A universal test for gravitational decoherence

Quantum mechanics and the theory of gravity are presently not compatible. A particular question is whether gravity causes decoherence. Several models for gravitational decoherence have been proposed, not all of which can be described quantum mechanically. Since quantum mechanics may need to be modified, one may question the use of quantum mechanics as a calculational tool to draw conclusions from the data of experiments concerning gravity. Here we propose a general method to estimate gravitational decoherence in an experiment that allows us to draw conclusions in any physical theory where the no-signalling principle holds, even if quantum mechanics needs to be modified. As an example, we propose a concrete experiment using optomechanics. Our work raises the interesting question whether other properties of nature could similarly be established from experimental observations alone—that is, without already having a rather well-formed theory of nature to make sense of experimental data.

An e↵ect respresents a measurement outcome. If a system in a state ! is measured with respect to a measurement M = {e 1 , . . . , e n }, then e k (!) is the probability that the measurement yields the outcome associated with e k . According to Definition III.2, this yields the states ⌦ = {⇢ 2 Pos(H) | tr(⇢) = 1}, which are precisely the density operators on H. Analogously, ⌦  are the subnormalized density operators. The e↵ects are the functionals induced by POVM elements via the trace, E = {tr(P · ) | P 2 Pos(H), P  I H }. Accordingly, the measurements are the sets of functionals that are induced by POVMs, M = {{tr(P k · ) | P k 2 {P k } k } | {P k } k is a POVM}. This precisely reproduces the structure of measurement statistics in quantum theory. For further details, see [Pfi12]. ⌅ By our definition, E is the set of all linear functionals e such that 0  e(!)  1 for all ! 2 ⌦. The underlying assumption that every such linear functional represents a physical measurement outcome has been called the norestriction hypothesis [Udu12]. A priori, there seems to be no immediate physical reason for this assumption, and some authors have argued about how to weaken this assumption [JL13]. For our purposes here, it is not relevant whether the no-restriction hypothesis holds, and weakening the assumption complicates the definitions. Thus, we assume it for simplicity.
In Section III B, we will define a decoherence quantity Dec(A|E) ! analogous to the quantum min-entropy H min (A|E) ⇢ . We will take our inspiration from expression (I.14) for the quantum min-entropy, which involves the fidelity as a measure of closeness of quantum states. Therefore, it is desirable to have a generalization of the fidelity to states in abstract state spaces. Such a generalization is easily found once it is noticed that the quantum fidelity of two states can be expressed as the Bhattacharyya coe cient (or classical fidelity) of the two probability distributions that the two states induce, minimized over all measurements. More precisely, the quantum fidelity satisfies [NC00] X p  We are now going to motivate an expression for the central quantitiy Dec(A|E) ! for our decoherence analysis for GPTs. We take our inspiration from expression (I.14) for the quantum min-entropy, which we repeat here for the reader's convenience: (I.14 revisited) There are two issues that prevent us from directly translating expression (I.14) into our framework. The first issue is that in Section III A 2, to keep our framework as general as possible, we have defined a tripartite scenario with an overall state space (V, V + , u) with tripartite states ⌦. We do not have notions of individual state spaces at hand. Thus, we do not have an analogue of a reduced state ⇢ AE or of a transformation R E!A 0 from one state space to another.
The second issue is that we do not know what the analogue of a maximally entangled state AA 0 in our framework is. We resolve the first issue here in Section III B 1, arriving at an expression for Dec(A|E) ! . In Section III B 2, we will then define what a maximally entangled state is in our framework.
Expression (I.14), which involves the state ⇢ AE and TPCPMs R E!A 0 , can be transformed to an expression in which both the state and the TPCPMs are purified (see Figure III.2). This expression will be our motivation for the expression for Dec(A|E) ! . The maximization over TPCPMs from E to A 0 is replaced by a maximization over unitaries from EE 00 to A 0 A 00 , where E 00 and A 00 are ancilla systems extending system E and A 0 , respectively. This is precisely the purification (or Stinespring dilation) of a channel as in Section I. Since systems EE 00 and A 0 A 00 have the same dimension, we can identify their Hilbert spaces and regard the resulting Hilbert space as the Hilbert space of a system E tot . This system involves all subsystems that the third party needs to control in order to bring itself as close as possible to maximal entanglement with Alice. Since U Etot is a transformation on system E tot alone, we can translate it into our generalized framework. . Part (a) shows the system and maps involved in expression (I.14) for the quantum min-entropy. In expression (III.20), we purify this situation, as shown in (b), to arrive at a situation with three parties A, B and Etot, and with a map UE tot which acts on one system Etot alone rather than mapping from one system to another.
The state ⇢ AE is replaced by a purification ⇢ ABE . We choose the purifying system B to be the channel's output system, which gives us the overall picture of our decoherence analysis as shown in Figure III  We are now going to motivate an expression for the central quantitiy Dec(A|E) ! for our decoherence analysis for GPTs. We take our inspiration from expression (I.14) for the quantum min-entropy, which we repeat here for the reader's convenience: (I.14 revisited) There are two issues that prevent us from directly translating expression (I.14) into our framework. The first issue is that in Section III A 2, to keep our framework as general as possible, we have defined a tripartite scenario with an overall state space (V, V + , u) with tripartite states ⌦. We do not have notions of individual state spaces at hand. Thus, we do not have an analogue of a reduced state ⇢ AE or of a transformation ⇤ E!A 0 from one state space to another.
The second issue is that we do not know what the analogue of a maximally entangled state AA 0 in our framework is. We resolve the first issue here in Section III B 1, arriving at an expression for Dec(A|E) ! . In Section III B 2, we will then define what a maximally entangled state is in our framework.
Expression (I.14), which involves the state ⇢ AE and TPCPMs ⇤ E!A 0 , can be transformed to an expression in which both the state and the TPCPMs are purified (see Figure III.2). This expression will be our motivation for the expression for Dec(A|E) ! . The maximization over TPCPMs from E to A 0 is replaced by a maximization over unitaries from EE 00 to A 0 A 00 , where E 00 and A 00 are ancilla systems extending system E and A 0 , respectively. This is precisely the purification (or Stinespring dilation) of a channel as in Section I. Since systems EE 00 and A 0 A 00 have the same dimension, we can identify their Hilbert spaces and regard the resulting Hilbert space as the Hilbert space of a system E tot . This system involves all subsystems that the third party needs to control in order to bring itself as close as possible to maximal entanglement with Alice. Since U Etot is a transformation on system E tot alone, we can translate it into our generalized framework. 20 B. A decoherence quantity for GPTs

Motivation of an expression that quantifies decoherence
We are now going to motivate an expression for the central quantitiy Dec(A|E) ! for our decoherence analysis for GPTs. We take our inspiration from expression (I.14) for the quantum min-entropy, which we repeat here for the reader's convenience: (I.14 revisited) There are two issues that prevent us from directly translating expression (I.14) into our framework. The first issue is that in Section III A 2, to keep our framework as general as possible, we have defined a tripartite scenario with an overall state space (V, V + , u) with tripartite states ⌦. We do not have notions of individual state spaces at hand. Thus, we do not have an analogue of a reduced state ⇢ AE or of a transformation ⇤ E!A 0 from one state space to another.
The second issue is that we do not know what the analogue of a maximally entangled state AA 0 in our framework is. We resolve the first issue here in Section III B 1, arriving at an expression for Dec(A|E) ! . In Section III B 2, we will then define what a maximally entangled state is in our framework.
Expression (I.14), which involves the state ⇢ AE and TPCPMs ⇤ E!A 0 , can be transformed to an expression in which both the state and the TPCPMs are purified (see Figure III.2). This expression will be our motivation for the expression for Dec(A|E) ! . The maximization over TPCPMs from E to A 0 is replaced by a maximization over unitaries from EE 00 to A 0 A 00 , where E 00 and A 00 are ancilla systems extending system E and A 0 , respectively. This is precisely the purification (or Stinespring dilation) of a channel as in Section I. Since systems EE 00 and A 0 A 00 have the same dimension, we can identify their Hilbert spaces and regard the resulting Hilbert space as the Hilbert space of a system E tot . This system involves all subsystems that the third party needs to control in order to bring itself as close as possible to maximal entanglement with Alice. Since U Etot is a transformation on system E tot alone, we can translate it into our generalized framework. . Part (a) shows the system and maps involved in expression (I.14) for the quantum min-entropy. In expression (III.20), we purify this situation, as shown in (b), to arrive at a situation with three parties A, B and Etot, and with a map UE tot which acts on one system Etot alone rather than mapping from one system to another.
The state ⇢ AE is replaced by a purification ⇢ ABE . We choose the purifying system B to be the channel's output system, which gives us the overall picture of our decoherence analysis as shown in Figure III  . Part (a) shows the system and maps involved in expression (I.14) for the quantum min-entropy. In expression (III.20), we purify this situation, as shown in (b), to arrive at a situation with three parties A, B and Etot, and with a map UE tot which acts on one system Etot alone rather than mapping from one system to another.
The state ⇢ AE is replaced by a purification ⇢ ABE . We choose the purifying system B to be the channel's output system, which gives us the overall picture of our decoherence analysis as shown in Figure III Supplementary Fig. 6: Purification of equation (14). Part (a) shows the system and maps involved in expression (14) for the quantum min-entropy. In expression (92), we purify this situation, as shown in (b), to arrive at a situation with three parties A, B and Etot, and with a map UE tot which acts on one system Etot alone rather than mapping from one system to another.
The state ⇢ AE is replaced by a purification ⇢ ABE . We choose the purifying system B to be the channel's output system, which gives us the overall picture of our decoherence analysis as shown in Figure III (III.20) is not specified, we can choose it such that it fits our situation for the decoherence analysis.
The following gives a precise formulation of the purification of expression (I.14). It can be proved using purification and Stinespring dilation.
The state ⇢ AE is replaced by a purification ⇢ ABE . We choose the purifying system B to be the channel's output system, which gives us the overall picture of our decoherence analysis as shown in Figure III (III.20) is not specified, we can choose it such that it fits our situation for the decoherence analysis.
The following gives a precise formulation of the purification of expression (I.14). It can be proved using purification and Stinespring dilation. The inequalities define a convex polytope over which the convex function ( ) is minimized, so the minimum is attained. It is straightforward to bring these inequalities into the standard form of linear programming. We solved the resulting linear program using standard linear programming routines in Mathematica and Octave.    7: Values of the decoherence quantity for some Pauli channels. The strength of the shown Pauli channels depends on the number of Pauli operators that are involved. The dephasing channel, with one involved Pauli operator, achieves a decoherence of 1/2, the 2-Pauli channel achieves a decoherence of 3/4, and the depolarizing channel and the BB84 channel (with three involved Pauli operators) achieve a decoherence of 1.

Example application of Pauli decoherence to a model for gravitational decoherence
In this subsection, we apply the knowledge about the decoherence of Pauli channels that we gained in the last subsection to derive a closed-form expression for the decoherence quantity for some gravitational decoherence channel that has been suggested in the literature. We emphasize that the calculations presented here are only one example application of our decoherence formalism. Many other models for gravitational decoherence or other decoherence processes can be studied in our formalism as well.
We consider a model for gravitational decoherence that has been derived by Kok and Yurtsever [45]. Their model is particularly appealing for our purposes as it leads to a qubit-to-qubit channel, therefore facilitating the analysis. It describes quantum metric fluctuations of qubits that are gravitationally coupled to a background spacetime. The metric is assumed to be described by a quantum state ⇢ g that evolves like a regular quantum system. The evolution ) is shown as a parametric plot for 0 6 t 6 1/⌦. The inward spiralling is a consequence of the phase rotation ↵(t), together with the exponential decay of the entry's absolute value. On the right, the decoherence quantity (5.44) of the channel is plotted as a function of the time t. It is a dephasing channel with a deformed strength parameter p. The plots have been made for the parameters v = 9/10, = 3/2, which we have chosen purely for illustration purposes. The analysis presented here is valid for any such parameters.
The conditional max-entropy of (5.47) can easily be solved using a semidefinite program (SDP) solver software such as SeDuMi and YALMIP for Matlab (see the SDP formulation of the conditional max entropy in Appendix 2.2.3). A plot of the decoherence quantity of C AD is shown on the left hand side of Figure 9 (see the graph with p = 1) as a function of . It achieves a maximal decoherence of 1. On the left hand side, a plot of the decoherence quantity of the amplitude damping channel is shown. It achieves full decoherence for = 1. In that maximally decoherent case, the channel simply replaces any input state by the |0i state. The erasure channel, in contrast, replaces any input state by the orthogonal |2i state for p = 1. As the plot on the right hand side shows, this decoheres the system much less, as measured by our decoherence quantity.

The decoherence quantity for erasure models
The next family of channels that we want to look at is the family of erasure channels. It is a family of channels parametrized by a probability p 2 [0, 1]. We denote the erasure channel with parameter p by C E p . For every such parameter, the channel C E p is a qubit-to-qutrit channel. The idea is that a qubit is transmitted via this channel, and with probability 1 p, it is left unchanged, and with probability p it is erased. In the case of an erasure, the receiver Supplementary Fig. 10: Plots for gravitational decoherence due to flat spacetime fluctuations. On the left, the off-diagonal element κ(t), equation (200), of the Choi matrix (192) is shown as a parametric plot for 0 t 1/Ω. The inward spiralling is a consequence of the phase rotation α(t), together with the exponential decay of the entry's absolute value. On the right, the decoherence quantity (214) of the channel is plotted as a function of the time t. It is a dephasing channel with a deformed strength parameter p. The plots have been made for the parameters v = 9/10, σ = 3/2, which we have chosen purely for illustration purposes. The analysis presented here is valid for any such parameters.  .22) is shown as a parametric plot for 0 6 t 6 1/⌦. The inward spiralling is a consequence of the phase rotation ↵(t), together with the exponential decay of the entry's absolute value. On the right, the decoherence quantity (5.44) of the channel is plotted as a function of the time t. It is a dephasing channel with a deformed strength parameter p. The plots have been made for the parameters v = 9/10, = 3/2, which we have chosen purely for illustration purposes. The analysis presented here is valid for any such parameters.
The conditional max-entropy of (5.47) can easily be solved using a semidefinite program (SDP) solver software such as SeDuMi and YALMIP for Matlab (see the SDP formulation of the conditional max entropy in Appendix 2.2.3). A plot of the decoherence quantity of C AD is shown on the left hand side of Figure 9 (see the graph with p = 1) as a function of . It achieves a maximal decoherence of 1. In that maximally decoherent case, the channel simply replaces any input state by the |0i state. The erasure channel, in contrast, replaces any input state by the orthogonal |2i state for p = 1. As the plot on the right hand side shows, this decoheres the system much less, as measured by our decoherence quantity.

The decoherence quantity for erasure models
The next family of channels that we want to look at is the family of erasure channels. It is a family of channels parametrized by a probability p 2 [0, 1]. We denote the erasure channel with parameter p by C E p . For every such parameter, the channel C E p is a qubit-to-qutrit channel. The idea is that a qubit is transmitted via this channel, and with probability 1 p, it is left unchanged, and with probability p it is erased. In the case of an erasure, the receiver ) is shown as a parametric plot for 0 6 t 6 1/⌦. The inward spiralling is a consequence of the phase rotation ↵(t), together with the exponential decay of the entry's absolute value. On the right, the decoherence quantity (5.44) of the channel is plotted as a function of the time t. It is a dephasing channel with a deformed strength parameter p. The plots have been made for the parameters v = 9/10, = 3/2, which we have chosen purely for illustration purposes. The analysis presented here is valid for any such parameters.
The conditional max-entropy of (5.47) can easily be solved using a semidefinite program (SDP) solver software such as SeDuMi and YALMIP for Matlab (see the SDP formulation of the conditional max entropy in Appendix 2.2.3). A plot of the decoherence quantity of C AD is shown on the left hand side of Figure 9 (see the graph with p = 1) as a function of . It achieves a maximal decoherence of 1. In that maximally decoherent case, the channel simply replaces any input state by the |0i state. The erasure channel, in contrast, replaces any input state by the orthogonal |2i state for p = 1. As the plot on the right hand side shows, this decoheres the system much less, as measured by our decoherence quantity.

The decoherence quantity for erasure models
The next family of channels that we want to look at is the family of erasure channels. It is a family of channels parametrized by a probability p 2 [0, 1]. We denote the erasure channel with parameter p by C E p . For every such parameter, the channel C E p is a qubit-to-qutrit channel. The idea is that a qubit is transmitted via this channel, and with probability 1 p, it is left unchanged, and with probability p it is erased. In the case of an erasure, the receiver Supplementary Fig. 11: Values of the decoherence quantity for the amplitude damping channel and the erasure channel. On the left hand side, a plot of the decoherence quantity of the amplitude damping channel is shown. It achieves full decoherence for λ = 1. In that maximally decoherent case, the channel simply replaces any input state by the |0 state. The erasure channel, in contrast, replaces any input state by the orthogonal |2 state for p = 1. As the plot on the right hand side shows, this decoheres the system much less, as measured by our decoherence quantity.
Raman single photon source Fig. 12: Illustration of the optomechanical setup. Two cavities each contain a Raman single photon source controlled by an external laser 'write field' E(t). The Raman sources are first prepared in an entangled state. Only one cavity contains a mechanical element coupled by radiation rouser to the cavity field.   fals , which is the minimal value that needs to be exceeded in the measurement of the CHSH value in order to rule out the gravitational decoherence model, is plotted as a function of time for the same materials and temperatures as above. In addition, the value mech is plotted, which is the CHSH value that can actually be measured using the standard CHSH measurement in the case where gravitational decoherence is absent and only mechanical heating contributes to the decoherence.
Supplementary Fig. 15: Optimal measurement times for ruling out the gravitational decoherence model. The three plots are identical to the ones in the leftmost box in Supplementary Fig. 14, i.e. for T = 1 nK. In addition, the time tmax at which the gap g(t) between β mech and β fals is maximal is indicated for the two cases where the material of the mechanical element has the density of aluminum or rhenium.

Supplementary Note 1: Background -Decoherence in quantum theory
In this appendix, we give a short introduction to decoherence in quantum theory. It consists of concepts, results and quantities that are well-established in quantum information science [1]. The topics are chosen to facilitate the understanding of our contributions in Supplementary Notes 2 and 3 rather than to give a full introduction to the subject of decoherence. In the first subsection, we explain that the dynamical evolution (e.g. decoherence) can equivalently be treated in a picture where the system is entangled with another system. The purification of that system after the evolution is a tripartite system. This tripartite state plays a central role in our later analysis. We emphasize that working in this picture is not a loss of generality but merely a picture that is equivalent to the single-system treatment, but which is more suitable for our analysis [2]. In the second subsection, we explain why the min-entropy is the relevant quantity in the information theoretic analysis of decoherence. The min-entropy is the quantity that we use for our analysis in Supplementary Note 2. It is also the quantity that serves as our motivation to define a decoherence quantity for generalized probabilistic theories in Supplementary Note 3. We note that a generalization of quantum theory by, for example, introducing additional terms into the Schrödinger equation fall under the regime of generalized theories in our discussion.
For this document, we make the following conventions.
• The logarithm is with respect to base 2, i.e. log ≡ log 2 .
• Hilbert spaces are assumed to be finite-dimensional, unless otherwise stated.
• We denote the set of density operators (states) on a Hilbert space H by S(H).
• We identify operators on Hilbert spaces with their reordered versions resulting from permutations of systems. For example, for Hilbert spaces • For a state ρ ABE ∈ S(H A ⊗ H B ⊗ H E ), we denote its reduced states by according changes of the subscript, e.g.

Dynamical evolution and its purification
Interaction and non-unitary evolution: Suppose that a system 1 A , initially in a state described by a density operator ρ A ∈ S(H A ), undergoes a dynamical evolution over some time interval. If A undergoes this evolution as a closed system, then according to one of the postulates of quantum mechanics, the state transforms as Fig. 1 (a)). In general, however, the system A may be open, i.e. it may interact with another system E that is called the environment. We consider the environment E to consist of all the systems that interact with system A . Taken together, the combined system A E then forms a closed system and hence evolves as where ρ E is the initial state of the environment and U A E→A E : We may be ignorant about the environment E and only have access to system A . Our description would then treat the state of the subsystem A after the evolution as a function of the state ρ A of A before the evolution. We arrive at this description by taking the partial trace over E in expression (2): (3) is easily shown to be a trace-preserving completely positive map (TPCPM). Thus, the evolution of an open system A , when the environment E is not visible, is described by a TPCPM Θ A →A (see Supplementary Fig. 1 (b)).
In a yet more general case, it may be that after the evolution of the system A E, we do not have access to system A but to a different subsystem B of A E. An example would be a two-particle system A interacting with another twoparticle system, where we only have access to one particle (B) of the four particles after the evolution. Mathematically speaking, the fact that we see a different subsystem before and after the evolution means that our factorization of the overall Hilbert space changes: Before the evolution, we write H = H A ⊗ H E and after the evolution, we write H = H B ⊗ H E (see Supplementary Fig. 1 (c)). Thus, the unitary evolution of the closed overall system is described by a unitary U A E →BE : Describing only the accessible part before and after the evolution, we end up with a TPCPM Thus, the evolution of a system A to a system B, when being ignorant about the environment, is described by a TPCPM Θ A →B .
Stinespring dilation: We have demonstrated that unitaries on two (or more) systems give rise to TPCPMs on one system. It is well-known that the converse is also true: such that This (or an equivalent statement) is the Stinespring dilation theorem [3]. For more details see [1]. Textbook definitions of decoherence: From now on, we will take the viewpoint that the TPCPM Θ A →B is what we are given in the first place. Physically speaking, we assume that we are in the setting where all that we observe is a process in which a system A in some state ρ A transforms into a state ρ B of some system B. We think of this as a channel Θ A →A , into which we input a system A and get a system B as an output. Our goal in this Supplementary Note is to find a precise mathematical formulation of the following question in the quantum theoretical framework: How much does the channel decohere the system? For the case where A = B, the standard quantum mechanics literature gives some simple descriptions of what the decoherence of a system under a dynamical evolution is. As an example, consider the case where A is a spin-1/2 particle, initially in the spin "up" state in the x-direction, If the channel Θ A →A is given by a measurement of the spin in the z-direction, then, written in the z-basis, the state of the system transforms as One possible observation one can make in (8) is that the spin measurement in the z-direction causes the off-diagonal terms of the density matrix to vanish. This is an extreme case of the dephasing channel in the z-basis, which causes a loss of the phase information of the superposition (7). This loss of phase information is often equated with decoherence. Another feature of (8) that is often said to be the characteristic of decoherence is that Θ A →A turns an initially pure state into a mixed state. These descriptions of decoherence, valid in their own right, are not favored by us for mainly three reasons. Firstly, these are no quantitative measures of decoherence. Secondly, they lack a clear operational meaning. Thirdly, they rely on the quantum mechanical formalism, in which states are expressed as density operators. It is not clear how to express them in more general cases that are not described by quantum theory.
The min-entropy as a measure for decoherence A lot can be learned about the channel if one takes an additional system into the picture. From now on, we consider the case where the input system A is purified by a system A. Then, while system A undergoes the channel evolution, system A remains unchanged. For example, one such case is the case where the system AA is in a maximally entangled state. This leads to the situation shown in Supplementary Fig. 2. The output system is now a tripartite system ABE. In quantum information science, it is very popular to think of the systems as being controlled by parties with intentions and interests rather than just being dead physical objects. We will follow this spirit and from now on use the language of a game and speak of parties Alice, Bob and Eve, that we think of as agents controlling the systems A, B and E. In quantum information science, it has been realized that important quantitative measures of the channel are functions of this tripartite state ρ ABE .
The coherent information: One such measure quantifying decoherence is the coherent information [4]. It is defined in terms of the conditional von Neumann entropy where H(AB) ρ = −Tr(ρ AB log(ρ AB )) and H(B) = −Tr(ρ B log(ρ B )) is the von Neumann entropy of the reduced state ρ AB and ρ B , respectively. The coherent information is defined as The coherent information I(A B) ρ has been shown to be related to the quantum channel capacity Q(Θ A →B ) of Θ A →B , which is known as the Lloyd-Shor-Devetak (LSD) theorem [5][6][7]. It says that where results from the n-fold use of the channel Θ A →B to transmit (A ) n , i.e. n copies of system A , while the purification A n of (A ) n remains unchanged. Thus, the r.h.s. of (11) is the coherent information in the limit of infinitely many channel uses. Likewise, the quantum capacity Q(Θ A →B ) is the limit of the achievable rate for quantum data transmission in the limit of infinitely many channel uses. One says that the quantum capacity, and therefore the coherent information, is an asymptotic quantity. This has the disadvantage that from the coherent information, only very limited statements about finitely many uses of the channel can be made. The min-entropy: More insight about the behavior of the channel under finitely many uses can be gained by considering the corresponding single-shot quantity. To formulate it, note that the state ρ ABE is pure, in which case the duality relation H(A|B) ρ = −H(A|E) ρ for the conditional von Neumann entropy holds. This gives us The corresponding single-shot quantity for the conditional von Neumann entropy H(A|E) ρ is the conditional minentropy, or just min-entropy, H min (A|E) ρ [8]. It is defined as where the maximum is taken over all subnormalized density operators on H E , i.e. all positive operators on H E with trace between 0 and 1. The min-entropy quantifies the maximal size of a subsystem of A that can be decoupled from E [9], and thus tells us how many EPR pairs between Alice and Bob can be created [10] given a noisy output state ρ AB . To obtain the single-shot capacity of n channel uses we are -as in the asymptotic case -allowed to optimize over input states ρ A n (A ) n . Clearly, however, the resulting expression can be lower bounded using a particular input state given by n copies of the maximally entangled state. This is the test state we employ here, and hence our test also provides a bound on the single shot capacity. For instance if A is a 2 level system, then the min-entropy readily quantifies the number of EPR-pairs we can recover, given that we started with n EPR pairs as an input. The min-entropy thus has a very appealing operational interpretation. For our purposes, another expression for the min-entropy is more useful. In the following, we use the symbol to denote that two Hilbert spaces are isomorphic, i.e. H A H A means that the two spaces have the same dimension. It has been shown [11] that the min-entropy can be expressed as where d A is the dimension of the Hilbert space H A of system A, A is a system with H A H A , the maximization is carried out over all TPCPMs R E→A from system E to system A , F (ρ, σ) = Tr ρ 1/2 σρ 1/2 is the fidelity and Φ AA is a maximally entangled state on AA , i.e. Φ AA is an element of the set The choice of Φ AA ∈ Γ AA , i.e. the choice of bases for H A and H A , is irrelevant for the value of H min (A|E) ρ . Since every Φ AA = |φ φ| AA ∈ Γ AA is pure, we have that F (Φ AA , σ AA ) = φ|σ|φ AA for any state σ AA on AA . The expression (14) provides an intuition for the min-entropy. We think of the system ABE, which is in the pure state ρ ABE , as being distributed between Alice, Bob and Eve. Imagine that Eve tries to perform operations on her share of the system with the intention to bring the reduced state between her and Alice as close as possible to the maximally entangled state Φ AA , where the square of the fidelity is the measure of closeness. The closer Eve can bring the state to the maximally entangled state, the smaller the min-entropy H min (A|E) ρ . The overall situation of our decoherence analysis is shown in Supplementary Fig. 3.
The min-entropy is strictly more informative than the conditional von Neumann entropy in the following sense. In the iid limit (which stands for independent and identically distributed ), where many identically prepared systems go through the channel and end up in a state ρ ⊗n AB , the min-entropy converges to the conditional von Neumann entropy: where > 0 is an arbitrary smoothing parameter. This is known as the asymptotic equipartition property [12]. Thus, in the limit of infinitely many channel uses, where the asymptotic quantity is relevant, the min-entropy reproduces the conditional von Neumann entropy.
To gain some intuition for H min (A|E) ρ , we now have a look at some special cases. For these special cases, we assume that H A H A H B H E . Assume that initially, the state ρ AA is maximally entangled, i.e. ρ AA = Φ AA for some Φ AA ∈ Γ AA analogous to (15). We think of the channel purification U A E →BE as being controlled by Eve.
• If the adversary Eve leaves system A untouched, i.e. the channel Θ A →B is the identity channel (or any other unitary channel), then In that case, H min (A|E) ρ = log d A , and we say that there is no decoherence.
• In the other extreme case, Eve snatches away the system A and forwards an uncorrelated system to Bob. In this case, ρ AE ∈ Γ AE with Γ AE analogous to equation (15) (maximal entanglement between A and E). Then, H min (A|E) ρ = − log d A , and we say that we have full decoherence.
• As an intermediate case, we might consider the case where Eve interferes such that she does not end up with maximal entanglement with Alice but such that she is classically correlated with Alice in some basis, i.e.
In that case, H min (A|E) ρ = 0, and we speak of partial decoherence.

Introduction
Our goal is to show that Alice and Bob can estimate the decoherence by performing a Bell experiment. We pose it as a feasibility problem: is it possible to observe certain statistics in a Bell experiment given a certain level of decoherence? Solving this problem allows us to determine and plot the feasible region in the space of suitably chosen parameters.
We look at the simplest Bell experiment, known as the Clauser-Horne-Shimony-Holt (CHSH) [13] scenario. If ρ AB is the state that Alice and Bob share and A j , B k for j, k ∈ {0, 1} are the observables they perform, then the CHSH value equals As explained previously the min-entropy H min (A|E) defined in Eq. (13) captures the notion of decoherence between Alice and Bob (although note that high min-entropy corresponds to low decoherence and vice versa). Since the range of values that the min-entropy takes depends on the dimension of Alice's system (denoted by d A ), it is only meaningful to compare scenarios in which d A is fixed. For simplicity, we consider the simplest non-trivial scenario in which the subsystems held by Alice and Bob are qubits, d A = d B = 2.
We define the feasible region S as follows. • The conditional min-entropy of A given E equals u: H min (A|E) = u.
First note that a CHSH value of v 2 can be achieved using trivial measurements (namely {1 1, 0}) acting on an arbitrary state. Therefore, for v 2 all values of u ∈ [−1, 1] are allowed. For the remainder of the argument we implicitly assume that v > 2 and the following intuitive argument shows why certain pairs (u, v) must indeed be forbidden. Consider a point u ≈ −1 and v > 2. According to the operational meaning of the min-entropy (14), u ≈ −1 means that Eve can recover the maximally entangled state with Alice with fidelity close to unity, which clearly allows Alice and Eve to violate the CHSH inequality. On the other hand, since v > 2 Alice also observes a CHSH violation with Bob. This violates the monogamy relation for tripartite three-qubit states proved in Ref. [14], which states that Alice can violate the CHSH inequality with at most one party (even if she is allowed to use different measurements for different scenarios). This simple argument leads to the conclusion that the region u ≈ −1 and v > 2 is forbidden. In the remainder of this section we show that the non-trivial part of the feasible region S can be fully characterized by a single inequality. where where the maximization is taken over While the definition of f might seem complicated, it is straightforward to see that f is monotonically increasing in v and evaluating f (v) numerically for a particular value of v is straightforward since the function to be maximized is concave. The feasible region S is plotted in Supplementary Fig. 4. The proof of Theorem 1 is conceptually simple, but it requires a wide array of technical tools, which we present in the Preliminaries subsection. In the two subsequent subsections, we prove the direct and converse parts of Theorem 1, respectively.

Preliminaries
Lemma 3: Let A 0 be a positive semi-definite operator, let Π A be the projector on its support and let |b be a normalized vector. Then A |b b| iff Note that since A might not be invertible, A −1 is only defined on the support of A.

Two-qubit states
A two-qubit state written in the Pauli basis takes the form where all the summations go over {x, y, z}. It is known that for every state there exists a local unitary U A ⊗ U B which diagonalizes the correlation tensor (i.e. ensures that T jk = 0 for j = k) and since all the properties we consider are invariant under local unitaries we can make this assumption without loss of generality. We denote these diagonal entries T xx , T yy and T zz by c x , c y and c z , respectively, which simplifies the expression to Without loss of generality, we assume that |c x | |c y | |c z | and c x , c y 0. As shown in Ref. [15] every Bell-diagonal state of two qubits (up to local unitaries which, again, we can safely ignore) can be written as where {p j } 4 j=1 is a probability distribution and |Φ 1,2 = |00 ±|11 . It is easy to verify that where

Non-locality
Definition 4: For a bipartite quantum state ρ AB the maximum CHSH value is defined as where the maximization is taken over all Hermitian, binary observables.
Note that for all states β max 2 and we say that the state violates the CHSH inequality if β max > 2. It was shown in Ref. [16] that if ρ AB is a state of two qubits then the value of β max is fully determined by the correlation tensor. Adopting the convention |c x | |c y | |c z | we have

Entropic measures of entanglement
To derive a bound on the min-entropy H min (A|E) ρ , we will use a closely related quantity, namely the max-entropy.
Definition 5: For a bipartite quantum state ρ AB the conditional max-entropy (or just max-entropy) is defined as where π A is the maximally mixed state on A and the maximization is taken over all states on B.
The proof uses the following known properties of the min-and max-entropies.
Lemma 6 (Duality, [11]): Let ρ ABC be a tripartite state. Then and the equality holds iff ρ ABC is pure.
Lemma 7 (Data-processing inequality, [8]): For an arbitrary tripartite state ρ ABC we have Lemma 8 (Conditioning on classical information, Proposition 4.6 of [17]): Let ρ ABK be a tripartite state where K is a classical register: Then Finally, we need an explicit expression for the max-entropy of a Bell-diagonal state. Note that by assumption Lemma 9: Let ρ AB be a Bell-diagonal state of form (23). Then the conditional max-entropy equals To prove Lemma 9 we use the fact that the optimization problem which appears in the definition of the max-entropy (32) can be written as a semidefinite program (SDP) [18]. More specifically, given ρ AB we have H max (A|B) = log λ, where λ is the value of the following SDP for ρ ABC being an arbitrary purification of ρ AB PRIMAL : minimize µ subject to µ1 1 B tr A (Z AB ) where P(H) denotes the set of positive semi-definite operators acting on H. By providing feasible solutions for the PRIMAL and the DUAL we show that for Bell-diagonal states which is precisely the statement of Lemma 9.
For the PRIMAL consider Clearly, Z AB 0, µ 0 and since Tr A (Z AB ) = 1 d j √ p j 2 1 1 B the first constraint is easy to check. The last inequality we need to check is We apply Lemma 3 to A = Z AB ⊗ 1 1 C and |b = |ψ ABC . The projector on the support of Z AB ⊗ 1 1 C equals and it is easy to verify that Π|ψ ABC = |ψ ABC . Moreover, since Showing that Z AB and µ constitute a valid solution to the PRIMAL implies that λ 1 Note that Y ABC is proportional to a rank-1 projector. The first constraint gives and the remaining ones are easily verified to be true. The value of this solution equals

Sufficiency of considering Bell-diagonal states
To prove the converse part of Theorem 1, we will use the following argument, which is similar in spirit and inspired by the symmetrization argument presented in Ref. [19].
Lemma 10: Let ρ AB be an arbitrary state of two qubits. Then, there exists a Bell-diagonal state σ AB which satisfies Proof. We present an explicit construction of σ AB which meets the requirements. According to Eq. (26), ρ AB can be written as Moreover, consider the following random unitary channel where U 1 = 1 1, U 2 = σ x , U 3 = σ y and U 4 = σ z . It is easy to verify that for j ∈ {x, y, z} because each Pauli operator commutes with identity and itself but anticommutes with the other two unitaries. This implies that σ AB = Λ(ρ AB ) is Bell-diagonal. Moreover, one can check that the map preserves the correlation tensor, i.e. for j ∈ {x, y, z} which implies that β max (ρ AB ) = β max (σ AB ). To check the last property consider the following state By the data processing inequality, we have H max (A|B) σ H max (A|BK) σ and by conditioning on classical information we have where Since the max-entropy is invariant under local unitaries we have H max (A|B) τ j = H max (A|B) ρ for j ∈ {x, y, z} which implies that The final technical lemma concerns the problem of maximizing the max-entropy of a Bell-diagonal state of two qubits whose maximal CHSH violation is fixed.
Lemma 11: Let ρ AB be a Bell-diagonal state of two qubits, whose maximal CHSH violation equals β ∈ (2, 2 √ 2]. Then, the max-entropy of ρ AB satisfies the following inequality for function f defined in Eq. (19). Moreover, there exists a state which saturates this inequality.
Proof. According to Lemma 9 the max-entropy of a Bell-diagonal state of two qubits equals Here, it is convenient to express the probabilities through the correlation coefficients c x , c y , c z . Inverting Eqs. (29) gives which allows us to write where In the space of correlation coefficients the feasible set are the triples (c x , c y , c z ) for which the function g(c x , c y , c z ) is well-defined (the expressions under the roots must be non-negative). As before, we assume without loss of generality that |c x | |c y | |c z | and c x , c y 0. Then, the maximal CHSH violation (we are only interested in states that violate the CHSH inequality) is given by Eq. (31) Since in our case β is fixed, the angular parametrization takes the form where q = β √ 2 and φ ∈ [0, π/4] (which ensures c x c y 0). Note that It is easy to check that the allowed range of c z is Note that we should also impose the condition |c z | |c y | but as it turns out the optimal solution will satisfy it even if we do not include it explicitly. To maximize the max-entropy it is sufficient to maximize function g defined in Eq. (62), which in the angular parametrization equals over The maximum is achieved either in the interior (denoted by R int ) or at the boundary. Let us start by ruling out the first option. Function g is differentiable everywhere in R int and the partial derivatives are To prove that there is no maximum in the interior, it suffices to show that there is no (φ, c z ) ∈ R int such that both derivatives vanish ∂g ∂cz = ∂g ∂φ = 0. To do this we consider the following linear combination and show that s(φ, c z ) = 0 has no solution in R int . Since the last term of s(φ, c z ) is negative, a necessary condition for s(φ, c z ) = 0 is that the sum of the first two terms is non-negative, which is equivalent to This can be rearranged to give which contradicts the second inequality in the definition of R int as shown below.
c z q 2 cos φ − 1 and 1 − q cos φ > c z (67) It is easy to check that the left-hand side of the final inequality is always at least √ 2, while the right hand side is always at most √ 2. This proves that the final (strict) inequality is always false, which implies that s(φ, c z ) = 0 has no solutions in R int and that g(φ, c z ) has no maximum in R int .
The boundaries c z = q sin φ − 1 and c z = 1 − q cos φ correspond to one of the expression under the roots being zero. Since the square root function has infinite slope at 0, such solutions cannot be optimal. Therefore, the maximum must be achieved at the boundary φ = 0. Combining Equations (61) and (63) and setting φ = 0 leads directly to the statement of the lemma.
To show that the solution of the optimization problem satisfies |c z | |c y |, it is sufficient to show that for φ = 0 and c z = −c y = −q/2 the partial derivative ∂g/∂c z is strictly positive.

The direct part
Here, we show (by an explicit construction) that points described by v ∈ (2, 2 √ 2] and f (v) u 1 are allowed. Lemma 11 shows that for v ∈ (2, 2 √ 2] there exists a Bell-diagonal state of two qubits whose max-entropy equals By duality (Lemma 6), if ρ ABE is an arbitrary purification, the conditional min-entropy equals In this example u = f (v), which corresponds to a point lying precisely on the boundary defined in Theorem 1. In order to obtain higher values of u (all the way up to 1), it suffices to apply noise of appropriate strength to subsystem E.

The converse part
Here, we show that every feasible point (u, v) must satisfy u f (v). Consider a state ρ ABE for which H min (A|E) ρ = u and which for some measurements achieves the CHSH value of v. Clearly, β max (ρ AB ) v and by Lemma 6 H max (A|B) ρ −u. Applying the symmetrization argument (Lemma 10) gives rise to a Bell-diagonal state σ AB such that H max (A|B) σ −u and β max (σ AB ) v. By Lemma 11 these quantities must satisfy which implies that where the last inequality follows from the fact that f is monotonically increasing.

Supplementary Note 3: Decoherence estimation through CHSH tests in GPTs
In this appendix, we are going to develop a framework for decoherence analysis in analogy to Supplementary Note 2, but without assuming that nature is correctly described by quantum theory. Instead, we will work in a framework that makes only minimal assumptions about the probabilistic structure of measurements. This allows to make statements in cases where quantum theory might not be a correct description of nature.
In the first subsection, we define a framework for probabilistic theories that has become a standard one in the literature. Besides defining the core structure, we explain how we extend this framework to make it suitable for analyzing tripartite states, in a way that allows us to make a decoherence analysis that is analogous to the quantum case.
In the second subsection, we will define a decoherence quantity Dec(A|E) ω for GPTs as an analogue of the quantum min-entropy H min (A|E) ρ . This will be our quantity of interest for the decoherence analysis for GPTs. We will first motivate an expression for Dec(A|E) ω , inspired by expression (14) for the min-entropy in the quantum case. This expression will require us to say what a maximally entangled state in a GPT is.
The third subsection is devoted to finding a bound on our decoherence quantity in terms of the CHSH winning probability for Alice and Bob. This is a measurable quantity in the case where the channel is an iid (for independent and identically distributed ) channel, meaning that it behaves identically in repeated uses of the channel without building up correlations amongst systems going through the channel in different uses of it. This is a practically relevant case, giving our bound a practical meaning. This bound allows us to infer non-trivial statements about decoherence from measured data when, apart from the iid assumption, we assume only very little about the behavior of nature. We approach our bound by first bounding our fidelity-based decoherence quantity by a trace distance-based quantity. We will then bound this trace distance-based quantity in terms of the CHSH winning probability for Alice and Bob by a quantity that can be expressed as a linear program.
Finally, in the fourth subsection, we show how our bound can be expressed as a linear program and present the numerical results. This is followed by a discussion of the physical interpretation of our numerical findings.

A basic framework for GPTs
Frameworks for probabilistic theories in which quantum theory and classical theory can be formulated as special cases have already been considered some decades ago [20][21][22]. After some period of oblivion, a seminal paper by Hardy [23] caused a revival in the interest in such frameworks (see, for example, [24][25][26][27][28][29] and references therein). Today, they are generally referred to as frameworks for generalized probabilistic theories [30].
We formalize our decoherence analysis for GPTs in the abstract state space framework [31][32][33][34]. It is one rigorous formalization of what a generalized probabilistic theory is, amongst a few equivalent or closely related ones that can be found in the literature (see the references cited above). We prefer it for its concise and precise formulation. For the sake of brevity, we will not go far beyond the mere mathematical definitions related to abstract state spaces here. For a detailed introduction to abstract state spaces, see [35].
Definition 12: An abstract state space is a triple (V, V + , u), where V is a finite-dimensional real vector space, V + is a cone 2 in V which is closed 3 and generating 4 and u ∈ V * is a linear functional 5 on V such that u(ω) > 0 for all ω ∈ V + \ {0}. The functional u is called the unit effect.
The subnormalized states are the elements of the set The effects are the elements of the set 3 We assume the standard topology on V , i.e. the only linear Hausdorff topology on V .
This precisely reproduces the structure of measurement statistics in quantum theory. For further details, see [35].
By our definition, E is the set of all linear functionals e such that 0 e(ω) 1 for all ω ∈ Ω. The underlying assumption that every such linear functional represents a physical measurement outcome has been called the norestriction hypothesis [28]. A priori, there seems to be no immediate physical reason for this assumption, and some authors have argued about how to weaken this assumption [36]. For our purposes here, it is not relevant whether the no-restriction hypothesis holds, and weakening the assumption complicates the definitions. Thus, we assume it for simplicity.
In the second subsection, we will define a decoherence quantity Dec(A|E) ω analogous to the quantum min-entropy H min (A|E) ρ . We will take our inspiration from expression (14) for the quantum min-entropy, which involves the fidelity as a measure of closeness of quantum states. Therefore, it is desirable to have a generalization of the fidelity to states in abstract state spaces. Such a generalization is easily found once it is noticed that the quantum fidelity of two states can be expressed as the Bhattacharyya coefficient (or classical fidelity) of the two probability distributions that the two states induce, minimized over all measurements. More precisely, the quantum fidelity satisfies [37] F (ρ, σ) = min where the minimization runs over all POVMs {P k } k on the Hilbert space on which ρ and σ are defined. The sum in (77) is precisely the Bhattacharyya coefficient of the probability distributions that the POVM {P k } k induces on the states ρ and σ. This motivates us to define the fidelity for abstract state spaces as follows.
The quantity b(ω, τ |M ) is the Bhattacharyya coefficient (or sometimes called the classical fidelity) of the probability distributions that the measurement M induces on the states ω and τ .
The fidelity as defined in Definition 15 precisely reduces to the quantum fidelity in the case where the abstract state space is a quantum state space. In addition to the fidelity, in third subsection we will also consider a generalization of the quantum trace distance D(ρ, σ) = 1 2 Tr|ρ − σ| in order to formulate a bound on Dec(A|E) ω . Somewhat analogously to the fidelity, the quantum trace distance is equal to the total variation distance (or classical trace distance) between the two probability distributions that the two states induce, maximized over all measurements [37]: This motivates the following definition.
Definition 16: Let (V, V + , u) be an abstract state space with normalized states Ω and measurements M. For states ω, τ ∈ Ω, the trace distance between ω and τ is given by The quantity d(ω, τ |M ) is the total variation distance (or sometimes called the classical trace distance) between the probability distributions that the measurement M induces on the states ω and τ .
Note that the fidelity and the trace distance take values between 0 and 1 for all states. For squares of the quantities F , b, D and d, we will write the square sign right after the letter, e.g. we will write F 2 (ω, τ ) instead of (F (ω, τ )) 2 .

A tripartite framework for GPTs
In the second subsection, we will consider a tripartite situation for the decoherence analysis, analogous to Supplementary Note 1. This requires us to model a tripartite scenario mathematically since such a structure is not induced by an abstract state space (V, V + , u) alone. We need to specify it as additional structure. Our goal here is to do this with the weakest possible assumptions, resulting in a very general validity of the bounds we derive.
Instead of assuming individual state spaces for every party, we only consider their overall combined state space, modeled by an abstract state space (V, V + , u) and all its induced structure as in Definitions 12 and 13. This has the advantage that we do not have to make assumptions about how individual state spaces combine to multipartite state spaces, keeping our assumptions weak. For our purposes, the only structure that we need to add to an abstract state space (V, V + , u) to make it suitable for the description of a tripartite scenario are the local transformations that each individual party can perform. The local measurements of the three parties are then induced by these local transformations.
We consider three parties, which we call Alice (A), Bob (B) and Eve (E) as before. We begin our considerations by assuming that there are three sets T A , T B and T E , containing all the transformations that Alice, Bob and Eve can perform, respectively. By a transformation, we mean a linear map T : V → V which maps states to subnormalized states, i.e. T (Ω) ⊆ Ω (or, equivalently, T (V + ) ⊆ V + and (u • T )(ω) u(ω) for all ω ∈ V + ). We can consider the case where several transformations are applied because compositions of transformations are transformations again: If T , T are linear maps V → V which map Ω inside Ω , then the same is true for the composition T • T (we denote the composition of maps by a • symbol).
We assume that the three parties act individually at spatially separated locations. Relativistic considerations lead to the consistency requirement that transformations performed by different parties must commute, e.g. if Alice performs a transformation T A ∈ T A and Bob performs a transformation T B ∈ T B , then the total transformation must satisfy For our purposes, we do not need to specify the sets T A , T B and T E any further; the only requirement is that transformations of distinct parties commute. The sets T A , T B and T E define the systems A, B and E, i.e. we define the individual parties via the transformations that they can perform. This leads us to the following definition.

Definition 17: A tripartite scenario is a quadruplet
where (V, V + , u) is an abstract state space, and where are such that for all P, P ∈ {A, B, E} with P = P , it holds that T P • T P = T P • T P for all T P ∈ T P and for all T P ∈ T P . We call the elements of T A , T B and T E the local transformations of A, B and E, respectively.
It is absolutely natural to define tripartite scenarios via commuting transformations rather than via a tensor product structure. In quantum theory, the two approaches are equivalent in finite dimensions (we will talk about this below). In more general infinite-dimensional cases, where it is not known whether the two approaches are equivalent, things are usually formalized in a commutative way rather than via tensor products (see [38], for example). Knowing about the equivalence in finite dimensions, we will formulate some quantum examples in the tensor product structure below.
where CPM stands for completely positive map. Having tensor product form, the local transformations of different parties commute.
For our purposes, Definition 17 is all the structure one needs to specify. The local measurements are induced by the local transformations. We formalize this via the notion of a local instrument [22]. To get an intuition for what an instrument is, consider a Stern-Gerlach experiment. A spin-1/2 particle enters a magnet and undergoes one of two transformations: It either gets deflected upwards or downwards. Which of the two transformations it undergoes is determined probabilistically. Then it hits a screen, which reveals which of the two transformations the particle has undergone. This way, a measurement has been performed in two stages: a probabilistic application of a transformation and a detection. The sum of the probabilities of detecting the particle at the top or the bottom of the screen is one. If the state of the particle is described by a state ω ∈ Ω of an abstract state space, we may model this by a set of two transformations {T up , T down }. Such a set is an instrument. The norm u(T up (ω)) is the probability that the particle is deflected upwards, and likewise for u(T down (ω)). Thus, u can be seen to play the role of the screen, detecting the particle. The requirement that the particle must undergo one of the two deflections reads u • T up + u • T down = u. The transformation T up is the analogue of the transformation ρ → P up ρP up in quantum theory, where P up is the projector onto the spin-up state. Since u is given by the trace in quantum theory, the probability for the upward-deflection to occur is given by Tr(P up ρP up ) = Tr(P up ρ), which is precisely the Born rule.
A local instrument is such a set of transformations where all the transformations are the local transformations of one party. This motivates the following definition.
for a trace non-increasing CPM R A on Herm(H A ). However, for every such CPM, there is a POVM element P A on This recovers the Born rule. Analogously, a composite measurement where Alice and Bob each perform local measurements consists of local effects of the form for POVM elements P A , P B on H A , H B . Thus, in our tripartite quantum example, local measurements reduce to POVM measurements of product form.
In Examples 18, 20 and 22, instead of choosing a tensor factorization for H and setting the local transformations to be acting non-trivially on one tensor factor, we could have chosen sets of transformations that merely commute, without a tensor product structure. The question of whether the resulting measurement statistics in that case would be different from the case with the tensor factor structure is known as Tsirelson's problem [39,40]. More precisely, the question is the following. Let H be a Hilbert space, let ρ be a density operator on H, let {P k } k , {Q l } l be POVMs on H such that P k Q l = Q l P k for all k, l. Tsirelson's problem is: Does there necessarily exist Hilbert spaces H A , H B , a density operator σ on H A ⊗ H B and POVMs {R k } k on H A and {S l } l on H B such that Tr(P k Q l ρ) = Tr((R k ⊗ S l )σ) for all k, l? In the case where H is finite-dimensional, the answer is known to be affirmative. For infinite-dimensional Hilbert spaces, the answer is still unknown.
Thus, for finite-dimensional quantum systems, we can restrict ourselves to the case with the tensor product structure without loss of generality. For abstract state spaces, however, an analogous restriction might cause a loss of generality. The advantage of our weak definition of a tripartite scenario is that we do not need to know the answer to an equivalent of Tsirelson's problem for generalized probabilistic theories. The downside is that it makes defining an equivalent of the min-entropy more difficult. We will deal with this issue in the next subsection.
Notation: From now on, whenever we speak of a tripartite scenario S ABE , we implicitly assume that all its parts and induced structures are denoted as in Definitions 12, 13, 17 and 19 without restating it, i.e. instead of writing "Let S ABE = ((V, V + , u), T A , T B , T E ) be a tripartite scenario, let Ω be its set of normalized states, . . . ", we will only write "Let S ABE be a tripartite scenario".

Motivation of an expression that quantifies decoherence
We are now going to motivate an expression for the central quantity Dec(A|E) ω for our decoherence analysis for GPTs. We take our inspiration from expression (14) for the quantum min-entropy, which we repeat here for the reader's convenience: There are two issues that prevent us from directly translating expression (14) into our framework. The first issue is that in in the first subsection, to keep our framework as general as possible, we have defined a tripartite scenario with an overall state space (V, V + , u) with tripartite states Ω. We do not have notions of individual state spaces at hand. Thus, we do not have an analogue of a reduced state ρ AE or of a transformation R E→A from one state space to another. The second issue is that we do not know what the analogue of a maximally entangled state Φ AA in our framework is. We resolve the first issue in the following paragraphs, arriving at an expression for Dec(A|E) ω . Further below, we will then define what a maximally entangled state is in our framework.
Expression (14), which involves the state ρ AE and TPCPMs R E→A , can be transformed to an expression in which both the state and the TPCPMs are purified (see Supplementary Fig. 6). This expression will be our motivation for the expression for Dec(A|E) ω . The maximization over TPCPMs from E to A is replaced by a maximization over unitaries from EE to A A , where E and A are ancilla systems extending system E and A , respectively. This is precisely the purification (or Stinespring dilation) of a channel as in Supplementary Note 1. Since systems EE and A A have the same dimension, we can identify their Hilbert spaces and regard the resulting Hilbert space as the Hilbert space of a system E tot . This system involves all subsystems that the third party needs to control in order to bring itself as close as possible to maximal entanglement with Alice. Since U Etot is a transformation on system E tot alone, we can translate it into our generalized framework.
The state ρ AE is replaced by a purification ρ ABE . We choose the purifying system B to be the channel's output system, which gives us the overall picture of our decoherence analysis as shown in Supplementary Fig. 7.
The following gives a precise formulation of the purification of expression (14). It can be proved using purification and Stinespring dilation. Let H A , H E be finite-dimensional Hilbert spaces of dimensions d A , d E , respectively, let ρ AE ∈ S(H A ⊗ H E ). Then, for any purification ρ ABE ∈ S(H A ⊗ H B ⊗ H E ) of ρ AE , any Hilbert spaces H A , H A and A , respectively, any maximally entangled state Φ AA ∈ Γ AA and any pure state |0 0| E ∈ S(H E ), it holds that where ρ ABEtot = ρ ABE ⊗ |0 0| E and where the first maximization ranges over unitaries and the second maximization ranges over pure states σ BA ∈ S(H B ⊗ H A ). Now we translate expression (92) into our generalized framework. We interpret the system E tot as the system controlled by Eve, and therefore rename E tot → E.
• Since we want to arrive at an expression that does not make unnecessary assumptions about the mathematical description of the physical situation, we avoid the factor d A present in (92). We look for a GPT analogue of As a consequence, we will have H min (A|E) ρ = − log d A Dec(A|E) ρ in quantum theory (see Example 25).
• We replace the maximization over all unitaries U Etot acting on system E tot by a supremum 7 over all local transformations T E ∈ T E . 8 • We generalize the quantum fidelity to the fidelity in abstract state spaces as defined in Definition 15.
• If we look at the state Φ AA ⊗ σ BA , we see that it is a state of maximal entanglement between Alice (A) and Eve (A A ) in the sense that by performing measurements with elements of the form P A ⊗ P A ⊗ 1 1 B ⊗ 1 1 A , they can get any statistics that two parties A and A would be able to get by performing local measurements on the maximally entangled state Φ AA . We translate this into our framework by assuming that there is a set Ψ AE of "states with maximal correlation between Alice and Eve". Instead of minimizing over states Φ AA ⊗ σ BA , we then minimize over the set Ψ AE .
We postpone the discussion of how such a set Ψ AE looks like. We will give a definition of such a set further below. For now, we write down an expression for our decoherence quantity Dec(A|E) ω that depends on the choice of such a set Ψ AE ⊆ Ω. According to what we have just discussed, the expression is We interpret the decoherence to be high when this quantity is high and vice versa, which is the opposite of H min (A|E) ρ (see the end of Supplementary Note 1). Before we can define Dec(A|E) ω , however, we need to specify what a maximally entangled state in a GPT is.

Definition of maximal correlation in GPTs
The expression (94) for our decoherence quantity Dec(A|E) ω contains a maximization over a set Ψ AE ⊆ Ω which we interpret to be the set of states with maximal correlation between Alice and Eve. We now define this set.
Definition 23: For a tripartite scenario S ABE , we define the set Ψ AE of states with maximal correlation between Alice and Eve by For every binary local instrument Definition 23 can be read as follows. The superscripts 0 and 1 of the elements of the instruments I A and I E stand for measurement outcomes, so (u is the probability that Alice and Eve both get outcome 0 or both get outcome 1, respectively, when they measure with respect to I A , I E , respectively. Thus, the sum of these probabilities is the probability that Alice's and Eve's measurement outcomes are perfectly correlated. This means that for a state ψ ∈ Ψ AE , it holds that for every binary measurement of Alice, there is a binary measurement for Eve such that their measurement outcomes are perfectly correlated.
A closer look at some subtleties is advisable here, both to avoid confusion and to see the advantages of the weak assumptions that define our framework. With reference to Example 22, one may point out that that the set is empty. This may seem to make our definition of Ψ AE incompatible with quantum theory. Note, however, that the set is not empty as long as dim then this set contains all the states of the form Φ AA ⊗ σ BA with Φ AA ∈ Γ AA as in (92). The advantage of our weak definition of the local transformations is that it does not force to see T A as the analogue of the set of all CPMs of the form R A ⊗ 1 1 B ⊗ 1 1 E , but that it can be considered to be the analogue of all such CPMs which induce a functional of the form σ → Tr(P σ), where P is a projector. Example 22 can be modified accordingly (see Example 25 below). This makes our definition of Ψ AE compatible with quantum theory.
With Definition 23 at hand, we are finally ready to define the decoherence quantity.
Definition 24: Let S ABE be a tripartite scenario, let ω ∈ Ω. We define the decoherence quantity of ω by Example 25: We consider a special case of a tripartite scenario in quantum theory. Consider and analogously for T B and T E . In addition, we assume for simplicity that H A H E . In this case, For every binary projective measurement {P 0 where Γ AE is the set of maximally entangled states on S(H A ⊗ H E ) analogous to (15). For a pure state ρ ABE ∈ S(H A ⊗ H B ⊗ H E ), this gives us Bounds on the decoherence quantity for GPTs The goal of this subsection is to derive an upper bound on Dec(A|E) ω in terms of the CHSH winning probability of Alice and Bob. This is a practically relevant bound: On the premise that the channel behaves identically in multiple uses and does not build up correlations between different uses (such a channel is said to be iid, for independent and identically distributed ), this winning probability can be estimated through repeated measurements on Alice's and Bob's side. What we show is that this estimate in turn gives a bound on Dec(A|E) ω . In this section, we formulate this bound as a minimization problem which we solve and interpret in the last part of this Supplementary Note.
In the following, we derive a lower bound on − log Dec(A|E) ω . We make the convention that − log 0 = ∞, where ∞ is a symbol for which we accept the inequality ∞ r for every real number r. This lower bound on − log Dec(A|E) ω then gives us an upper bound on Dec(A|E) ω . In a first step, we bound the fidelity-based quantity − log Dec(A|E) ω by a trace distance-based quantity. This has the advantage that the resulting optimization problems which give us the bounds can be solved using linear programming.
Proposition 26: Let S ABE be a tripartite scenario, let ω ∈ Ω. Then The following lemma is useful for the proof of Proposition 26 below.
Proof. We have that − log(x 2 ) = −2 log(x), so the claim is equivalent to The functions F (x) = log(x) and G(x) = x − 1 are differentiable on R >0 . Thus, by the fundamental theorem of calculus, it holds that for all x ∈ R >0 , so for all x ∈ (0, 1], we have that This proves the claim. Proof of Proposition 26. Since the right hand side of (107) is a finite real number, the inequality trivially holds if Dec(A|E) ω = 0 by the above convention. Thus, we assume in the following that Dec(A|E) ω > 0. We have that 26 For x ∈ (0, 1], it holds that − log x 2 2(1 − x) (see Lemma 27). Thus, since we get that − log Dec(A|E) ω 2 1 − sup For the Bhattacharyya coefficient b and the total variation distance d, it has been shown [41] that for any two probability distributions distributions, it holds that 2(1 − b) d 2 . Since this is true in particular for the two probability distributions that the measurement M induces on the states ψ and T E (ω), we get that as claimed.
The idea that the fidelity and the trace distance are related is not new. In quantum theory, the Fuchs-van de Graaf inequalities (FvdG) relate the two quantities [42]. Inequality (107) is not completely analogous to the FvdG inequalities: It makes use of the logarithm in (107), which allows to apply classical relations that lead to a stronger bound than with the application of the FvdG inequalities.
For the bounds that we are going to derive, the notion of a non-signalling distribution is central. Our bounds are essentially minimizations of functions over sets of non-signalling distributions Pr[a, b, c|x, y, z] ω and Pr[a, c|x, z] ψ with certain additional properties.
The interpretation of equations (120) to (122) is that it is impossible for each of the three parties to signal to the other two parties by influencing their measurement statistics with the choice of the measurement setting. These one-party no-signalling constraints imply all the multi-party no-signalling constraints, saying that no collection of parties can signal to the remaining parties [43], so we do not need to require these constraints separately. Now we are going to formulate the bound on − log Dec(A|E) ω in terms of the CHSH winning probability of Alice and Bob. Assume that Alice, Bob and Eve are in a situation described by a tripartite scenario S ABE . Suppose that Alice and Bob have estimated that for the state ω ∈ Ω that they are analyzing, their CHSH winning probability is at least λ for some λ ∈ [0, 1]. Formulated in our tripartite scenario language, this means that they have found out that for local instruments it holds that In that case, what can Alice and Bob infer about − log Dec(A|E) ω ? We have seen in Proposition 26 that this quantity is lower bounded by inf T E ∈T E inf ψ∈Ψ AE D 2 (ψ, T E (ω)). Alice's and Bob's estimate on their CHSH winning probability can be translated into a bound on this quantity. This is shown by the following proposition.
where D ω (λ) is the set of non-signalling distributions for Alice, Bob and Eve such that Alice and Bob have a CHSH winning probability of at least λ, i.e.
Pr[a, b, c|x, y, z] ω is a non-signalling distribution such that and where D ψ is the set of non-signalling distributions for Alice and Eve such that their measurement outcomes are perfectly correlated when they choose the same measurement setting, i.e.
Proposition 29 reduces our problem of lower bounding the decoherence quantity for GPTs to an optimization over non-signalling distributions. This allows us to use linear programming techniques, which in similar ways have been used in [44] to answer questions about non-signalling distributions.
We need the following lemma for the proof of Proposition 29 below.
Lemma 30: Let S ABE be a tripartite scenario, let ω, ψ ∈ Ω. Then, for all local instruments (I A , I B , Proof. It is sufficient to show that for all ω, ψ ∈ Ω, for all T E ∈ T E and for all (I A , I B , I E ) ∈ I A × I B × I E , 28 This is what we are going to show now. Let ω, ψ ∈ Ω, let T E ∈ T E , let (I A , I B , I E ) ∈ I A × I B × I E . Then If instead of taking the supremum over M, we only evaluate the expression for a particular element of M, we get a lower bound on (134). We choose the element (c.f. Remark 21) Hence, By the definition of a local instrument, u = T B ∈I B u • T B . Thus, where in the last equality, we made use of the fact that transformations of different parties commute.
Proof of Proposition 29. It is sufficient to show that for every ψ ∈ Ψ AE , the claimed inequality holds without the minimization over Ψ AE , i.e. inf By means of Lemma 30, we know that for all x, y ∈ {0, 1} and every Let ψ ∈ Ψ AE , let I 0 E } ∈ I E be local instruments for Eve such that which exist according to the definition of Ψ AE (Definition 23). It holds that for every x, y, z ∈ {0, 1}, where Hence, For the no-signalling condition, note that for all c, z ∈ {0, 1}, it holds that and that for all a, x ∈ {0, 1}, it holds that where in the second equality, we made use of the fact that local transformations of different parties commute. Analogously, for every T E ∈ T E , Pr[a, b, c|x, y, z] ω is a non-signalling distribution. Moreover, Pr[a, b, c|x, y, z] ω satisfies where the inequality is one of the assumptions of the proposition. Furthermore, Pr[a, c|x, 1] ψ satisfies (where we made use of (141) and (142) Since Pr[a, b, c|x, y, z] ω satisfies the no-signalling property, the right hand side of (159) is independent of y, so the minimization only needs to be performed over x and z. Moreover, the infimum over the sets D ω (λ) and D ψ is a minimum because it is the infimum of a continuous function over a convex polytope, which is always attained (see the last part of this Supplementary Note for more details). This completes the proof.
Corollary 31 (The bound): Let S ABE be a tripartite scenario, let ω ∈ Ω be a state. If the CHSH winning probability of Alice and Bob is at least λ (in the above sense), then where δ(λ) = min Proof. This is a direct consequence of Propositions 26 and 29.

Formulation of the bound as a linear program
In this subsection, we evaluate the bound (160). To this end, we rewrite (161) in terms of linear programs.  (131) The inequalities define a convex polytope over which the convex function δ(λ) is minimized, so the minimum is attained. It is straightforward to bring these inequalities into the standard form of linear programming. We solved the resulting linear program using standard linear programming routines in Mathematica and Octave.

Solution of the linear program and discussion of the results
We plot the result in Supplementary Fig. 8. The bound 2 −δ 2 (λ) is non-trivial for values λ ∈ (3/4, 1]. This is a very satisfactory result as one cannot expect the bound to be non-trivial for λ ∈ [0, 3/4]: A CHSH winning probability of at least λ ∈ [0, 3/4] for Alice and Bob is always compatible with Dec(A|E) ω = 1. To see this, note that the requirement for a state to yield a CHSH winning probability for Alice and Bob of at least λ ∈ [0, 3/4] is trivial: Alice and Bob can choose trivial measurements that always yields 1 as an outcome, independently of the state. More precisely, in our tripartite scenario language, we can express this as follows. Certainly, there are tripartite scenarios in which the identity map 1 1 V and the zero map 0 V are in T A , T B and T E . 9 For such tripartite scenarios, the condition (c.f. (126) to (128)) is always satisfied because for all ω ∈ Ω, This means that the requirement that the CHSH winning probability for Alice and Bob is at least λ ∈ [0, 3/4] does not exclude the case ω ∈ Ψ AE . In that case, Dec(A|E) ω = sup T E ∈T E sup ψ∈Ψ AE F 2 (ψ, T E (ω)) = 1.

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Supplementary Note 4: General approach for the application of the decoherence formalism In the previous part of this Supplementary Information, we developed our decoherence estimation formalism. In this part, we demonstrate how our formalism can be used to devise tests for given models of decoherence. We will calculate the decoherence quantity for some standard models of decoherence and for a model of gravitational decoherence in Supplementary Note 5, and we will derive a full test for a model of gravitational decoherence in Supplementary Note 6.
We emphasize that the material presented in Supplementary Notes 5 and 6 constitutes example applications of our formalism. We want to see our formalism in action for some examples of decoherence models, and how one can derive a test for a given model. However, our formalism is universal, and its application is not limited to the particular models considered in Supplementary Notes 5 and 6. Therefore, before diving into the applications, we want to give a general picture for how to derive a test from our formalism for a given model for (gravitational) decoherence.
In the literature, there are lots of different theories about how gravitational decoherence affects physical systems. Many of them eventually describe gravitational decoherence as an evolution of a quantum system that is coupled to another system (see the dark gray part in Fig. 1 of the main article). This is effectively describing gravitational decoherence as a quantum channel. In order to test such theories, one would perform measurements on the affected system to see whether the outcomes are compatible with the tested theory. Such a theory and its test rely on the assumption that quantum theory is a correct description for the affected system and the gravitational decoherence process that it undergoes.
Our decoherence estimation formalism allows to take such a theory and turn its test into a universal test which does not rely on the validity of quantum theory (see the light gray part in Fig. 1 of the main article). In this universal test, the system undergoing gravitational decoherence is part of a maximally correlated pair. Measurements on both parts of this maximally entangled pair allow to estimate the CHSH parameter β, which in turn yields a bound h(β) on the decoherence that took place in the experiment. This bound h(β) can be formulated in any theory in which there is a notion of a maximally correlated pair, rather than just quantum theory.
To reach at such a universal test, some work needs to be done, which is divided into two parts. Firstly, one needs to decide on the range of theories within which one wants to test. The range of theories is given by the constrants that one imposes on the theories. Different constraints lead to different functions h that relate the measured parameter β to the bound on the decoherence of the process. We have seen two extreme cases of such constraints: In Supplementary Note 2, we have derived h for the case where the constraints are given by quantum theory, in which case we get with f given in equation (19). The other extreme case that we look at is the case where the only constraint is given by the no-signalling property. This leads to a test within a much larger range of theories. As we have seen, the bound that we get in that case is given by Equation (160) and Supplementary Fig. 8. However, these are just two out of many possible ranges of theories within which one can test decoherence processes with our framework. Secondly, one needs to determine how the existing theory affects one half of a pair of systems. Then, given an input state of the pair, one can calculate Dec(A|E) for this pair. This will give a value for the decoherence quantity that one can test in the experiment.
The first part (determining h) is what we demonstrated in Supplementary Notes 2 and 3. In Supplementary Notes 5 and 6, we will demonstrate how the second part is done. In Supplementary Note 5, we calculate Dec(A|E) for the case where the input state of the pair is the maximally entangled state which leads to the final state The state ρ AB (Λ) as in (167) is known in the literature as the Choi representation of the channel Λ (c.f. channelstate-duality). Written out as a matrix in the product computational basis, the state ρ AB (Λ) becomes the Choi matrix of the channel. The decoherence quantity is then a function of its Choi representation, namely These are calculations within quantum theory, which are just here to show the relation between a decoherence model (which in this case is a quantum channel) and the decoherence quantity associated with it. We emphasize again that this part can alternatively be done in any probabilistic theory. Some of our calculations also concern models for gravitational decoherence that have been considered in the literature. In Supplementary Note 6, we derive a full test for a model for gravitational decoherence. We consider a model first proposed by Bouwmeester [45]. It is a model that describes a photon in an optomechanical cavity. We extend this model where two entangled photons are located in two different cavities, one of which is coupled to a movable mirror introducing gravitational decoherence. This way, we extend the single-photon experiment to a two-photon experiment that can be tested in our formalilsm. We calculate Dec(A|E) for this experiment. Then, we calculate the minimal β that needs to be measured in order to failsify this model, for the case where h is chosen to be the quantum bound. We emphasize, however, that this choice of the bound can be modified within our framework in order to arrive at tests for other ranges of theories. The parameter β measured in the experiment can be reconsidered in all possible ranges of theories.
Supplementary Note 5: The decoherence quantity for some models of decoherence In this appendix, we calculate the decoherence quantity for some important standard classes of channels, for some of which we give closed-form expressions. As we will see, some models in the literature predict that gravitational decoherence acts like one of these standard channels. This allows us to derive closed-form expressions for the predicted values of the decoherence quantity for some models for gravitational decohrence.
More precisely, we calculate the decoherence quantity for the following channels: • Pauli channels, including the special cases depolarizing channel, dephasing channel, -2-Pauli channel, • gravitational decoherence due to metric fluctuations in flat space time [46], which we will reduce to the decoherence of a Pauli channel, • amplitude damping channel, • erasure channel.
In each of these cases, we look at the state that results from the action of the channel on one half of a maximally entangled state, equation (167), and calculate its decoherence quantity, (168). This quantity can be calculated as where we got the first equality from equation (106) and the second equality from Lemma 6. Hence, calculating the decoherence quantity for a channel reduces to calculating the max-entropy of its Choi matrix. We will make use of this in all of the following examples.
The decoherence quantity for Pauli channels Here we consider a class of qubit-to-qubit channels which is called the class of Pauli channels. Many decoherence models turn out to be Pauli channels or turn out to have the same strength of decoherence as the Pauli channels. We will see two examples for that: our calculations in the next subsection and in Supplementary Note 6 will reduce the gravitational decoherence of some models to the decoherence of Pauli channels.
The Pauli channels get their names from the fact that they correspond to the probabilistic application of the Pauli operators σ x , σ y and σ z . They are defined as follows.
Definition 32: For probabilities p x , p y , p z ∈ [0, 1] with p x + p y + p y 1, the Pauli channel C Pauli p with respect to p = (p x , p y , p z ) is defined as the channel In the following, we restrict ourselves to the scenario of metric fluctuations in flat spacetime, and denote the resulting channel by C fluc t . Kok and Yurtsever calculate the effect of the channel on a qubit in the state We modify their analysis and consider the action of the channel C fluc t on one half of a maximally entangled state instead, i.e. we determine the Choi representation (c.f. Supplementary Note 4). For the metric fluctuations in flat spacetime, Kok and Yurtsever consider a uniformly moving qubit with velocity dx/dt = v in the x-direction in a two-dimensional flat Minkowski space. The state of the metric is given by where f ( a) determines the quantum fluctuations of the metric around a = 0, which are assumed to Gaussian. A derivation along the lines of [46] shows that for the resulting channel C fluc t , the Choi matrix reads where and where η * (t) is the complex conjugate of η(t), with the following real parameters: • the elapsed time t, • the particle speed v, • the transition frequency Ω = ω 1 − ω 0 , • the variance σ of the fluctuation.
A comparison of (192) with (172) may make the reader wonder whether the decoherence of this channel can be expressed by the decoherence of an appropriately parametrized dephasing channel, i.e. a Pauli channel with p x = p y = 0 and some p x (t) ∈ [0, 1] for all t ∈ [0, 1]. This is indeed the case. To see that, we express the off-diagonal terms of the Choi matrix (192) in polar form. The factor e −Γ(t) 2 is already real and the factor e i(γΩt+δ(t)) already has polar form. For η(t), we get = |η| e i arg(η) (198) = 1 + 1 4 (1 + v 2 ) 4 Ω 2 t 2 σ 2 e i arctan( 1 2 (1+v 2 ) 2 Ωtσ 2 ) .
It turns out that the Choi matrix (192) can be expressed as ρ AB C fluc t = (1 1 ⊗ U −α(t) ) (1 1 ⊗ C Pauli (0,0,pz(t)) )|Φ + Φ + | (1 1 ⊗ U α(t) ) (202) = (1 1 ⊗ U −α(t) )ρ AB (C Pauli (0,0,pz(t)) )(1 1 ⊗ U α(t) ) where U α(t) = 1 0 0 e iα(t) , This means that, up to a local unitary, the channel C fluc t is a dephasing channel with a deformed strength parameter p. This is good news for us, because local unitaries do not change the decoherence quantity. This can be seen as follows. First note that where we got the first equality from equation (170), and the second equality from the definition of the max-entropy, see equation (32). Furthermore, we have that log d A F 2 (ρ AB (C Pauli (0,0,pz(t)) ), π A ⊗ U −α(t) σ B U α(t) ) (210) = max σ B log d A F 2 (ρ AB (C Pauli (0,0,pz(t)) ), π A ⊗ σ B ) , where we used equation (203) for the first equality, the invariance of the fidelity under unitaries in the second equality, and the fact that a unitary does not change the maximization over all states σ B on B for the last equality. Hence, the decoherence quantity for the channel C fluc t is the same as the decoherence quantity for the channel C Pauli A plot of the decoherence quantity (214) as a function of the time t is shown in Supplementary Fig. 10.
The decoherence quantity for amplitude damping where Its Choi matrix is given by The conditional max-entropy of (217) can easily be solved using a semidefinite program (SDP) solver software such as SeDuMi and YALMIP for Matlab (see the SDP formulation of the conditional max entropy in Supplementary Note 2). A plot of the decoherence quantity of C AD λ is shown on the left hand side of Supplementary Fig. 11 (see the graph with p = 1) as a function of λ. It achieves a maximal decoherence of 1.
The decoherence quantity for erasure models The next family of channels that we want to look at is the family of erasure channels. It is a family of channels parametrized by a probability p ∈ [0, 1]. We denote the erasure channel with parameter p by C E p . For every such parameter, the channel C E p is a qubit-to-qutrit channel. The idea is that a qubit is transmitted via this channel, and with probability 1 − p, it is left unchanged, and with probability p it is erased. In the case of an erasure, the receiver is notified about the erasure, which is modelled by saying that the qubit is mapped to the state | 2 2 | of the qutrit. We can write this channel as where ρ is the image of the embedding which maps the qubit onto the subspace of the qutrit spanned by |0 and |1 .
We get that A plot of the decoherence quantity for this channel can be seen on the right hand side of Supplementary Fig. 11.
Supplementary Note 6: An example test for gravitational decoherence An optomechanical setting and its model for gravitational decoherence We have seen how to calculate the decoherence quantity for several models of decoherence in the previous section. Now we make things more concrete by devising a full test of a particular model of gravitational decoherence. This includes not only the calculation of the decoherence quantity, but also the minimal CHSH quantity β that one would need to measure in order to falsify the model, and an estimate of the point in time at which it is most likely to measure such a value.

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The objective here is to create two entangled photonic qubits in which one photon is prepared in an opto-mechanical system that is itself subject to gravitational decoherence -if there is any -and the other photon is prepared in an identical cavity except the mirrors are fixed and cannot move. This model is a modification of the model first proposed by Bouwmeester [45] in which an itinerant single photon pulse is injected into a cavity rather than created intra-cavity as here. Our modification avoids the problem that the time over which the photons interact with the mechanical element is stochastic and determined by the random times at which the photons enter and exit the cavity through an end mirror. In the new scheme, the cavities are assumed to have almost perfect mirrors -very narrow line width (see for example [47]).
The intracavity single photon Raman source is described in Nisbet-Jones, et al. [48]. In this scheme (see Supplementary Fig. 12) a control pulse can quickly and efficiently prepare a cavity mode in a single photon state by driving a Raman transition between two hyperfine levels we label as |g , |e . In our scheme there are two optical cavities otherwise identical except in one of the cavities a mechanical element can respond to the radiation pressure force of light.
We will assume that we can prepare the atomic sources in an arbitrary entangled state |g, e + |e, g , for example, using the trapped ion schemes of Monroe [49]. In addition we will assume that we can make arbitrary rotations in the g, e subspace of each source and also make fast efficient single shot readout of the state of each source, for example using fluorescence shelving. This means we can readout the atomic qubit in each cavity in any basis.
The write laser implements the Hamiltonian H w = i Ω(t)(a † |e g| − a|g e|)/2. This is a rotation in the state space {|g |0 , |e |1 }. We can thus prepare arbitrary states of the form cos θ/2|g |0 + sin θ/2|e |1 , where θ is determined by the pulse area. We will refer to the case of θ = π as a π-pulse. Note that if the source is in the excited state |e and the cavity is in the vacuum, no photon is excited.
Starting with the cavities in the vacuum state the protocol proceeds as follows: 1. prepare the source atoms in the state |g, e + |e, g .
2. apply the write laser with a π-pulse 3. free evolution of the OM systems for a time T 4. apply the write laser with a π-pulse 5. readout the atomic state in each cavity.
At the end of Step 2, the state of the sources and the cavities is |ψ 2 = |e, e ⊗ (|1, 0 + |0, 1 ) where |n, m = |n ⊗ |m with each factor being a photon number eigenstate.

Gravitational decoherence
We will use Diosi's theory of gravitational decoherence [50]. This is equivalent to the decoherence model introduced in Kafri et al. [51]. One mirror of the opto-mechanical cavity is free to move in a harmonic potential with frequency ω m . The master equation for a massive particle moving in a harmonic potential, including gravitational decoherence is withx,p the usual canonical position and momentum operators. The gravitational decoherence rate Λ grav is given by with G the Newton gravitational constant and ∆ the density of the mechanical element. As one might expect Λ grav is quite small, of the order of 10 −8 s −1 for suspended mirrors ( as in LIGO) with ω m ∼ 1. Form a phenomenological perspective the effect of gravitational decoherence is analogous to a Browning heating effect. To see this we note that the average vibrational quantum number increases diffusively