (q, p). The quantum degrees of freedom are described by a Hilbert space H. Given the Hilbert space, we denote the set of positive semi-definite operators with trace at most unity as S ≤1 (H). Then the CQ object defining the state of the CQ system at a given time is a map : M → S ≤1 (H) subject to a normalization constraint M dzTr H [ ] = 1. To put it differently, we associate to each classical degree of freedom a sub-normalized density operator, (z), such that Tr H [ ] = p(z) ≥ 0 is a normalized probability distribution over the classical degrees of freedom and M dz (z) is a normalized density operator on H. An example of such a CQ-state is the CQ qubit depicted as a 2 × 2 matrix over phase space [24]. More generally, we can define any CQ operator f (z) which lives in the fibre bundle with base space M and fibre H.
Just as the Lindblad equation is the most general evolution law which maps density matrices to density matrices, we can ask, what is the most general evolution law, which preserves the quantum-classical state-space. Any such dynamics, if it is to preserve probabilities, must be completely positive, norm preserving, and linear in the CQ-state 2 . We will also require the map to be Markovian on the combined classical-quantum system, which is equivalent to requiring that there is no hidden system which acts as a memory. This is natural if the interaction is taken to be fundamental, but is the assumption which one might want to remove if one thinks of the hybrid theory as an effective description. We thus take these as the minimal requirements that any fundamental classical-quantum theory must satisfy if it is to be consistent.
The most general CQ-dynamics, which maps CQ states onto themselves can be written in the form [11] (z, where the L µ are an orthogonal basis of operators and Λ µν (z|z , δt) is positive semi-definite for each z, z . Henceforth, we will adopt the Einstein summation convention so that we can drop µν with the understanding that equal upper and lower indices are presumed to be summed over. The normalization of probabilities requires The choice of basis L µ is arbitrary, although there may be one which allows for unique trajectories [24]. Equation (1) can be viewed as a generalisation of the Kraus decomposition theorem.
In the case where the classical degrees of freedom are taken to be discrete, Poulin [71] used the diagonal form of this map to derive the most general form of Markovian master equation for bounded operators, which is the one introduced in [20]. When the classical degrees of freedom are taken to live in a continuous configuration space, we need to be a little more careful, since (z) may only be defined in a distributional sense; for example, (z) = δ(z,z) (z). In this case (1) is completely positive if the eigenvalues of Λ µν (z|z , δt), λ µ (z|z , δt), are positive so that dzdz P µ (z, z )λ µ (z|z , δt) ≥ 0 for any vector with positive components P µ (z, z ) [72].
One can derive the CQ master equation by performing a short time expansion of (1) in the case when the L µ are bounded [11]. To do so, we first introduce an arbitrary basis of traceless Lindblad operators on the Hilbert space, L µ = {I, L α }. Now, at δt = 0 we know (1) is the identity map, which tells us that Λ 00 (z|z , δt = 0) = δ(z, z ). Looking at the short time expansion coefficients, by Taylor expanding in δt 1, we can write By substituting the short time expansion coefficients into (1) and taking the limit δt → 0 we can write the master equation in the form where {, } + is the anti-commutator, and preservation of normalisation under the trace and dz defines W µν (z) = dz W µν (z |z).
We see the CQ master equation is a natural generalisation of the Lindblad equation and classical rate equation in the case of classical-quantum coupling. We give a more precise interpretation of the different terms arising when we perform the Kramers-Moyal expansion of the master equation at the end of the section. The positivity conditions from (1) transfer to positivity conditions on the master equation via (3). We can write the positivity conditions in an illuminating form by writing the short time expansion of the transition amplitude Λ µν (z|z , δt), as defined by equation (3), in block form and the dynamics will be positive if and only if Λ µν (z|z , δt) is a positive matrix. It is possible to introduce an arbitrary set of Lindblad operatorsL µ and appropriately redefine the couplings W µν (z|z ) in (4) [11]. For most purposes, we shall work with a set of Lindblad operators which includes the identity L µ = (I, L α ); this is sufficient since any CQ master equation is completely positive if and only if it can be brought to the form in (4), where the matrix (6) is positive.

A. The CQ Kramers-Moyal expansion
In order to study the positivity conditions it is first useful to perform a moment expansion of the dynamics in a classical-quantum version of the Kramers-Moyal expansion [11]. In classical Markovian dynamics, the Kramers-Moyal expansion relates the master equation to the moments of the probability transition amplitude and proves to be useful for a multitude of reasons. Firstly, the moments are related to observable quantities; for example, the first and second moments of the probability transition amplitude characterize the amount of drift and diffusion in the system. This is reviewed in Subsection II B. Secondly, the positivity conditions on the master equation transfer naturally to positivity conditions on the moments, which we can then relate to observable quantities. In the classical-quantum case, we shall perform a short time moment expansion of the transition amplitude Λ µν (z|z , δt) and then show that the master equation can be written in terms of these moments. We then relate the moments to observational quantities, such as the decoherence of the quantum system and the diffusion in the classical system.
We work with the form of the dynamics in (4), using an arbitrary orthogonal basis of Lindblad operators L µ = {I, L α }. We take the classical degrees of freedom M to be d dimensional, z = (z 1 , . . . z d ), and we label the components as z i , i ∈ {1, . . . d}. We begin by introducing the moments of the transition amplitude W µν (z|z ) appearing in the CQ master equation (3) The subscripts i j ∈ {1, . . . d} label the different components of the vectors (z − z ). For example, in the case where d = 2 and the classical degrees of freedom are position and momenta of a particle, z = (z 1 , z 2 ) = (q, p), then we . The components are then given by (z − z ) 1 = (q − q ) and (z − z ) 2 = (p − p ). M µν n,i1...in (z , δt) is seen to be an n'th rank tensor with d n components. In terms of the components D µν n,i1...in the short time expansion of the transition amplitude Λ µν (z|z ) is given by [72] and the master equation takes the form [11] ∂ where we define the Hermitian operator H(z) = i 2 (D µ0 0 L µ − D 0µ 0 L † µ ) (which is Hermitian since D µ0 0 = D 0µ * 0 ). We see the first line of (9) describes purely classical dynamics, and is fully described by the moments of the identity component of the dynamics Λ 00 (z|z ). The second line describes pure quantum Lindbladian evolution described by the zeroth moments of the components Λ α0 (z|z ), Λ αβ (z|z ); specifically the (block) off diagonals, D α0 0 (z), describe the pure Hamiltonian evolution, whilst the components D αβ 0 (z) describe the dissipative part of the pure quantum evolution. Note that the Hamiltonian and Lindblad couplings can depend on the classical degrees of freedom so the second line describes action of the classical system on the quantum one. The third line contains the non-trivial classical-quantum back-reaction, where changes in the distribution over phase space are induced and can be accompanied by changes in the quantum state.

B. Physical interpretation of the moments
Let us now briefly review the physical interpretation of the moments which will appear in our trade-off relation. In particular, the zeroth moment determines the rate of decoherence (and Lindbladian coupling more generally), the first moment gives the force exerted by the quantum system on the classical system, and the second moment determines the diffusion of the classical degrees of freedom. For this discussion we shall take the classical degrees of freedom to live in a phase space Γ = (M, ω), where ω is the symplectic form. Consider where we have used cyclicity of trace and integration by parts, to bring the equation of motion into a form which would enable us to write a CQ version of the Heisenberg representation [11] for a CQ operator. If we are interested in the expectation value of phase space variables O(z) = z i I then Equation (10) gives with all higher order terms vanishing, and we see that µν =00 D µν 1,i L † ν L µ governs the average rate at which the quantum system moves the classical system through phase space, and with the back-reaction is quantified by the Hermitian matrix D αµ 1 := (D br 1 ) αµ . The force of this back-reaction is especially apparent if the equations of motion are Hamiltonian in the classical limit as in [11]. I.e. if we define H I (z) := h αβ L † β L α and take D αβ 1,i = ω j i d j h αβ with ω the symplectic form and d j the exterior derivative. Then Equation (11) is analogous to Hamilton's equations, and the CQ evolution equation after tracing out the quantum system has the form of a Liouville's equation to first order and in the classical limit, with ρ(z) := Tr [ (z)]. The significance of the second moment is also seen via Equation (10) to be related to the variance of phase space variables σ zi 1 zi 2 : In the case when D 1,zi 1 is uncorrelated with z i2 and D 1,zi 2 uncorrelated with z i1 , then the growth of the variance only depends on the diffusion coefficient. The zeroth moment D αβ 0 is just the pure Lindbladian couplings. The simplest example is the case of a pure decoherence process with a single Hermitian Lindblad operator L and decoherence coupling D 0 . Then we can define a basis {|a } via the eigenvectors of L and and we see that the matrix elements of which quantify coherence between the states |a ,|b decay exponentially fast with a decay rate of D 0 (L(a) − L(b)) 2 . For a damping/pumping process of a quantum harmonic oscillator with Hamiltonian H = ωa † a, L ↓ = a, L ↑ = a † , a the creation operator, and D ↑↑ 0 , D ↓↓ 0 the non-zero couplings, then standard calculations [24,73] show that an initial superposition 1 √ 2 |n + m with n,m large and n m will initially decohere at a rate of approximately (D ↑↑ 0 + D ↓↓ 0 )(m + n)/2, and the state will eventually thermalise to a temperature of ω/ log (D ↓↓ 0 /D ↑↑ 0 ). So in this case, the Lindblad couplings not only determine the rate of decoherence, but also the rate at which energy is pumped into the harmonic oscillator. In the next section we will derive the trade-off between Lindblad couplings and the diffusion coefficients. Although we will sometimes refer to this as a trade-off between decoherence and diffusion, this terminology is only strictly appropriate for pure decoherence processes, while more generally, it is a trade-off between Lindblad couplings and diffusion coefficients.

III. A TRADE OFF BETWEEN DECOHERENCE AND DIFFUSION
In this section we use positivity conditions to prove that the trade off between decoherence and diffusion seen in models such as those of [11,21,24] are in fact a general feature of all classical-quantum interactions. We shall also generalise this, and derive a trade-off between diffusion and arbitrary Lindbladian coupling strengths. The trade-off is in relation to the strength of the dynamics and is captured by Equation's (20), (23) and (25). In section IV we extend the trade-off to the case where the classical and quantum degrees of freedom can be fields and use this to show that treating the metric as being classical necessarily results in diffusion of the gravitational field.
There are two separate possible sources for the force (or drift) of the back-reaction of the quantum system on phase space -it can be sourced by either the D 0α 1,i components or the Lindbladian components D αβ 1,i . We shall deal with both sources simultaneously by considering a CQ Cauchy-Schwartz inequality which arises from the positivity of for any vector of CQ operators O µ . One can verify that this must be positive directly from the positivity conditions on Λ µν (z|z ) and we go through the details in appendix A. A common choice for O µ would be the set of operators L µ = {I, L α } appearing in the master equation. The inequality in Equation (15) turns out to be especially useful since it can be used to define a (pseudo) inner product on a vector of operators with components O µ via (16) where ||Ō|| = Ō ,Ō ≥ 0 due to (15). Technically this is not positive definite, but this shall not be important for our purpose. Taking the combination and as long as ||Ō 2 || = 0 we have a Cauchy-Schwartz inequality We can use (18) to get a trade-off between the observed diffusion and decoherence by picking O 2µ = δ α µ L α and O 1µ = b i (z − z ) i L µ , where L µ = {I, L α } are the Lindblad operators appearing in the master equation. In this case ||Ō 2 || = dzTr D αβ 0 L α L † β and one can verify using CQ Pawula theorem [72] 3 that in order to have non-trivial back-reaction on the quantum system complete positivity demands that ||Ō 2 || > 0, meaning the Cauchy-Schwartz inequality in Equation (18) must hold. By using the short time moment expansion of Λ µν (z|z ) defined in Equation (8) and using integration by parts, we then arrive at the observational trade-off between decoherence and diffusion which must hold for any positive CQ state (z). Stripping out the b i vectors, (19) is equivalent to the matrix positivity condition where we define Since (20) holds for all states, the tightest bound is provided by the infimum over all states The quantities D 2 and D 0 appearing in Equation (20) are related to observational quantities. In particular D 2 is the expectation value of the amount of classical diffusion which is observed and D 0 is related to the amount of decoherence on the quantum system. The expectation value of the back-reaction matrix D br 1 quantifies the amount of back-reaction on the classical system. In the trivial case D br 1 = 0, Equation (20) places little restriction on the diffusion and Lindbladian rates appearing on the left hand side. We already knew from [25,26] that the D αβ 0 must be a positive semi-definite matrix, and we also know that diffusion coefficients must be positive semi-definite. However, in the non-trivial case, the larger the back-reaction exerted by the quantum system, the stronger the trade-off between the diffusion coefficients and Lindbladian coupling. Equation (20) gives a general trade-off between observed diffusion and Lindbladian rates, but we can also find a trade-off in terms of a theory's coupling coefficients alone. We show in appendix A 1 that the general matrix trade-off holds for the matrix whose elements are the couplings D µν 2,ij , D αµ 1,i , D αβ 0 for any CQ dynamics. Moreover, (I − D 0 D −1 0 )D br 1 = 0, which tells us that D 0 cannot vanish if there is non-zero back-reaction. Equation (23) quantifies the required amount of decoherence and diffusion in order for the dynamics to be completely positive. In Equation (23), and throughout, D −1 0 is the generalized inverse of D αβ 0 , since D αβ 0 is only required to be positive semi-definite. In the special case of a single Lindblad operator α = 1 and classical degree of freedom, and when the only non-zero couplings are D 11 0 := D 0 , D 00 2,pp := 2D 2 and D 0 1,q = 1 this trade-off reduces to the condition D 2 D 0 ≥ 1 used in [21]. It is also useful to try to obtain an observational trade-off in terms of the total drift due to back-reaction as calculated in Equation (11) It follows directly from Equation (20) that when the back-reaction is sourced by either D 0µ 1,i or D αβ 1,i we can arrive at the observational trade-off in terms of the total drift 4 where the quantities appearing in Equation (25) are now all observational quantities, related to drift, decoherence and diffusion as outlined in the previous subsection II B.
In the case where the back-reaction is Hamiltonian at first order in the sense of Equation (12), then (25) can be written as As a result, we can derive a trade-off between diffusion and decoherence for any theory which reproduces this classical limit and treats one of the systems classically.
To summarize, whenever back-reaction of the quantum system on the classical system induces a force on the phase space, then we have a trade-off between the amount of diffusion on the classical system and the strength of decoherence on the quantum system (or more precisely the strength of the Lindbladian couplings D αβ 0 ). This can be expressed both as a condition on the matrix of coupling co-efficients in the master equation, via Equation (23) or in terms of observable quantities using Equation's (20) and (25). In the case when the back-reaction is Hamiltonian, we further have Equation (26). We would like to apply this trade-off to the case of gravity in the non-relativistic, Newtonian limit. In order to do so, we will need to generalise the trade-off to the case of quantum fields interacting with classical one, which we do in Section IV. The goal will be to understand the implications of treating the metric (or Newtonian potential) as being classical by using the trade-off when the quantum back-reaction induces a force on the gravitational field which, on expectation, is the same as the weak field limit of general relativity.

IV. TRADE OFF IN THE PRESENCE OF FIELDS
We would like to explore the trade-off in the gravitational setting and explore the consequences of treating the gravitational field as being classical and matter quantum. Since gravity is a field theory, we must first discuss classical-quantum master equations in the presence of fields. In the field theoretic case, both the Lindblad operators and the phase space degrees of freedom can have spatial dependence, z(x), L µ (x) and a general bounded CP map which preserves the classicality of the two systems can be written [11] where, as is usually the case with fields, in Equation (27) it should be implicitly understood that a smearing procedure has been implemented. We elaborate on some of details when fields are introduced in appendix B. The condition for (27) to be completely positive on all CQ states is meaning that Λ µν (x, y) can be viewed as a positive matrix in µν and a positive kernel in x, y. In the field theoretic case one can still perform a Kramers-Moyal expansion and find a trade-off between the coefficients D 0 (x, y), D 1 (x, y), D 2 (x, y) appearing in the master equation. The coefficients now have an x, y dependence, due to the spatial dependence of the Lindblad operators. The coefficients D 1 (x, y), D 2 (x, y) still have a natural interpretation as measuring the amount of force (drift) and diffusion, whilst D 0 (x, y) describes the purely quantum evolution on the system and can be related to decoherence. Using the positivity condition in (28) we find the same trade of between coupling constants in Equation (23) but where now D 2 (x, y) is the (p + 1)n × (p + 1)n matrix-kernel with elements D µν 2,ij (x, y), D br 1 (x, y) is the (p + 1)n × p matrix-kernel with rows labeled by µi, columns labelled by β, and elements D µβ 1,i (x, y), and D 0 (x, y) is the p × p decoherence matrix-kernel with elements D αβ 0 (x, y). 5 In the field theoretic trade off we are treating the objects in Equation (23) as matrix-kernels, so that for any position dependent vector . Explicitly, we find that positivity of the dynamics is equivalent to the matrix condition which should be positive for any position dependent vectors b i µ (x) and a α (x). This is equivalent to trade-off between coupling constants in Equation (23) if we view (23) as a matrix-kernel equation.
Though we make no assumption on the locality of the Lindbladian and diffusion couplings, we shall hereby assume that the drift back-reaction is local, so that D br 1 (x, y) = δ(x, y)D br 1 (x). As we shall see in the next section, this is a natural assumption if we want to have back-reaction which is given by a local Hamiltonian. However, one might not want to assume that the form of the Hamiltonian remains unchanged to arbitrarily small distances. With this locality assumption, Equation (29) gives rise to the same trade-off of Equation (23), where the trade-off is to be interpreted as a matrix kernel inequality. Writing this out explicitly we have where asking that this inequality holds for all vectors α i µ (x) is equivalent to the matrix-kernel trade-off condition of Equation (23). We give two examples of master Equations satisfying the coupling constant trade-off in appendix D. The decoherence-diffusion trade-off tells us how much diffusion and stochasticity is required to maintain coherence when the quantum system back-reacts on the classical one. If the interaction between the classical and quantum degrees of freedom is dictated by unbounded operators, such as the mass density, then there can exist states for which the back-reaction can be made arbitrarily large. This is the case for a quantum particle interacting with its Newtonian potential through its mass density at arbitrarily short distances. Hence, if one considers a particle in a superposition of two peaked mass densities, then there can be an arbitrarily large response in the Newtonian potential around those points, and either there must be an arbitrary amount of diffusion, or the decoherence must occur arbitrarily fast. The former is unphysical, while the latter turns out to be the case in simple examples of theories such as those discussed in Appendix E.
Since our goal is to experimentally constrain classical-quantum theories of gravity, we shall hereby ask that the map (27) is CP when acting on all physical states ρ. If one allows for arbitrarily peaked mass distributions then the coupling constant trade-off of Equation (30) should be satisfied. In the field theoretic case, we can similarly find an observational trade-off, relating the expected value of the diffusion matrix D 2 (x, y) to the expected value of the drift in a physical state as we did in Section III. This is done explicitly in Appendix B, using a field theoretic version of the Cauchy-Schwartz inequality given by Equation (B16), we find where equation (33) is to be understood as a matrix inequality with entries Similarly, when the back-reaction is sourced by either D 0µ 1,i or D αβ 1,i it follows from Equation (31) we can arrive at the observational trade-off in terms of the total drift due to back-reaction where We shall now use the trade-off to study the consequences of treating the gravitational field classically. We will consider the back-reaction of the mass on the gravitational field to be governed by the Newtonian interaction (or more accurately, a weak field limit of General Relativity). We shall then find that experimental bounds on coherence lifetimes for particles in superposition require large diffusion in the gravitational field in order to be maintained and this can be upper bounded by gravitational experiments.
To summarise this section, we have derived the trade-off between decoherence and diffusion for classical-quantum field theories, both in terms of coupling constants of the theory and in terms of observational quantities. This trade-off puts tight observational constraints on classical theories of gravity which we now discuss.

V. PHYSICAL CONSTRAINTS ON THE CLASSICALITY OF GRAVITY
In this section we apply the trade-off of Equation (29) to the case of gravity. Since the trade-offs derived in the previous section depend only on the back-reaction, or drift term, they are insensitive to the particulars of the theory. We shall consider the Newtonian, non-relativistic limit of a classical gravitational field which we reproduce in Appendix C. It is in taking this limit where some care should be taken, since one is gauge fixing the full general relativistic theory. We denote Φ to be the Newtonian potential and in the weak field limit of General Relativity, it has a conjugate momenta we denote by π Φ . We assume (i) The theory satisfies the assumptions used to derive the master equation as in Sections IV; in particular that the theory be a completely positive norm-preserving Markovian map, and that we can perform a short-time Kramers-Moyal expansion as in Appendix B.
(ii) We apply the theory to the weak field limit of General Relativity, where as recalled in Appendix C the Newtonian potential interacts with matter through its mass density m(x), and the conjugate momentum to Φ satisfiesπ where in the c → ∞ limit we recover Poisson's equation for the Newtonian potential. We assume this limit of General Relativity is satisfied on expectation, at least to leading order.
(iii) In relating D 0 to the decoherence rate of a particle in superposition, we shall assume that the state of interest is well approximated by a state living in a Hilbert space of fixed particle number. We believe this is a mild assumption: ordinary non-relativistic quantum mechanics is described via a single particle Hilbert space, and we frequently place composite massive particles in superposition and they do not typically decay into multiple particles.
(iv) We will assume that the diffusion kernel D 2 (Φ, x, x ) does not depend on π Φ i.e. it is minimally coupled. This is reasonable, since in the purely classical case matter couples to the Newtonian potential and not its conjugate momenta.
With these assumptions, and treating the matter density as a quantum operatorm(x), this tells us that in order for the back-reaction term to reproduce the Newtonian interaction on average then we must pick meaning that the back-reaction matrix D µα 1,πΦ is non vanishing. In Appendix C we give examples of master equations for which (38) is satisfied, but their details are irrelevant since we only require the expectation of the back-reaction force to be the expectation value of the mass -a necessary condition for the theory to reproduce Newtonian gravity.
As a consequence of the coupling constant and observational trade-offs derived in Equations (30) and (31), a nonzero D 1,πΦ implies that there must be diffusion in the momenta conjugate to π Φ . This diffusion is equivalent to adding a stochastic random process J(x, t) (the Langevin picture), to the equation of motion (36) to givė where we allow some colouring to the noise via a function u(Φ,m) which can depend on Φ, and the matter distribution m (assumption (iv)). The noise process satisfies where we have defined D 2 (x, y, Φ) = Tr D µν 2 (x, y, Φ)L µ (x)ρL † ν (y) , and ρ is the quantum state for the decohered mass density. Here the m, Φ subscripts of E m,Φ allow for the possibility that the statistics of the noise process can be dependent on the Newtonian potential and mass distribution of the particle. The restriction on E m,Φ [uJ(x, t)] follows from assumption (ii). If uJ(x, t) is Gaussian, Equation (40) completely determines the noise process, but in general, higher order correlations are possible, although they need not concern us here, since we are only interested in bounding the effects due to D 2 (x, y, Φ).
In the non-relativistic limit, where c → ∞, we can takeπ Φ to be small in comparison to the other terms, and we recover Poisson's equation for gravity, but with a stochastic contribution to the mass. This is precisely as expected on purely physical grounds: in order to maintain coherence of any mass in superposition, there must be noise in the Newtonian potential and this must be such that we cannot tell which element of the superposition the particle will be in, meaning the Newtonian potential should look like it is being sourced in part by a random mass distribution. In other words, the trade-off requires that the stochastic component of the coupling obscures the amount of mass m at any point.
The solution to Equation (39) is given by and a formal treatment of solutions to non-linear stochastic integrals of the form of Equation (39) can be found in [74]. A higher precision calculation would involve a full simulation of CQ dynamics and in Appendix C we show in full detail the evolution the Newtonian potential looks like for general continuous CQ theories using the continuous unraveling of CQ dynamics introduced in [75]. We find the effects are qualitatively and quantitatively the same as Equation (41).
In [72] it was shown that there are two classes of CQ dynamics, at least in the sense that there are those with continuous trajectories in phase space and those which contain discrete jumps. For the class of continuous CQ models (see [27] and appendix C 1 a), we know that J(x, t) should be described by a white noise process in time, and its statistics should be independent of the mass density of the particle. We go through the full CQ calculation for the continuous models in Appendix C27.
For the discrete class (see [11,76] and Appendix C 1 b), J(x, t) can involve higher order moments, and will generally be described by a jump process [24,72]. It's statistics can also depend on the mass density, since in general the diffusion matrix D µν 2,ij couples to Lindblad operators. It is worth noting that the discrete CQ theories considered in [11,24,34] generically suppress higher order moments, and often we expect that we can approximate the dynamics by a Gaussian process, but this need not be the case in general.
This variation in Newtonian potential leads to observational consequences which can be used to experimentally test and constrain CQ theories of gravity for various choices of kernels appearing in the CQ master equation. One immediate consequence is that the variation in Newtonian potential leads to a variation of force experienced by a particle or composite mass via F tot = − d 3 xm(x)∇Φ(x). We can also estimate the time averaged force via 1 T T 0 F tot where T is the time over which the force is measured and is the useful quantity when comparing with experiments. Using Equation (41), in Appendix F 1 we find that the variance of the magnitude of the time averaged force experienced by a particle in a Newtonian potential is given by where the variation is averaged over a time period T . We will use this to estimate the variation in precision measurements of mass, such as modern versions of the Cavendish experiement for various choices of D 2 (x , y , Φ) . On the other hand, experimentally measured decoherence rates can be related to D 0 . We explore the calculation of decoherence rates in gravity in detail in [77]. The important point is that the decoherence rate is dominated by the background Newtonian potential Φ b due to the Earth. In Appendix E, we show that for a mass whose quantum state is a superposition of two states |L and |R of approximately orthogonal mass densities m L (x), m R (x), and whose separation we take to be larger than the correlation range of D 0 (x, y), the decoherence rate is given by Via the coupling constant trade-off, Equations (42) and (43) then give rise to a double sided squeeze on the coupling D 2 . Equation (42) upper bounds D 2 in terms of the uncertainty of acceleration measurements seen in gravitational torsion experiments, whilst the coupling constant trade-off Equation (43) lower bounds D 2 in terms of experimentally measured decoherence rates arising from interferometry experiments.
We now show this for various choices of diffusion kernel, with the details given in Appendix F 1. The diffusion coupling strength will be characterized by the coupling constant D 2 , which we take to be a dimension-full quantity with units kg 2 sm −3 , and is related to the rate of diffusion for the conjugate momenta of the Newtonian potential. We upper bound D 2 by considering the variation of the time averaged acceleration σ a = σ F M for a composite mass M which contains N atoms which we treat as spheres of constant density ρ with radius r N and mass m N . We lower bound D 2 via the coupling constant trade-off of Equation (29) and then by considering bounds on the coherence time for particles with total mass M λ , which have typical length scale when in superposition R λ and volume V λ .
For continuous dynamics D 2 (x, y, Φ) = D 2 (x, y, Φ) since the diffusion is not associated to any Lindblad operators. Let us now consider a very natural kernel, namely D 2 (x, y; Φ) = D 2 (Φ)δ(x, y) which is both translation invariant, and does not create any correlations over space-like separated regions. In general, the squeeze will depend on the functional choice of D 2 (Φ) on the Newtonian potential. However, in the presence of a large background potential Φ b , such as that of the Earth's, we will often be able to approximate D 2 (Φ) = D 2 (Φ b ). This is true for kernels which depend on Φ and ∇Φ, though the approximation does not hold for all kernels, for example D 2 ∼ −∇ 2 Φ of Equation (D13) which creates diffusion only where there is mass density. For diffusion kernels D 2 (Φ b ) where the background potential is dominant we find the promised squeeze on where V b is the volume of space over which the background Newtonian potential is significant. V b enters since the variation in acceleration is found to be where the d 3 x integral is over all space. This immediately rules out continuous theories with noise everywhere, i.e, with a diffusion coefficient independent of the Newtonian potential since the integral will diverge. Standard Cavendish type classical torsion balance experiments measure accelerations of the order 10 −7 ms −2 , so a very conservative bound is σ a ∼ 10 −7 ms −2 , whilst for a kg mass N ∼ 10 26 and r N ∼ 10 −15 m. Conservatively where r E is the radius of the Earth and h is the atmospheric height gives D 2 ≤ 10 −41 kg 2 sm −3 . The decoherence rate λ is bounded by various experiments [78]. Typically, the goal of such experiments is to witness interference patterns of molecules which are as massive as possible. Taking a conservative bound on λ, for example that arising from the interferometry experiment of [54] which saw coherence in large organic fullerene molecules with total mass M λ = 10 −24 kg over a timescale of 0.1s, gives an upper bound on the decoherence rate λ < 10 1 s −1 . Fullerene molecules have typical size 10 −9 m. After passing through the slits the molecule becomes delocalized in the transverse direction on the order of 10 −7 m before being detected. Since the interference effects are due to the superposition in the transverse x direction, which is the direction of alignment of the gratings, it seems like a reasonable assumption to take the size of the wavepacket in the remaining y, z direction to be the size of the fullerene, since we could imagine measuring the y, z directions without effecting the coherence. We therefore take the volume V λ ∼ 10 −9 10 −9 10 −7 m 3 = 10 −25 m 3 , which gives D 2 ≥ 10 −24 kg 2 sm −3 , and suggests that classical-quantum theories of gravity with local continuous noise need to have a dependence on the Newtonian potential which will suppress the diffusion by 20 orders of magnitude. This happens to be the case for the kernel from Section D 3 and whose motivation comes from the constraint algebra [76].
On the other hand, the discrete models appear less constrained due to the suppression of the noise away from the mass density. For example consider the local discrete jumping models, such as the one given in Section C 1 b which is the Planck mass and l P = G c 3 is the Planck length, required to ensure D 2 has the units of kg 2 sm −3 . We find the squeeze on D 2 and plugging in the numbers tells us that discrete theories of classical gravity are not ruled out by experiment and we find 10 −1 kgs ≥ l 3 P m P D 2 ≥ 10 −25 kgs. We can also consider other noise kernels, with examples and some discussion given in Section D. A natural kernel is The inverse Lindbladian kernel satisfying the coupling constants trade-off is to zeroeth order in Φ(x), the Diosi-Penrose kernel D 0 (x, y, |x−y| . For this choice of dynamics, we find the squeeze for D 2 in terms of the variation in acceleration Using the same numbers as for the local continuous model, with R λ ∼ V 1/3 λ ∼ 10 −9 m we find that classical torsion experiments upper bound D 2 by 10 −9 kg 2 sm −1 ≥ l 2 P D 2 , whilst interferometry experiments bound D 2 from below via l 2 P D 2 ≥ 10 −40 kg 2 sm −1 . Equations (44), (46) and (47) show that classical theories of gravity are squeezed by experiments from both ways. We have here been extremely conservative, and we anticipate that further analysis, as well as near term experiments, can tighten the bounds by orders of magnitude. There are several proposals for table-top experiments to precisely measure gravity, some of which have recently been performed, and which could give rise to tighter upper bounds on D 2 . Some of these experiments involve millimeter-sized masses whose gravitational coupling is measured via torsional pendula [56,57], or rotating attractors [58]. With such devices, the gravitational coupling between small masses can be measured while limiting the amount of other sources of noise. There are proposals for further mitigating the noise due to the environment, including the inertial noise, gas particles collisions, photon scattering on the masses, and curvature fluctuations due to other sources [79][80][81]. Other experiments are based on interference between masses; for example, atomic interferometers allow for the measurement of the curvature of space-time over a macroscopic superposition [82,83].
We can get stronger lower bounds via improved coherence experiments. Typically, the goal of such experiments is to witness interference patterns of molecules which are as massive as possible, while here, we see that the experimental bound on CQ theories is generically obtained by maximizing the coherence time for massive particles with as small wave-packet size V λ .
Thus far in this section we have considered local effects on particles due to the diffusion. While this enables us to rule out some types of theories, the bounds are generally weak if one wanted to rule out all of them. However, it may be possible to do so via cosmological considerations. In attempting to place experimental constraints on this diffusion, it is also worth considering other regimes, such as longer range effects which might be detected by gravitational wave detectors such as LIGO.
In Appendix F 2, we begin a study of the cosmological effects of the diffusion by studying the observational trade-off in Equation (33). For the class of CQ theories sourced by either the D 0µ 1,i term, or the D αβ 1,i this gives a lower bound for the diffusion of the conjugate momentum in terms of the mass density of the particle and its decoherence rate λ which leads to an estimate of the production rate of stochastic waves in terms of their gravitational kinetic energy. This can be lower bounded in terms of experimentally decoherence rates The diffusion is akin to the stochastic production of gravitational waves, but these waves need not be transverse (see Appendix F). One advantage of studying this regime, is that Equation (49) is a bound which holds very generally, and which is independent of the choice of kernel, since the kinetic energy rate is lower bounded in terms of a experimentally measured decoherence rate. Since fullerene interference experiments require d∆Ē dt ∼ 10 −19 Js −1 over the wave-packet size of nucleons, this implies that the energy density of waves must be produced at a rate of at least ∼ 10 5 Js −1 m −3 . If this was produced over all space, as required by the diffusion kernel D 2 (x, x ) = D 2 δ(x, x ) (already ruled out by precision Cavendish experiments), this gives an apparent energy density in the ball park of 10 22 J/m 3 accumulated over the age of the universe. Since the observed expansion rate of the universe puts its energy density at 10 −9 J/m 3 , this discrepancy would appear to rule out continuous realisations of classical gravity with the diffusion kernel δ(x, x ). This can be seen as a specific instance of diffusion kernels D 2 (x, x ) which diverge as x → x , something we comment on in Section D. However, even here, care should be taken, partly because our estimate is non-relativistic 6 , and partly because our understanding of cosmology requires some degree of modesty -after all, quantum field theory predicts an energy density of 10 113 J/m 3 , and yet we do not see this reflected in the acceleration of the universe.
We leave a detailed study of the effect of gravitational diffusion on LIGO to future work. It suffices to mention that the effect will again depend on the form of the kernel D 2 (x, x ). Our estimates [84] suggest that local effects from tabletop experiments currently place stronger bound on gravitational theories than LIGO currently does. In particular, unlike for gravitational wave measurements, which are reasonably high frequency events requiring extraordinary high precision in relative displacement of the arm length from its average, it is preferential to have a lower precision measurement, but which occurs over a longer time period to allow for the diffusion in path length to build up, and with a smaller uncertainty in the average length of the arm itself. Furthermore, since the LIGO arm is kept in a vacuum, we do not expect strong bounds on discrete models where the diffusion is associated to an energy density.

VI. DISCUSSION
A number of direct proposals to test the quantum nature of gravity are expected to come online in the next decade or two. These are based on the detection of entanglement between mesoscopic masses inside matter-wave interferometers [59][60][61][62][63][64][65]67]. For these experiments, some theoretical assumptions are needed: one requires that it is only gravitons which travel between the two masses and mediate the creation of entanglement. If this is the case, then the onset of entanglement implies that gravity is not a classical field. These can be thought of as experiments which if successful, would confirm the quantum nature of gravity (although other alternatives to quantum theory are possible [85]).
Here, we come from the other direction, by supposing that gravity is instead classical, and then exploring the consequences. Theories in which gravity is fundamentally classical were thought to have been ruled out by various nogo theorems and conceptual difficulties. However, these no-go theorems are avoided if one allows for non-deterministic coupling as in [11, 20-24, 27, 34, 72]. We have here proven that this feature is indeed necessary, and made it quantitative by exploring the consequences of complete positivity on any dynamics which couples quantum and classical degrees of freedom. Complete positivity is required to ensure the probabilities of measurement outcomes remain positive throughout the dynamics. We have shown that any theory which preserves probabilities and treats one system classically, is required to have fundamental decoherence of the quantum system, and diffusion in phase space, both of which are signatures of information loss. Using a CQ version of the Kramers-Moyal expansion, we have derived a trade-off between decoherence on the quantum system, and the system's diffusion in phase space. The trade-off is expressed in terms of the strength of the back-reaction of the quantum system on the classical one. We have derived the trade-off both in terms of coupling constants of the theory, and in terms of observational quantities that can be measured experimentally.
In the case of gravity, the observational trade-off places a lower bound on the rate of diffusion of the gravitational degrees of freedom as expressed by Equation (48) in terms of the decoherence rate of particles in superposition. We find that theories which treat gravity as fundamentally classical, are not ruled out by current experiments, however we have been able to rule out a broad parameter space of such theories. This is done partly through table-top observations via Equations (44), (46) and (47). Given any diffusion kernel, we can compute the inaccuracy of mass measurements due to fluctuations in the gravitational field, and using the trade-off, we can derive a bound on the associated decoherence rate. This allows us to rule out broad classes of theories in terms of their diffusion kernel. For example, we are able to rule out a number of theories which are continuous in phase space. Then, using the trade-off of Equation (49), we saw that there was some tension with cosmological observations and kernels such as that of Equation (D11) and (D12), which produce diffusion over all space. However, we are not confident enough in our understanding of cosmology in CQ theories to rule these out.
Any theory which treats gravity classically has fairly limited freedom to evade the effects of the trade-off. There is freedom to choose the diffusion or decoherence kernels D 2 (x, x ) and D 0 (x, x ), but the trade-off restricts one in terms of the other. Then, because of the results proven in [72], one can consider two classes of theory, those which are continuous realisations and whose diffusion can only depend on the gravitational degrees of freedom, and discrete theories whose diffusion can also depend directly on the matter fields. Examples of both classes of theory are given in Appendix C. Finally, one could consider theories which do not reproduce the weak field limit of General Relativity to all distances, namely we could imagine that the interaction Hamiltonian of Equation (35) does not hold to arbitrarily short distances, or arbitrarily high mass densities. This would correspond to modifying D 1 (x, x ) in some way, either by making it slightly non-local, or by disallowing arbitrarily high mass densities, or by including an additional contribution such as the friction term discussed in C 1 a. All of these modifications would seem to violate Lorentz invariance in some way 7 .
Here, we have only given an order of magnitude estimate of when gravitational diffusion will lead to appreciable deviations from Newtonian gravity or General Relativity. We have done so in a number of regimes. The most promising being table-top experiments which precisely measure the mass of an object. This is an area which is important from the perspective of weight standards, for example those undertaken by NIST on the 1kg mass standard K20 and K4 [86]. The increased precision and measuring time of Kibble Balances [87] and atomic interferometers [82,83,88,89] would make such measurements an ideal testing ground, both to further constrain the diffusion kernel, and to look for diffusion effects, whose dependence on the test mass is outlined in Appendix F. Here, we have found that the time T over which results of the measurement are made, affects the strength of the bound, and it would be helpful if future experiments reported this value. Since we have found that CQ theories predict an uncertainty in mass measurements it is perhaps intriguing that different experiments to measure Newton's constant G yield results whose relative uncertainty differ by as much as 5 · 10 −4 m 3 kg −1 s −2 , which is more than an order of magnitude larger than the average reported uncertainty [47][48][49]. If one were to try and explain the discrepancy in G measurements via gravitational diffusion, then for all the kernels we studied in Section V we find that the variation in acceleration depends on 1 √ N the number of nucleons in the test mass, so that masses with smaller volume should yield larger uncertainty and this would be the effect to look for in measurement discrepancies. The relatively large uncertainty in such measurements, also makes it challenging for table-top experiments to place strong upper bounds on gravitational diffusion.
We have also estimated the effect that this diffusion would have on the energy density of the universe, and in the production of stochastic waves in terms of gravitational kinetic energy in the weak field limit. We have found that spatially uncorrelated and continuous realisations of classical gravity which reproduce General Relativity at short distances, appear to be ruled out by cosmological considerations as well, since the energy density of the stochastic wave contribution is high enough that it should effect the expansion rate of the universe. However, this is a regime where we do not understand the theory well, and so we are cautious about making too strong a claim. We have also found that the stochastic production of gravitational kinetic energy waves is in a regime which could be detectable by LIGO, an effect which constrains the form of D 2 (x, x ). However, initial estimates suggest that this is less constraining than table top experiments. For this to be definitive, a more precise understanding of gauge artifacts and of the dynamics that the diffusion induces on the Newtonian potential is required, especially over longer time scales. How this diffusion might effect dynamics over galactic scales and longer times, requires a fuller General Relativistic treatment, a study which we undertake in [90] where we find that it causes deviations from what general relativity predicts.
Turning to the other side of the trade-off, improved decoherence times would further squeeze theories in which gravity remains classical. While a current experimental challenge is to demonstrate interference patterns using larger and larger mass particles, we here find that some of our bounds depend on the expectation of the particle's mass density, either in terms of m 2 (x) /λ, or in ways which depend on the particular kernel. Thus interference experiments with particles of high mass density rather than mass can be preferable. There are also kernels, for which the relevant quantity is the expectation of the mass density, which will depend on the size of the wave-packet used in the interference experiment, a quantity which is rarely obtainable from most papers which report on such experiments. While this dependence might initially appear counter-intuitive, it follows from the fact that in order to relate the trade-off in terms of coupling constants to observational quantities, and in particular, the decoherence rate, we took expectation values of the relevant quantities to get a trade-off in terms of only averages. And indeed the decoherence rate, which is an expectation value, can easily depend on the wave-packet density, as we see from examples is Section E.
Since we here show that all theories which treat gravity classically necessarily decohere the quantum system, another constraint on theories which treat gravity classically is given by constraints on fundamental decoherence. These are usually constrained by bounds on anomalous heating of the quantum system [91]. However, these constraints are not in themselves very strong, since fundamental decoherence effects can be made arbitrarily weak. In the simplified model in Appendix C, the strength of the decoherence depends on the strength of the gravitational field, thus, constraints due to heating [91][92][93][94][95][96][97][98][99][100][101][102][103][104] can be suppressed, either by scaling the Lindbladian coupling constants, or by having strong decoherence effects more pronounced near stronger gravitational fields such as near black holes where one expects information loss to occur. The necessity for decoherence to heat the quantum system is further weaked by the fact that the dynamics are not Markovian on the quantum fields, if one integrates out the classical degrees of freedom, space-time acts as a memory. This potentially captures some of the non-Markovian features advocated in [105], who recognised that Markovianity is a key assumption in attempts to rule out fundamental decoherence or information loss. Here however, we see that there is less freedom than one might imagine. If the Lindbladian coupling constants are made small to reduce heating, the gravitational diffusion must be large. Thus, heating constraints which place bounds on D 0 (x, x ) place additional constraints on D 2 (x, x ).
While the absence of diffusion could rule out theories where gravity is fundamentally classical, the presence of such deviations, at least on short time scales, might not by itself be a confirmation of the classical nature of gravity. Such effects could instead be caused by quantum theories of gravity whose classical limit is effectively described by [11] or perhaps [106]. In other words, one might expect some gravitational diffusion, because from an effective theory point of view, one is in a regime where space-time is behaving classically. However, the trade-off we have derived is a direct consequence of treating the background space-time as fundamentally classical. In a fully quantum theory of gravity, the interaction of the gravitational field with particles in a superposition of two trajectories will cause decoherence, but coherence can then be restored when the two trajectories converge. This is what happens when electrons interact with the electromagnetic field while passing through a diffraction grating, yet still form an interference pattern at the screen. This is a non-Markovian effect, and the trade-off we derived is a direct consequence of the positivity condition, which is a direct consequence of the Markovian assumption. In the non-Markovian theory where General Relativity is treated classically, one still expects the master equation to take the form found in [11], but without the matrix whose elements are D µν n needing to be positive semi-definite [107,108].
In this section, we will introduce two forms of positivity conditions used to prove the decoherence diffusion trade-off. The first inequality we would like to introduce is which holds for any A µ (z, z ) for which (A1) is well defined: i.e, so that the distributional derivatives in (A1) are well defined. We can derive the positivity condition (A1) from the positivity of Λ µν (z|z ), which must be a positive semi-definite matrix in µν. More precisely, the eigenvalues of Λ µν (z|z ), which we denote by λ µ (z|z ) must be positive. They must be positive in the distributional sense, since we allow for the case that λ µ (z|z ) is a positive distribution, for example λ 0 (z|z ) ∼ δ(z − z ). Hence we require dzdz λ µ (z|z )P (z, z ) (A2) is positive for any positive smearing function P (z, z ). Since each λ µ must be positive, we can also pick a different smearing function for each µ, so that should be positive for any vector P µ (z, z ) with all positive entries. We can then write the matrix Λ µν (z|z ) in terms of its eigenvalues We can then see the positivity of (A1) directly since which is positive as a consequence of (A3). As a consequence of Equation (A1) being positive, we also know that will be positive for any vector of operators (potentially phase space dependent) O µ (z, z ). This follows from the cyclicity of the trace and the fact that Λ µν (z|z )O † ν (z, z )O µ (z, z ) will be a positive operator so long as (A1) holds. A common choice of O µ would be the Lindblad operators L µ appearing in the master equation.
The inequality in Equation (A1) proves useful to derive positivity conditions on the coupling constants appearing in the master equation, whilst (A6) is useful in deriving the observational trade-off for the continuous master equation as we shall now discuss.

General trade-off between decoherence and diffusion coefficients
We can get a general trade-off between the decoherence and diffusion coefficients which appear in the master equation, arriving at a trade-off between the decoherence and diffusion coefficients in terms of the back-reaction drift coefficient D µα 1,i . Consider (A1), with A µ = δ α µ a α + b i µ (z − z ) i . By integrating by parts over the phase space degrees of freedom, we find Taking i ∈ {1, . . . , n} α ∈ {1, . . . , p} and µ ∈ {1, . . . , p + 1}, we can write this as a matrix positivity condition where D 2 is the (p + 1)n × (p + 1)n matrix with elements D µν 2,ij , D br 1 is the (p + 1)n × p matrix with rows labeled by µi and columns labelled by β with elements D µβ 1,i and D 0 is the p × p decoherence matrix with elements D αβ 0 . D br 1,i describes the quantum back-reacting components of the drift. Equation (A8) is equivalent to the condition that the ((p + 1)n + p) × ((p + 1)n + p) matrix Since we know D 2 and D 0 must be positive semi-definite, we know from Schur decomposition that  In this section, we describe CQ dynamics in the case where the Lindblad operators and the phase-space degrees of freedom can have spatial dependence z(x), L µ (x).
For the case of fields, operators O(x) constructed out of local fields φ(x) will in general be unbounded and hence the Stinespring dilation theorem does not hold. This problem is a common one in the study of algebraic quantum field theory and we can get around it by considering the case in which operators are of interest are obtained by smearing the local fields over bounded functionals F . For example, we can first smear the local fields fields over a smearing function f , φ f = dxφ(x)f (x) and then consider bounded functions of φ f such as F (φ f ) = e iφ f . In doing this we can write a CQ version of the Stinespring dilation theorem exactly and proceed along the lines of [11] to show that any completely positive CQ map can be written in the form where the positvity condition states We shall assume that we deal with dynamics which can be written in Lindblad form, as is usually assumed in the unbounded case [109].
The characteristic function then takes the form C µν (u, z ; x, y) = Dze i dwu(w)·(z(w)−z (w)) Λ µν (z|z ; x, y) (B5) and expanding out the exponential this takes the form performing the inverse Fourier transform, allows us to write the transition amplitude in terms of functional derivatives of the delta function and we can use this to write a CQ master equation in the form D µν n,i1...in (z, w 1 , . . . w n ; x, y)L µ (x) (z)L † ν (y) .

Trade-off between diffusion and decoherence couplings in the presence of fields
In the field theoretic case the positivity condition is given by Equation (B2) and we can find a trade-off between decoherence and diffusion by considering where we use the shorthand notation D µν 2,ij (z, x, y) := D µν 2,ij (x, y) and similarly D αµ 1,i (z; x, y) := D αµ 1,i (x, y). Taking i ∈ {1, . . . , n} α ∈ {1, . . . , p} and µ ∈ {1, . . . , p + 1}, we can write this as a matrix positivity condition where D 2 (x, y) is the (p + 1)n × (p + 1)n matrix-kernel with elements D µν 2,ij (x, y), D br 1 (x, y) is the (p + 1)n × p matrixkernel with rows labeled by µi and columns labelled by β with elements D µβ 1,i (x, y) and D 0 (x, y) is the p×p decoherence matrix-kernel with elements D αβ 0 (x, y). D br 1,i describes the quantum back-reacting components of the drift. Equation (B10) is equivalent to the condition that the ((p + 1)n + p) × ((p + 1)n + p) matrix of operators be positive semi-definite. Here we are viewing the objects of (B11) as matrix-kernels, so that for any position . Since we know D 2 and D 0 must be positive semi-definite, we know from Schur decomposition that where D −1 0 is the generalized inverse of D 0 . Furthermore, from Equation (B13), we see if D 0 vanishes, then clearly D br 1 must also vanish in order for (B11) to be positive semi-definite.

Observational trade-off in the presence of fields
We can use the same methods to arrive at an observational trade-off using the field theoretic version of the Cauchy-Schwartz inequality in (18). This arises from the positivity of for any local vector of CQ operators O µ (z, z , x). We have to be careful, since (B14) is not in general well defined since O µ may not be trace-class. We hence assume that we consider states ρ(z) and operators O µ (z, z , x) for which (B14) is well defined. Since we are interested in getting an observational trade-off we expect this to always be the case for physical classical-quantum states ρ(z).
We shall use Equation (B14) to arrive at a (pseudo) inner product on a vector of operators O µ via where ||Ō|| = Ō ,Ō ≥ 0 due to (B14). Technically this is not positive definite, but again, this will not worry us. Hence, so long as ||Ō 2 || = 0, which holds due to the CQ inequality derived in the derivation of the Pawula theorem [72], we again have a Cauchy-Schwartz inequality , we arrive at a local trade-off between the diffusion, drift and the total decoherence. In particular, using B17, the definitions of the expectation values of couplings defined in (32) and the fact that for back-reaction the expectation value of D 0 cannot vanish, we arrive at the observational trade-off of Equation (33) b which we write in matrix form as It then follows directly from Equation (B19) that when the back-reaction is sourced by either D 0µ 1,i or D αβ 1,i components we can arrive at the observational trade-off in terms of the total drift where in Equation (B20) recall that the definition of D T 1 (x) † is given by Equation (34) in the main body.

A spatially averaged observational trade-off
It is also useful to note that one can arrive at a spatially averaged observational trade-off which can be used to bound all of the elements of the diffusion matrix, not just its diagonals. Specifically, taking Equation (B17) with b i (x) = b i a constant, we arrive at the trade-off where we define the expectation matrix For the Newtonian limit discussed in the main body this bounds the diffusion in terms of the total mass of the particle We can also arrive at a trade-off in terms of the effective Newtonian potential sourced by the masses by taking b i (x) = 1 |x−x| . In this case, we find the trade-off which for the Newtonian limit gives a trade-off between the diffusion matrix and the effective Newtonian potential of the particle as sourced by its expectation value where we have defined the effective Newtonian potential as Φ = −G dx m(x) |x−x| .
the pure gravity Hamiltonian, and is the interaction Hamiltonian, from which we see that non-relativistic matter couples to the Newtonian potential through its mass density m(x). In the case where we have the state of matter being described by a point particle δ(x − x(t)) of mass m the pure matter Hamiltonian would be Let us review the classical deterministic constraints. In the Newtonian limit, the Hamiltonian and momentum constraints themselves become [110] These are modified in the the classical-quantum case [34], but we need not consider this here. If the static approximation is made, then the Hamiltonian constraint is solved by π Φ = 0, in which case the Hamiltonian constraint reduces to Poisson's equation We can also see this directly from Hamilton's equations which come from varying the Newtonian Hamiltonian. The equations of motion for the gravitational degrees of freedom readṡ which, for π Φ = 0 yields the Newtonian solution for a stationary mass density. In a Louivile formulation the dynamics for the density ρ(Φ, π Φ , x i , p i ) is given by where the Hamiltonian and momentum constraints tell us that ρ(Φ, π Φ , x i , p i ) should only have support over phase space degrees of freedom which satisfy the Hamiltonian and momentum constraints in Equation (C13). From Equation (C18) we can identify the classical drift associated to the back-reaction of the matter on the gravitational field from the m(x) δρ δπΦ(x) term, so that In the classical-quantum case, we promote m(x) to an operatorm. In this case Equation (C11) is the interaction Hamiltonian used in [27] to study CQ gravity. We see from Equation (C18) that in any theory whose first moment reproduces the Newtonian back-reaction on average must have a D br 1,πΦ given by from which the discussion at the beginning of section V follows.

Weak field master equations
Although the trade-off we derive does not depend on the particulars of the classical-quantum theory (provided it reproduces Newtonian gravity in the classical limit), we give two concrete examples for completeness. In [72] we show that there are two classes of classical-quantum dynamics, one which is continuous in phase space, and one which has discrete jumps in phase space. We will give examples of each. Although they are the weak field limit of [11], it is worth stressing that taking the Newtonian limit entails certain coordinate choices and restrictions on the metric. For example, here, we have restricted ourselves to metrics of the form of Equation (C7). Any gauge fixing of general relativity which is done before deriving the master equation, is generally not equivalent to taking the master equations of [11], and then taking the appropriate limit [111].

a. Continuous master equation
For the class of continuous master equation's, specifying that the first moment on average satisfies Equation (C21) is enough (up to drift terms which vanish under trace) to fix the general form of master Equation to be were H c is the purely classical gravity Hamiltonian. We have taken the dynamics, i.e, the drift to be local in x, while we allow for the decoherence and diffusion terms to have some range. In this case the evolution law is still local but correlations can be created [112]. This master equation is close to the one considered in [27], where the decoherence and diffusion kernels are chosen to be the ones discussed in D 2. This is the weak field limit of the simplest realisation in [11]. The case where the diffusion is spatially uncorrelated D 2 (x, y) = (x − x ) a regulator which approaches a scalar delta function corresponds to the Newtonian limit of the diffusion term ( , y), which can be understood as the Newtonian limit of the spatially diffeomorphism invariant kernel discussed in Section D 3. We find in Section F that the δ(x, x ) kernel leads to diverging diffusion in the Newtonian potential, so this choice would need to be supplemented by some mechanism to control the diffusion. For example, a friction term such as In the weak field limit, this would adds a term proportional to to the master equation of Equation (C22). Such a term would break Lorentz invariance since it sets a temperature scale, although this is not necessarily a deal breaker, since it is believed by many that quantum gravity is also likely to also have an anomaly. However, the friction term is a modification to D 1 (x), and if too large, could run afoul of precision tests of General Relativity, such as the orbital decay of binary pulsars.

b. Discrete master equation
An example of a discrete master equation satisfying Equation (C21) is with τ a dimensionless constant, andm(x) = ψ † ψ a peaked regulator with units of inverse volume. We have here included and c to make it easier to compare with experiments. To leading order, we could drop terms proportional to Φ(x)/c 2 in both the exponential and in N √ g ≈ 1 − 2Φ/c 2 inside the integral over x. This gives These dynamical equations are supplemented with modified constraint equations as outlined in [34]. In any case, the trade-off in Equation (F19) is a statement independent of constraints and constraint preservation, at least in the weak field limit.

Unraveling of continuous master Equations and an exact sourcing by a random mass
In [75] we study unravelings of continuous master Equations, and we can use this to illustrate Equation (41). In particular, the Newtonian CQ dynamics of Equation (C22) is equivalent to the unravelled set of coupled stochastic differential equations where ξ i (x) is a Wiener process in time satisfying and ρ is a normalized quantum state. We see that the evolution of the quantum state in Equation (C27) causes the quantum state to decohere stochastically into a mass density eigenbasis m(x) at a rate determined by D 0 . After the state has decohered, the expectation value of the mass appearing in the dynamics for the conjugate momenta will look like it is being sourced by decohered mass density m(x) directly.
In the presence of a background potential, and assuming that the theory is minimally coupled, so that σ only depends on Φ b , we can absorb dyσ(Φ b , x, y)dξ(y) into a random noise term dξ where E[dξ(x)dξ(y)] = dtD 2 (Φ b , x, y).
Note that since dξ(x) is a Wiener process in time, then the noise term in Equation (41) corresponds to J(x, t) = dξ(x) dt and is a white noise process in time and E[J(x, t)J(x, t )] = D 2 (Φ b , x, y)δ(t, t ). Hence, after the decoherence time, we see that in the c → ∞ limit we can solve Equation's (C27) for Φ to find Equation (F1), where the Newtonian potential is being sourced by a random mass term and so the Equation for the Newtonian potential will be given exactly by that of Equation (41). An unravelling equation for hybrid Newtonian dynamics was also given in the measurement and feedback approach of [30,32,113]. In that approach, the dynamics is equivalent to an unravelling of a Lindblad equation, since there are no independent classical (gravitational) degrees of freedom. The Newtonian potential is directly sourced by a measurement of spatial separation between two particles, and so will fluctuate directly with measurement results. Here, the conjugate momentum to the Newtonian potential allows the gravitational field to have independent degrees of freedom, and becauseπ Φ is stochastic, rather than Φ(x), the dynamics can be continuous on the phase space.

Appendix D: Examples of Kernels saturating the decoherence diffusion coupling constants trade-off
In this section, we give examples of kernels satisfying the decoherence diffusion coupling constant trade-off in Equation (23). For any choice of kernel, we can compute the degree of diffusion it induces in precision mass measurements (Section F) and decoherence experiments (Section E) which allows us to rule out certain kernels experimentally.

Gaussian Lindbladian kernel
As a first example we shall take the Lindbladian coupling to be Gaussian, taking where g N (x, y) is a normalized Gaussian distribution. The mass m 0 is a reference mass, and we shall take it equal to the mass of the nucleons which were considered in Section V, meanwhile λ αβ is a coupling constant which determines the strength of the Lindbladian. It should be noted that with this choice of smearing function the pure Lindbladian evolution appearing in (C22) can be taken to resemble the Lindbladian part of spontaneous collapse models [55,[114][115][116][117], except here, there is no need to think about any ad-hoc field, nor think of the collapse as being a physical process. Rather, one necessarily gets decoherence of the wave-function for free, via gravitationally induced decoherence [11,23,30,32,77,118].
We now find the diffusion kernel D 2 (x, y) using the coupling constants trade-off in (23). For simplicity, we shall assume the trade-off is saturated, and we will take the back-reaction to be local, so that (D br y). In this case we find where g −1 N (x, y) is the kernel inverse of a normalized Gaussian distribution. It is shown in [119], that the inverse distribution takes the form where and the limit N → ∞ is taken. In Equation (D4) c n (r 0 ) = (−1) n (r0) 2n n! 2 n and d is the spatial dimension, so that x = (x 1 , x 2 , . . . , x d ).
In total then, we arrive at the expression for the D 2 which saturates the bound If we further take the back-reaction that of the Newtonian limit in section C (D br 1 ) µα i (x, y) = 1 2 δ 0m δ πΦ i δ(x, y) then we find the D 2 which saturates the bound is

Diosi-Penrose Lindbladian coupling
In this section, we give another example of a Lindbladian coupling which is familiar in the literature, For a single Lindblad operator, this is the coupling introduced in [27] used to reproduce a CQ master equation of gravity with a decoherence rate given by the Diosi-Penrose formula [120][121][122]. An alternative interpretation is presented in [77]. Here we consider the special case where the x, y dependence of D 0 (x, y) is the same for all α, β which need not hold in general. The fact that it gives the same decoherence rate as Diosi-Penrose can be seen by plugging Equation (D7) into the classical-quantum master equation in Equation (C22).
To invert the kernel in (D7) we use the fact that from which one can immediately read of the generalized inverse (D −1 0 ) αβ (x, y) to be where (D −1 0 ) αβ are the matrix elements of the generalized inverse of D 0 . As a consequence, we find for this specific choice of kernel that the diffusion matrix saturating the coupling constants bound in Equation (30) is where we have also assumed the back-reaction is local. Taking the back-reaction to further be that of Newtonian limit of Equation (C22) (D br This diffusion kernel is argued for on the grounds of having the fluctuations satisfy a Poisson equation, in [27].

Diffeomorphism invariant kernel
Attempts to derive the constraint algebra of a generally covariant CQ theory [34,76], motivates the spatially diffeomorphism invariant kernel where ∆ x is the Laplace-Beltrami operator 9 . This kernel's weak field limit is which is close to that of Equation (D11), but with a correction term which turns out to be important. Using D 1 (x, x ) = − 1 2 N √ gδ(x, x ), the Lindbladian kernel in dimension d which saturates the trade-off for this diffusion kernel is with G(x, x ) the Green's function for −∆. It is a density in the x coordinate and a scalar in x. In the weak field limit, and to 0th order in Φ(x), this gives the Diosi-Penrose kernel, Equation (D7). One could also consider the kernel which in the weak field limit is

A comment on divergences
The examples given above give rise to divergent variance in the classical degrees of freedom, since in both cases the diffusion coefficient diverges when evaluated at the same point D 2 (x, x). Though we do not have a general proof, this seems to be a general feature of the coupling constant trade-off: for the examples where we can compute the kernel inverse, at least one of D 2 (x, x) and D 0 (x, x) diverge. A divergent D 2 (x, x) generally leads to a formally divergent classical energy production, whilst a divergent Lindbladian coupling D 0 (x, x) can lead to a divergent energy production in the matter degrees of freedom. The later is related to the BPS problem [91] of anomolous heating, although it isn't necessarily equivalent since some kernels may diverge and be well behaved from the point of view of energy production. This is not an issue from a conceptual point of view, since the only reason we expect energy to be conserved is due to Noether's theorem, and Noether's theorem doesn't apply when the evolution isn't unitary.
In the standard BPS problem, energy production in open quantum field theory can be made small by renormalizing the Lindbladian coefficient D 0 (x, y) appearing in the master equation. Thus the problem is merely one akin to the hierarchy problem, where we are required to introduce another energy scale. However, in the case of classical-quantum coupling, the coupling constant trade-off tells us that we cannot re-normalize D 0 (x, y) without effecting D 2 (x, y). In particular, tuning the diagonals D 0 (x, x) to be arbitrarily small (large) has the effect of tuning D 2 (x, x) to be arbitrarily (large) small: heuristically, one trades energy production in the classical system with energy production in the quantum system, and the relationship is fixed by the trade-off. On expectation, the total energy could be preserved, and the back-reaction can even slow down the flow of energy, but it's unclear if this is enough.
However, it is worth noting that while D 2 (x, x ) may appear to diverge at a single point as x → x , when integrated over test functions, dxdx D 2 (x, x )f (x)f (x ) is usually well behaved. The kernels discussed above have this property. When it comes to physically relevant quantities, such as measuring the gravitational diffusion in table top experiments, it is the smeared well behaved quantity which is physically relevant. However, in cosmology, we typically take the constraint equation of general relativity to be exactly satisfied at each point, and so one imagines that π 2 Φ (x) 10 , and hence D 2 (x, x) is the relevant quantity (see the discussion in Section F). If D 2 (x, x) is the relevant quantity, than it's divergence is a serious challenge which may require modifying the interaction at short distances, perhaps by introducing non-locality in D 1 (x, x ). This non-locality of the interaction is more serious than allowing D 0 (x, x ) or D 2 (x, x ) to not be delta functions, since this just allows for non-local correlations to be created and destroyed [112].
Studying this in detail is beyond the scope of this work, but it may be that classical-quantum field theory can only be made finite once a physical cut-off has been imposed. One possible method of studying this problem rigorously would be by studying the regularisation properties of the classical-quantum path integral which we introduce in [123].
We define the total quantum state ρ Q by integrating over the classical degrees of freedom and we shall relate D 0 appearing in the trade-off to the decoherence rate of the off diagonals of ρ Q . Integrating over the classical phase space in Equation (9), one finds the follows expression for the evolution of ρ Q In particular, one finds that the off-diagonals L| ∂ρ Q ∂t |R evolve in part according to the commutator, and in part due to the Lindbladian term (E4) Care must be taken however, because both the quantum Hamiltonian, and the Lindbladian coupling constants depend on the classical degrees of freedom which are effected by the quantum degrees of freedom, and thus the evolution on the quantum system is non-Markovian in general.
We shall now study the two terms appearing in Equation (E4) separately, starting with the first term. Since we assume that the state is well approximated by a state with fixed particle number then the contributions to the first term in (E4) only come from terms where L α (x) and L β (y) have the same number of creation and annihilation operators. To compute the expression, one commutes through the creation operators to act on the L| bra, and picks up a term f L (x). Similarly, one commutes the annihilation operators to the act on the |R ket, and picks up a term f R (y). As a consequence where the last equality follows from the fact that we are taking the masses to be well separated and the range of D 0 (x, y) is assumed to be much less than the separation between the masses. Hence, the evolution of the off-diagonals comes from the (off-diagonals) of the unitary evolution and the second term in (E4), the so called no-event term. The off-diagonals of the no-event term is which is negative definite and acts to exponentially suppress the coherence. To see this, note that expanding out (Φ, π Φ , t) in terms of the approximate 2 dimensional Hilbert space and using the fact that the range of D 0 (x, y) is much less than the separation between the left and right masses, we can write the off-diagonals of the no-event term as Equation (E8) already expresses the fact that the off-diagonal terms will decay, and the particle will decohere at a rate determined by the integrand of (E8). We can go slightly further when in the presence of a background Newtonian potential which is dominant, such as the Earth's Φ b . The Earth's background potential dominates over small fluctuations in Φ due to the particles [77] and we can approximate (E8) by where the coupling D αβ 0 (x, y) depends on the background Newtonian potential, but is otherwise phase-space independent. The result is to exponentially decrease the coherence L|ρ Q |R with a rate λ determined by Let us now show that the D 0 term appearing in the trade-off (F19) is always less than (twice) this decoherence rate when in the presence of a background potential. Specifically, we show that where we assume that we are in the prescence of a background potential. To see this, we first expand out the CQ state in terms of (E7) and use the fact that D 0 has range less than the separation of the masses. We then arrive at the following expression for the left hand side of (E11) In the presence of a background potential, this dominates the contribution to the decoherence and we are left with Due to the positivity of the CQ density matrix L|ρ Q |L and R|ρ Q |R must both be positive. Furthermore, they must sum to one due to normalization, from which (E10) directly follows.
It is also important to note that though λ is the decoherence rate of a particle in superposition of L/R states, the bound (E11) holds even for fully decohered masses in any mixture of |L L|, |R R| states. This can be seen directly from (E12) which depends only on u L , u R .

Decoherence rate example
In this section we give an explicit example of a decoherence rate calculation. Importantly, we see that in general the decoherence rate can depend on the probability density. This suggests that the terms appearing in the trade-off relation, will need to depend on expectation values such as the expectation value of the mass at a point x, rather than a stronger bound in terms of the mass density. This is perhaps not surprising, since the decoherence rate itself can be though of as an expectation value, being related to the average time it takes for off-diagonal elements to decay. In the conclusion, this motivates us to advocate for the volume of the wave-packet to be included in the figure of merit in future interference experiments.
We take the Newtonian limit master equation defined by Equation (C22). We ignore the unitary part of the evolution, since it will not directly contribute to the decoherence rate, and can be small for a free particle in superposition. From Equation (C22) we find the relevant evolution for the quantum state ρ Q , obtained by integrating over the classical degrees of freedom to be We now compute the off-diagonal elements for a particle in super-position of orthogonal |L , |R states where m L (x) ≈ L|m(x)|L and similarly for the right state. We see that the off-diagonals decay exponentially with a rate determined by In the main body, and the previous subsection, we have assumed that the superposition of the particle is much less than the typical scale of D 0 (x, y). In this example, this means that we take the particles sufficiently separated, so that we can approximate D 0 (x, y)m L (x)m R (y) ≈ 0, in which case Equation (E16) is precisely the decoherence rate calculated in (E10) with L(x) =m(x), as is to be expected.
As an example of a decoherence kernel, we can take D 0 (x, x ) to be the Diosi-Penrose decoherence kernel defined via D 0 (x, y) = D0 |x−y| , so that the off-diagonals decay exponentially with a rate proportional to the Diosi-Penrose decoherence rate In this example, taking the particles to be sufficiently separated means that we are approximating D0 |x L −x R | ≈ 0 in comparison with the rest of the terms appearing in (E17). We are then left with which for spherical distributions of radius R and total mass M is proportional to the average gravitational self-energy of each mass distribution λ = 6D0M 2 5R . Both Equation's (E17) and (E18) depend on the probability density of the mass. In particular, taking the probability density to be arbitrarily peaked, one finds that the decoherence rate also diverges. This has to be the case: recall from Section III that if one considers a particle in a superposition of two arbitrarily peaked probability densities, then there can be an arbitrarily large response in the Newtonian potential around those points. As a consequence, for such states, the decoherence must occur arbitrarily fast, or there must be an arbitrarily large amount of diffusion to cover up the back-reaction and maintain coherence. For the continuous master equation, such as that of Equation (C22) this diffusion must also occur throughout space, although it can depend on the gravitational degrees of freedom. Since divergent energy production throughout space is clearly unphysical, it must be the case that the decoherence rate must also depend on the expected mass density, as is the case for this example. This argument allows us to rule out continuous master equations which have pure Lindbladian terms which predict decoherence rates which which remain finite as the mass density becomes arbitrarily peaked, since the coupling constant trade-off will demand that an infinite amount of diffusion is required to cover up the back-reaction and maintain coherence. This is the case for the class of models with CSL type Lindbladian couplings given by Equation (D1).

Appendix F: Detecting gravitational diffusion
In this section we show how the diffusion induced on the Newtonian potential can be measured experimentally. As shown in the main body of the text, in the non-relativistic limit, c → ∞, the CQ dynamics can be approximated by sourcing the Newtonian potential by a random mass term, and that in order to maintain coherence of any mass is superposition, there must be noise in the Newtonian potential such that we cannot tell which element of the superposition the particle will be in with where D 2 (x, y, Φ) := Tr D µν 2 (x, y, Φ b )L µ (x)ρL † ν (y) and ρ is the quantum state for the decohered mass density. The diffusion coefficient in Equation (F2) is chosen in order for the dynamics to have the same moments as the CQ master equation (4). The solution to Equation (F1), having absorbed u into J is given by where the statistics of J are described by Equation (F2). A formal treatment of solutions to non-linear stochastic integrals of the form Equation (F1) can be found in [74]. One can also verify this behaviour in specific cases. In the continuous model of Section C 1 a, the noise is taken be Gaussian, and this, as well as the evolution of the quantum state, is what determines the diffusion in Equation (C27). For the class of discrete models, the higher order moments such as E m [J(x, t)J(y, t )J(z, t )] are suppressed by an order parameter [11,24,34] and that whenever this is true we expect we can approximate the dynamics of the Newtonian potential by a Gaussian process. Whether this is the case or not, it is the second order moment which enters into our discussion of the variance here. As such, for minimally coupled theories, the Newtonian potential will appear to be sourced by a random mass distribution.
In the discrete case, a precise understanding of the effects of the diffusion beyond the Gaussian approximation involve solving the full classical-quantum dynamics, perhaps using the methods of [24]. In Equation (F1) we are also taking the time-scale of the diffusion to be faster than the dynamics of the matter distribution. Likewise for the decoherence -we showed in Appendix C 2 for continuous models the evolution of the quantum state acts to decohere it into a mass density eigenbasis m(x). One could of course also include the quantum state evolution in a simulation of full CQ dynamics, but this is beyond the scope of the current work.

Table-top experiments
In this section we estimate the variation in force which would be seen in table-top experiments which bounds the diffusion of classical theories of gravity from above, giving a squeezed bound on D 2 due to lower bounds on diffusion arising from coherence experiments . We do this for dynamics in Equation (F1), but the methodology is general and could also be used in a full simulation of CQ dynamics.
The variation in force induced on a composite mass is found via Using the solution in Equation (F3), the total force can be written In reality, we measure time averaged force by measuring time averaged accelerations over a period T 1 T T 0 dtF tot . The total variation in the forces time averaged magnitude 11 σ 2 F := F tot · F tot can be written as We shall use Equation (F6) to provide an upper bound on coupling constants of CQ theories for different choices of kernels D 2 (x , y , Φ). Given a choice of functional form of the kernel, all that remains is the strength of the diffusion coupling, which for the translation invariant kernels we consider here takes the form of a single coupling constant D 2 . We take D 2 to be a dimension-full quantity with units kg 2 sm −3 which characterizes the rate of diffusion for the conjugate momenta of the Newtonian potential.
For a composite mass, we can approximate the mass density by summing over N individual atoms of mass density m i (x), m(x) = i m i (x). The total force is the given by F tot = i F i , where F i is the force on each individual atom F i = − V dxm i (x)∇Φ(x), and the total variation of force is then σ 2 In general, the squeeze will depend on the functional choice of D 2 (x, y, Φ) on the Newtonian potential. As mentioned in the main body, in the presence of a large background potential Φ b , such as that of the Earth's, we will often be able to approximate D 2 (x, y, Φ) = D 2 (x, y, Φ b ). This is true for the kernels with functional dependence of the form D 2 ∼ Φ n , D 2 ∼ ∇Φ, though the approximation does not hold for all kernels, for example D 2 ∼ ∇ 2 Φ which creates diffusion only where there is mass density. We hereby shall only consider diffusion kernels D 2 (x, y, Φ b ) where the background potential is dominant, leaving more general considerations to Section F 2 and future work.
For local translation invariant dynamics for which the background Newtonian potential is dominant, for example D 2 ∼ Φ n , we have D 2 (x, y, Φ b ) = D 2 (Φ b ) δ(x, y) and we arrive at the expression for the total variation in time averaged force To leading order, the integral in Equation (F7) is dominated by the self variation term where i = j, since nuclear scales 10 −15 m dominate over inter-atomic scales 10 −9 m, so that E[ ij F i F j ] ∼ i E[F 2 i ]. Approximating the mass 11 The full covariance matrix for various kernels is given in [90] density of the atoms as coming from their nucleus, and taking them to be spheres of constant density ρ with radius r N and mass m N , we find that the integral in Equation (F7) is approximately For the class of continuous dynamics D 2 (Φ b ) = D 2 (Φ b ), since the diffusion is not associated to any Lindblad operators. If there is noise everywhere throughout space, then the integral in Equation (F8) diverges, and gives evidence that continuous CQ theories with noise everywhere should be ruled out.
As such, we expect that continuous CQ theory must contain non-linear terms proportional to the Newtonian potential appearing in Equation (F1), in which case we can approximate dx D 2 by V b D 2 where V b is the volume of the region over which the background Newtonian potential is significant. In total then, we find for continuous local CQ dynamics From this, we can calculate D 2 in terms of the total variance of the acceleration σ 2 a = σ 2 F m 2 tot to get a lower bound Standard Cavendish type classical torsion experiments measure accelerations of the order 10 −7 ms −2 , and we can take the time over which the acceleration is averaged to be that of minutes T ∼ 10 2 s, so a very conservative bound is σ a ∼ 10 −7 ms −2 , whilst N will be N ∼ 10 26 and r N ∼ 10 −15 m. We take the background Newtonian potential to be that of the earths and we (conservatively) take V b to be V b ∼ r 2 E h ∼ 10 15 m 3 where r E is the Earths radius and h is the atmospheric height. We see that this bounds D 2 from above by D 2 ≤ 10 −41 kg 2 sm −3 .
On the other hand, D 2 is bounded from below from interferometry experiments which bound the decoherence rate. From Equation (E16) and the coupling constant trade-off, for the kernel D 2 (x, y) = D 2 δ(x, y) we see (ignoring constant factors) that the decoherence rate is found to be where M λ is the mass of the particle in the interferometry experiment and V λ is its volume. This gives rise to the squeeze Using the numbers from [54], with M λ ∼ 10 −24 kg and V λ ∼ 10 −9 10 −9 10 −7 m 3 = 10 −25 m 3 , λ ∼ 10 1 s −1 we find that D 2 ≥ 10 −24 kg 2 sm −3 . This suggests that the D 2 (x, y) = D 2 δ(x, y) kernel for classical gravity is already ruled out by experiment.
For the local discrete models, such as that of Equation (C25), the theory is less constrained due to the dependence of the diffusion on the mass density. In this case D 2 (Φ b ) = l 3 P m P D 2 (Φ b )m(x), where the factors of Planck length and Planck mass are to ensure that D 2 (Φ b ) has the required units. We arrive at the upper bound for D 2 Meanwhile, from Equation (E10), and coupling constant trade-off (29) the decoherence rate for local discreet jumping models goes as λ ∼ M λ m P l 3 P D2 , which gives rise to the lower bound for D 2 . From this we arrive at the squeeze and plugging in the numbers we find the bound given by Equation (46) which gives rise to the squeeze for local discrete models 10 −1 kgs ≥ l 3 P m P D 2 ≥ 10 −25 kgs.
We can also consider other diffusion kernels, for example that of Equation (D13). In this case, for continuous dynamics we have that D 2 (x, y) = −l 2 P D 2 (Φ b )∇ 2 δ(x, y). The Lindbladian kernel saturating the coupling constants trade-off at zeroeth order in Φ(x), is the Diosi-Penrose kernel D 0 (x, y, Φ b ) = D0(Φ b ) |x−y| , as we saw in Section (D 2). Approximating the masses as spheres of constant density we find from a substitution of the kernel into Equation (F6) that the variation in time averaged force is given by (F15) We therefore find a lower bound for D 2 in terms of the variation in acceleration which for classical torsion experiments σ a ∼ 10 −7 ms −2 , T ∼ 10 2 s, N ∼ 10 26 and r N ∼ 10 −15 m gives D 2 l 2 p ≤ 10 −9 kgsm −1 . On the other hand, for this kernel the decoherence rate can be calculated via Equation (E18) which gives the squeeze on D 2 T σ 2 a N r 3 For the numbers used in the main body of the text, M λ ∼ 10 −24 kg, R λ ∼ 10 −9 m, λ ∼ 10 1 s, this yields D 2 l 2 P ≥ 10 −40 kgsm −1 and so this model is not ruled out by experiment.
In general then, we expect that by simulating full CQ dynamics satisfying the decoherence diffusion trade-off we will be able to squeeze D 2 from above and below. We bound D 2 from above by studying the effects of diffusion on gravitational experiments, and we bound D 2 from below using the coupling constant trade-off and coherence experiments lower bounding the decoherence rate. As we have seen in this section, it appears that classes of continuous CQ hybrid theories of gravity, including models without spatial correlations, are already experimentally ruled out, whilst others, such as the kernels in Subsection D 3 require stronger bounds from both gravitational and coherence experiments. We have been very conservative in our estimates, and so we expect a more thourough analysis will tighten the bounds by orders of magnitute.

Kinetic energy produced by gravitational diffusion
In this section, we obtain a lower bound for the amount of energy production required in a coherence experiment in order to maintain coherence for masses in a superposition. By virtue of the coupling constant trade-off (29), theories of CQ gravity will generically involve energy production, but the amount of production will be theory dependent. The bound we derive here follows from the observational trade-off of Equation (33) and is a theory independent energy production which must be seen by any CQ theory for which the observational trade-off holds. This includes all continuous CQ dynamics aswell as the discrete models which back-react solely via D αβ 1,i , for example those given in [11,24,34]. Plugging (38) into the trade-off in Equation (33) we find We shall now relate the remaining quantities in (F19) to diffusion and decoherence. Experimentally measured decoherence rates can be related to D 0 . We explore the calculation of decoherence rates in gravity in detail in [77]. In appendix E, we show that for a mass in state LR (Φ, π Φ ) whose quantum state is a superposition of two states |L and |R of approximately orthogonal mass densities m L (x), m R (x), and whose separation we take to be larger than the non-locality scale of D 0 (x, y), the expectation value of D 0 entering in the trade-off of Equation (F19) is bounded above by (twice) the decoherence rate λ of the particle DΦDπ Φ dx dy Tr D αβ 0 (x , y )L † β (y )L α (x ) LR ≤ 2λ.
bound on λ, for example that arising from the interferometry experiment of [54] which saw coherence in large organic fullerene molecules with total mass 10 −24 kg over a timescale of 0.1s, gives an upper bound on the decoherence rate λ < 10 1 s −1 . We now use this to estimate a lower bound on the production of stochastic gravitational waves. The fullerene molecules had typical size r ∼ 10 −9 m. After passing through the slits the molecule becomes delocalized in the transverse direction on the order of 10 −7 m before being detected. Since the interference effects are due to the superposition in the transverse x direction, which is the direction of alignment of the gratings, it seems like a reasonable assumption to take the size of the wavepacket in the remaining y, z direction to be the size of the fullerene, since we could imagine measuring the y, z directions without effecting the coherence. We therefore can estimate the expectation of the mass density to be m(x) ∼ 10 −24 10 −9 10 −9 10 −7 ∼ 10 1 and one finds d∆Ē dt ∼ 10 −19 Js −1 . In comparison, the energy of the gravitational waves detected at LIGO, for example GW150914 [124], is of the order 10 −2 J/m 2 s.
Cosmological observations appear to rule out a stochastic production of waves over all space of this magnitude -as is required by the continuous realisations of classical-quantum dynamics discussed in Appendix C 1 a, where D 2 (x, x ) = D 2 δ(x, x ) doesn't depend on Φ. This assumption is required, if we want to extrapolate the diffusion required in a terrestrial interference experiment to that occurring throughout space. Since we require d∆Ē dt ∼ 10 −19 Js −1 over the wave-packet size of nucleons, this implies that the energy density of waves must be produced at a rate of at least ∼ 10 5 Js −1 m −3 . This gives an energy density in the ball park of 10 22 J/m 3 accumulated over the age of the universe.This would appear to rule out the D 2 (x, x ) = D 2 δ(x, x ), although we should be careful about extrapolating the theory to a regime we understand little about. We should also be mindful that our definition of gravitational kinetic energy may require some regularisation via point-splitting in this instance.