Introduction

Many-body quantum devices promise exciting applications in ultraprecise quantum metrology1, quantum computing2,3,4 and quantum simulators5,6,7,8,9. In the quest for their large-scale realization, impressive progress on a variety of quantum technologies has recently been made6,7,8,9. Among these technologies, optical platforms play a key role. For example, sophisticated manipulations of multi-qubit entangled states of up to eight parametrically downconverted photons10,11 have been demonstrated and continuous-variable entanglement among 60 stable12 and up to 10,000 flying13 modes has been verified in optical set-ups. In addition, small-sized simulations of BosonSampling14,15,16,17 and Anderson localization in quantum walks18,19 have been performed with on-chip integrated linear-optical networks.

This fast pace of advance, however, makes the problem of reliable certification an increasingly pressing issue20,21,22,23,24. From a practical viewpoint, further experimental progress on many-body quantum technologies is nowadays hindered by the lack of practical certification tools. At a fundamental level, certifying many-body quantum devices is ultimately about testing quantum mechanics in regimes where it has never been tested before.

Tomographic characterization of quantum states requires the measurement of exponentially many observables. Compressed-sensing techniques25 reduce, for states approximated by low-rank density matrices, the requirements significantly, but still demand exponentially many measurements. Efficient certification techniques, requiring only polynomially many measurements, for universal quantum computation26,27,28 and a restricted model of computation with one pure qubit29 exist in the form of quantum interactive proofs. However, these require either a fully fledged fault-tolerant universal quantum computer26,27,28 or an experimentally non-trivial measurement-based quantum device29. In addition, these methods involve sequential interaction rounds with the device26,27,28,29. In contrast, permutationally invariant tomography30, tensor network techniques31, Monte Carlo fidelity estimation32,33,34, and Clifford-circuit benchmarking techniques35 provide experimentally friendly alternatives for the efficient certification of preparations of permutationally invariant30 and qubit stabilizer or W states32,33,34,35, respectively. Nevertheless, none of these methods addresses continuous-variable systems, not even in Gaussian states.

Here we introduce an experimentally friendly technique for the certification of continuous-variable state preparations without estimating the prepared state itself. First, we discuss intuitively and define precisely reliable quantum-state certification tests. We do this for two notions of certification, differing in that in one of them robustness against small preparation errors is mandatory. Then, we present a certification test, based on single-mode homodyne and heterodyne detection, for arbitrary m-mode pure Gaussian states, pure non-Gaussian states resulting from passive Gaussian unitary operations on Fock-basis states with n photons, and pure states prepared by post-selecting states in the latter class with Fock-basis measurements on a<m ancillary modes. This covers, for instance, Gaussian quantum simulations such as those of refs 12, 13 as well as the non-Gaussian ones of refs. 6, 10, 11, 14, 15, 16, 17, 18, 19. Furthermore, both photon-added or -subtracted linear-optical network states36,37,38,39 as well as all non-Gaussian states accessible to qumode-encoded qubit40,41 quantum computers also lie within the range of applicability of our method. For all Gaussian states and all mentioned non-Gaussian states with constant n, the protocol is efficient in m and, for the cases with post-selection, also in the inverse post-selection success probability.

With high probability, our test rejects all experimental preparations with a fidelity with respect to the chosen target state lower than a desired threshold and accepts if the preparation is sufficiently close to the target. That is, the protocol is robust against small preparation errors. We upper-bound the failure probability in terms of the number of experimental runs and calculate the necessary number of measurement settings. Our method is built on a fidelity lower bound, based on a natural extremality property, that is interesting in its own right. Finally, the experimental estimation of this bound relies on non-Gaussian state nullifiers, which we introduce on the way.

Results

Certification notions

We present our results in terms of photons propagating through optical networks, but our methods apply to any bosonic platform with equivalent dynamics. We consider a sceptic certifier, Arthur, with limited quantum capabilities, who wishes to ascertain whether an untrusted quantum prover, Merlin, presumably with more quantum capabilities, can indeed prepare certain quantum states that Arthur cannot. This mindset is reminiscent to that of quantum interactive-proof systems26,27,28,29 of computer science, but our method has the advantage that no interaction apart from the measurements of the certifier on the single-run experimental preparations from the prover is required.

In particular, we consider the situation where Merlin possesses at least a network of active single-mode squeezers and displacers as well as passive beam-splitters and phase-shifters, sufficient to efficiently implement any m-mode Gaussian unitary42,43,44,45,46, plus single-photon sources. Arthur’s resources, in contrast, are restricted to classical computational power augmented with single-mode measurements. With that, he can characterize each of his single-mode measurement channels up to any desired constant precision. The task is for Merlin to provide Arthur with copies of an m-mode pure target state of Arthur’s choice. We assume that Merlin follows independent and identical state-preparation procedures on each experimental run, described by the density matrix . We refer to as a preparation of the target state . His preparation is unavoidably subject to imperfections and he might even be dishonest and try to trick Arthur. Thus, Arthur would like to run a test, with his own measurement devices, to certify whether is indeed a bona fide preparation of .

To measure how good a preparation of is, we use the fidelity between and , which we define as

where the last equality holds because is assumed to be pure. Another usual definition of the fidelity corresponds to the square root of the fidelity as defined above. All our results can be adapted to that definition and also to the trace distance D:=D(, ), which can be defined via the 1-norm distance in state space as D(,):=Tr[|−|]/2. Note that D can be bounded from both sides in terms of F, as defined in equation (1), through the well-known inequalities , where the first inequality holds because is pure.

Let us first discuss what properties an experimental test must fulfil to qualify as a state certification protocol. Different certification paradigms are schematically represented in Fig. 1. We start with the formal definition of certification in the sense of Fig. 1c.

Figure 1: Different certification paradigms.
figure 1

(a) Naive approach: To certify an untrusted experimental preparation of the target state , a certifier Arthur would like to run a statistical test that, for all , decides whether the fidelity F between and is greater or equal than a prespecified threshold FT<1 (inner green region, accept), or smaller than it (outer red region, reject). However, due to the preparations at the boundary of the two regions and experimental uncertainties, a test able to make such a decision does not exist. (b) The ideal scenario: A more realistic certification notion is to ask that the test rejects every for which F<FT (outer red region) and accepts every for which F⩾FT+Δ (inner green region), for some given Δ<1−FT. Here a buffer region of width Δ (in grey) is introduced within which the behaviour of the test can be arbitrary, but, in return, the certification is now feasible. This type of certification is thus robust against experimental infidelities as large as 1−FT−Δ. (c) The practical scenario: Finally, the least one can demand is that the test rejects every for which F<FT (outer red region) and accepts at least (green point). The former condition is sometimes called soundness and the latter one completeness. Here no acceptance is guaranteed for any with F⩾FT (grey region) other than itself, but any accepted by the test necessarily features F⩾FT. This certification notion is not necessarily robust against state deviations, but it can be more practical. In addition, in practice, the resulting tests succeed also in accepting many for which F⩾FT.

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Definition 1 (Quantum-state certification). Let be a target state, FT<1 a threshold fidelity, and α>0 a maximal failure probability. A test, which takes as input copies of a preparation and outputs ‘accept’ or ‘reject’, is a certification test for if, with probability at least 1−α, it both rejects every for which F(,) <FT and accepts if =. We say that any accepted by such a test is a certified preparation of .

Classes of target states

To specify the target states we need to introduce some notation. We denote m-mode Fock basis states by , with n:=(n1, n2,…,nm) being the sequence of photon numbers nj⩾0 in each mode j∈[m], where the short-hand notation [m]:={1, 2,…,m} is introduced, and call the total input photon number. In particular, we will pay special attention to Fock basis states with exactly one photon in each of the first n modes and the vacuum in the remaining m−n ones, that is, those for which n=1n, with

Note that is the Gaussian vacuum state . We denote the photon number operator corresponding to mode j by and the total photon number operator by .

In addition, for post-selected target states, we denote by , where each element labels a different mode, the subset of modes on which the post-selection measurements are made. We then identify the remaining m−a modes as the system subset , which carries the post-selected target state . The subindex emphasizes that represents an (m−a)-mode post-selected target state and distinguishes it from m-mode target states without post-selection, which we denote simply as . We denote by , with , an a-mode pure normalized Fock-basis state of total photons on the modes . We use the short-hand notations , where indicates partial trace over the Fock space of , denotes the identity on , and is the post-selection success probability, that is, the probability of measuring in a projective measurement on . Without loss of generality, we consider throughout only the non-trivial case .

With the notation introduced, we derive our results for: arbitrary m-mode pure Gaussian states, given by the class

m-mode pure linear-optical network states from the class

and (m−a)-mode pure locally post-selected linear-optical network states from the class

The three classes of target states are schematically represented in Fig. 2. The class is crucial within the realm of ‘continuous-variable’ quantum optics and quantum information processing. It encompasses, for instance, ‘twin-beam’ (two-mode squeezed vacuum) states under passive networks, which are used to simulate, upon coincidence detection, multi-qubit states6. The class includes all the settings sometimes referred to as ‘discrete variable’ linear-optical networks. This class covers, among others, the targets of several recent experimental simulations with on-chip integrated linear-optical networks14,15,16,17,18,19. The third class, , is the one of linear-optical network states locally post-selected with Fock-basis measurements. This class includes important non-Gaussian resources for quantum information and quantum optics. For instance, it encompasses both photon-added or -subtracted linear-optical network states36,37,38,39. Furthermore, when n is proportional to m, it also includes all the states prepared by probabilistic schemes of the type of refs 40, 41 for universal qumode-encoded qubit quantum computation.

Figure 2: Classes of target states.
figure 2

(a) is the class composed of all m-mode pure Gaussian states. These can be prepared by applying an arbitrary Gaussian unitary Û (possibly involving multimode squeezing) to the m-mode vacuum state . (b) The class includes all m-mode pure non-Gaussian states produced at the output of an arbitrary linear-optical network, which implements a passive Gaussian unitary Û (without squeezing), with the Fock-basis state containing one photon in each of the first n modes and zero in the remaining m−n ones as input. As the order of the modes is arbitrary, choosing the first n modes as the populated ones does not constitute a restriction. (c) The third class, encompasses all (m−a)-mode pure non-Gaussian states obtained by projecting a subset of a<m modes of an m-mode pure linear-optical network state onto a pure normalized product Fock-basis state . In practice, this is done probabilistically by measuring in a local basis that contains and post-selecting only the events in which is measured. Thus, the a modes in are used as ancillas, whereas the effective system is given by the subset containing the other m−a modes, which carries the final target state. For concreteness, but without any loss of generality, in the plot, the ancillary modes are chosen to be the last a ones. These three classes cover the target states considered in the vast majority of quantum photonic experiments.

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The certification test

The basis of the our certification scheme is a technique for the estimation of the quantity

with n the total input photon number. As shown in the Methods section, for all target states , F(n) is a lower bound on the fidelity F and, moreover, F(n)=F=1 if =. In addition, this bound is connected to a natural extremality property of Gaussian states, discussed also in the Methods section. Our test , summarized in Box 1, yields an estimate F(n)* of F(n). If F(n)* is sufficiently above the threshold FT, the preparation is accepted. Otherwise it is rejected. We introduce the measurement schemes and , which depend on the specific target state, to obtain the estimate F(n)*. Gaussian states can be estimated with the scheme , while linear-optical network states with . Both measurement schemes are summarized in the Methods section and described in detail in Supplementary Note 1. In turn, a fidelity bound for post-selected target states in similar to F(n) is presented in the Methods section. Its derivation, the adaptation of the test to post-selected targets, and the corresponding measurement scheme are provided in Supplementary Note 2.

Our theorems guarantee that the test from Box 1 is indeed a certification test and give a bound on the scaling of the number of samples that are needed for the test. To state them, we introduce some notation related to mode space descriptions of linear-optical networks first. Any Gaussian unitary transformation on Hilbert space can be represented by

an affine symplectic transformation in mode space, that is, by a symplectic matrix followed by a phase-space displacement (see equation (25) in the Methods section), where the real-symplectic group contains all real 2m × 2m matrices that preserve the canonical phase-space commutation relations42,43. By virtue of the Euler decomposition42,45, S can be implemented with single-mode squeezing operations and passive mode transformations. We denote the maximum single-mode squeezing of S by smax and define the mode range d≤m to be the maximal number of input modes to which each output mode is coupled (for details see Supplementary Note 1). Also, it will be useful to define

The displacement x can be implemented by a single-mode displacer at each mode j∈[m], with amplitude (x2j−1, x2j), where xk, for k∈[2m], is the kth component of x. The vector 2-norm is denoted by , that is, .

We take σi to be a uniform upper bound on the variances of any product of i phase-space quadratures in the state . If is Gaussian, then σ1 and σ2 are functions of the single-mode squeezing parameters of . In addition, we call the maximal ith variance of . Finally, we use the Landau symbol O to denote asymptotic upper bounds.

Theorem 2 (Quantum certification of Gaussian states). Let FT<1 be a threshold fidelity, α>0 a maximal failure probability, and 0<ɛ≤(1−FT)/2 an estimation error. Let have maximum single-mode squeezing smax⩾1, mode range d≤m, and displacement x. Test from Box 1 is a certification test for and requires at most

copies of a preparation with first and second variance bounds σ1>0 and σ2>0, respectively.

Theorem 3 (Quantum certification of linear-optical network states). Let FT<1 be a threshold fidelity, α>0 a maximal failure probability, and 0<ɛ ≤(1−FT)/2 an estimation error. Let have mode range d ≤m. Test from Box 1 is a certification test for and requires at most

copies of a preparation with maximal 2(n+1)-th variance σ≤2(n+1), where λ>0 is an absolute constant.

The proofs of all our theorems are provided in the Supplementary Information. The treatment of the class follows as a corollary of Theorem 3 and is also provided in Supplementary Note 2. Equations (8) and (9) are highly simplified upper bounds on the total number of copies of that requires. For more precise expressions see Supplementary Lemmas 6 and 9. Note that neither of the two theorems requires any energy cut-off or phase-space truncation. While the bound in equation (9) is inefficient in n, both for the Gaussian and linear-optical cases, the number of copies of scales polynomially with all other parameters, in particular with m. Thus, arbitrary m-mode target states from the classes and with constant n, are certified by efficiently.

Interestingly, since states in in general display negative Wigner functions, sampling from their measurement probability distributions cannot be efficiently done by the available classical sampling methods47,48,49. Furthermore, for Fock-state measurements, these distributions define BosonSampling, for which hardness results exist50 for m asymptotically lower bounded by n5.

Also, note that there are no restrictions on except that, in practice, to apply the theorems, one needs bounds on σ1, σ2, and σ≤2(n+1). These variances are properties of and are therefore a priori unknown to Arthur. However, he can reasonably estimate them from his measurements. Note that, for random variables that can take any real value, assuming that the variances are bounded is a fundamental and unavoidable assumption to make estimations from samples; and it is one that can be contrasted with the measurement results.

Robustness against preparation imperfections

To end up with, we consider certification in the sense of Fig. 1b:

Definition 4 (Robust quantum-state certification). Let be a target state, FT<1 be a threshold fidelity, α>0 a maximal failure probability, and Δ<1−FT a fidelity gap. A test, which takes as input copies of a preparation and outputs ‘accept’ or ‘reject’, is a robust certification test for if, with probability at least 1−α, it both rejects every for which F(, )<FT and accepts every for which F(, )⩾FT+Δ. We say that any accepted by such a test is a certified preparation of .

This definition is more stringent than Definition 1 in that it guarantees that preparations sufficiently close to are necessarily accepted, rendering the certification robust against preparation imperfections causing fidelity deviations as large as 1−(FT+Δ). We now show that our test from Box 1 is actually a robust certification test.

To this end, we first write as

where is an operator orthogonal to with respect to the Hilbert–Schmidt inner product, that is, such that . As is assumed to be pure, it follows immediately that is actually a state. In fact, multiplying both sides of equation (10) by and taking the trace, one readily sees that decomposition in equation (10) is just another way to express the fidelity in equation (1).

According to equation (6), the lower bound F(n) can be defined as an expectation value of the observable

with respect to . In a similar way, we define the quantity

By taking the expectation value of equation (10) with respect to the observable and using that and F(n)≤F, one finds that . The parameter turns out to quantify the robustness of our certification test.

Theorem 5 (Robust quantum certification). Under the same conditions as in Theorems 2 and 3, test from Box 1 is a robust certification test with fidelity gap

Since and FT<1, it is clear that Δ>0. On the other hand, note that can in general be arbitrarily smaller than zero. This happens, for instance, for preparations for which , with n1, n2,…,nn⩾1 and n arbitrarily large. In particular, in the limit , it holds that Δ→1−FT, so that the certification becomes less robust with decreasing , as one would expect. In contrast, as increases from −∞ to 0, the gap decreases to its minimal value Δ=2ɛ. Note that, since it depends on , cannot be directly estimated from measurements on alone. However, Theorem 5 guarantees the existence of an entire closed convex set of states around that are rightfully accepted and Δ lower bounds the size of that region. Furthermore, in experimentally relevant situations, is expected to be small, meaning that Δ is close to its optimal value 2ɛ.

Finally, a statement equivalent to Theorem 5 for target states follows as an immediate corollary of it and is Supplementary Note 2.

Discussion

Large-scale photonic quantum technologies promise important scientific advances and technological applications. So far, considerably more effort has been put into their realization than into the verification of their correct functioning and reliability. This imposes a serious obstacle for further experimental advance, specifically in the light of the speed at which progress towards many-mode architectures takes place. Here we have presented a practical reliable certification tool for a broad family of multimode bosonic quantum technologies.

We have proven theorems that upper bound the number of experimental runs sufficient for our protocol to be a certification test. Our theorems provide large-deviation bounds from a simple extremality-based fidelity lower bound that is interesting in its own right. Our theorems hold only for statistical errors, but the stability analysis on which they rely (see Supplementary Lemmas 5 and 8) holds regardless of the nature of the errors. In Supplementary Note 5, we show that our fidelity estimates are robust also against small systematic errors.

From a more practical viewpoint, our test allows one to certify the state preparations of most current optical experiments, in both the ‘continuous-variable’ and the ‘discrete-variable’ settings. This is achieved under the minimal possible assumptions: namely, only that the variances of the measurement outcomes are finite. Thus, the certification is as unconditional as the fundamental laws of statistics allow. In particular, no assumption on the type of noise is made. Despite the rigorous bounds on the estimation errors and failure probabilities, our methods are both experimentally friendly and resource efficient.

Notably, our test can efficiently certify multimode negative-Wigner-function states that define, via local measurements, sampling problems whose classical simulation is not known to be efficient47,48,49. For instance, it can be applied to the certification of optical circuits of the type used in BosonSampling: There, m-mode Fock-basis states of n photons are subjected to a linear-optical network described by a random unitary Û drawn from the Haar measure50 and, subsequently, each output mode is measured in the Fock basis. While the question of the certification of the classical outcomes of such samplers without assumptions on the device is still largely open20,21, with the methods described here the premeasurement non-Gaussian quantum outputs of BosonSampling devices14,15,16,17 can be certified reliably and, for constant n, even efficiently. In this sense, this work goes significantly beyond previously proposed schemes to rule out particular cheating strategies by the prover21,22,23,24. Furthermore, a variety of non-Gaussian states paradigmatic in quantum optics and quantum information are also covered by our protocol (see Supplementary Note 2 for details). These include, for instance, linear-optical network outputs post-selected though photon-number measurements, ranging from both photon-added or -subtracted linear-optical network states36,37,38,39 to all the states preparable with Knill–Laflamme–Milburn-like schemes40,41. For all such states, our test is efficient in the inverse post-selection success probability .

The present method constitutes a step forward in the field of photonic quantum certification, with potential implications on the certification of other many-body quantum-information technologies. Apart from that of BosonSamplers and optical schemes with post-selection, the efficient and reliable certification of large-scale photonic networks as those used, for instance, for multimode Gaussian quantum-information processing12,13, non-Gaussian Anderson-localization simulations18,19, and quantum metrology1, with a constant number of input photons, is now within reach.

Methods

Fidelity lower bound

Here we formalize the extremality notion and derive a lower bound on the fidelity F for non post-selected target states. All the non post-selected target states we consider are of the form

where is an arbitrary Gaussian unitary and an arbitrary Fock-basis state. First, we derive a fidelity lower bound for general states of the form given in equation (14) and then consider the linear-optical and Gaussian cases separately. Lower bounds for the post-selected target states are provided further below in the Measurement Scheme.

We start recalling that

where is the creation operator of the jth mode. Its Hermitian conjugated is the corresponding annihilation operator. These operators satisfy , where δj,j′ denotes the Kronecker delta of j and j′, and , for all j, j′∈[m]. The fidelity in equation (1) can be written as , where is the Heisenberg representation of with respect to . With this, equation (15), and the cyclicality property of the trace, we obtain that

where

is a (not necessarily normalized) positive-semidefinite operator.

To lower bound , we consider the expectation value . We write for the identity operator. From the facts and , it follows that

and hence,

For = it holds that and . Thus, for = the inequality in equation (18) becomes an equality and, therefore, the bound in equation (19) is then saturated, as announced earlier.

Next, we define the operator valued Pochhammer-symbol

for any integer t⩾0, and p−1(x):=1. In Supplementary Note 6 we show that

and

Inserting equation (17) into equation (19), using the cyclicity property of the trace, grouping the operators of each mode together, using equation (21a) and equation (21b), and that , we obtain the general fidelity lower bound

where n! :=n1!n2!…nm!.

In order to specialize to the linear-optical case , we take n=1n, i.e., nj=1 for all j∈[n] and nj=0 otherwise. With this, F(n) in equation (22) simplifies to the bound F(n) in equation (6). Finally, to restrict it to the Gaussian case , we take nj=0 for all j∈[m]. This yields the particularly simple expression

The last expression manifests the above-mentioned connection between our fidelity lower bound and an intuitive extremality property of Gaussian states. Namely, the lower the average number of photons of is, the closer to the vacuum it must be and, therefore, the closer to the target state .

Arthur does not have, in general, enough quantum capabilities to directly estimate by undoing the operation Û on Merlin’s outputs and then measuring in the Fock-state basis. However, we show in the next section that he can efficiently obtain , as well as the expectation values in equations (22) and (6), from the results of single-mode homodyne or heterodyne measurements.

Measurement scheme

First, we introduce some notation. By and we denote, respectively, the conjugated position and momentum phase-space quadrature operators of the jth mode in the canonical convention42,43, that is, with the commutation relations . The particle number operator of the jth mode can be written in terms of the phase-space quadratures as . In addition, it will be convenient to group all quadrature operators into a 2m-component column vector , with elements

As already mentioned, the action of Û on mode space is given by a symplectic matrix and a displacement vector . More precisely, under a Gaussian unitary Û, transforms according to the affine linear map42

Equivalently, the right-hand side of this equation defines the Heisenberg representation of with respect to Û. In addition, it will be useful to denote the Heisenberg representation of with respect to Û† by . Thanks to equation (25), we can write in terms of the symplectic matrix S and displacement vector x that define Û, as

The symbols and will represent, respectively, the scalar products of and with themselves. Also, we will use the same notation for the Heisenberg representations of each quadrature operator with respect to Û†, that is, and .

Next, for β∈{0, n, n}, we express our fidelity bounds in the general form

where is an observable decomposed explicitly in terms of the local observables to which Arthur has access. We start with the Gaussian case . To express the bound of equation (23) as in equation (27), we first write the total photon-number operator as

This, in combination with equation (23), yields

Note that, due to equation (26), each component of is a linear combination of at most 2m components of . This implies that Arthur can obtain by measuring at most 2m single-quadrature expectation values of the form and 4m2 second moments of the form . He can then classically efficiently combine them as dictated by S and x in equation (26). In Supplementary Note 1, we give the details of this measurement procedure, which we call , and show that measuring mκ second moments, instead of 4m2, is actually enough. Furthermore, in Supplementary Note 4, we show that only m+3 experimental settings suffice if homodyne detection is used and a single setting if heterodyne detection is used.

Now, proceeding in a similar manner with the generic bound of equation (22), we obtain

Note that the observable in equation (29) is contained as the special case n=0. For target states in the class , Û is assumed to be a passive Gaussian unitary. Such unitaries preserve the area in phase-space, that is, if it holds that (for details, see Supplementary Note 1). Hence, using this and specialising to the case n=1n, equation (30) simplifies to

Again by virtue of equation (26), Arthur can now obtain the expectation values of the observables in equations (30) and (31) by measuring 2jth moments of the form and then classically recombining them, which—for constant n—he can do efficiently. In Supplementary Note 1, we give the details of the measurement procedure to obtain F(n), which we call . In particular, we show that, to obtain , estimating a total of O(m(4d2+1)n) 2jth moments, with j∈[n+1], is enough. Also, we list which moments are the relevant ones in terms of . Furthermore, in Supplementary Note 4, we show that a single heterodyne experimental setting throughout suffices here too.

Finally, in Supplementary Note 2, we derive a bound analogous to that of equations (27) and (31) for post-selected target states . More precisely, we show that the fidelity between and an arbitrary, unknown (m−a)-mode system preparation is lower bounded as

Actually, the bound holds not only for target states projected onto but also for the more general target states of equation (14), with Û any Gaussian unitary and any Fock-basis state, projected onto any generic a-mode pure product state on . Apart from being experimentally more relevant, linear-optical network target states post-selected with Fock-basis measurements possess the peculiarity that the corresponding bound is tight for perfect preparations. That is, for these states, if then , just as in the cases without post-selection.

Non-Gaussian state nullifiers

It is instructive to mention that the observables

for j∈[m], correspond to the so-called nullifiers of the Gaussian states in . The nullifiers are commuting operators that, despite originally introduced51 as a tool to define Gaussian graph states, can be tailored to define any pure Gaussian state52,53: If a state is the simultaneous null-eigenvalue eigenstate of all m nullifiers of a given pure Gaussian state, then the former is necessarily equal to the latter. The bound F(0), given by equations (27) and (29), exploits the fact that if a preparation gives a sufficiently low expectation value for the sum of all m nullifiers, then its fidelity with the target state must be high. A similar intuition has been previously exploited12,13 to experimentally check for multimode entanglement of ultralarge Gaussian cluster states. Here we cannot only certify entanglement but the quantum state itself.

Analogously, in the non-Gaussian case, from the derivation of equation (30) and the fact that

equals 1 for =, we can identify the observable

as the jth nullifier of the m-mode non-Gaussian state of equation (14). Indeed, all m observables given by equation (36) for all j∈[m] commute and have as their unique, simultaneous null-eigenvalue eigenstate. To end up with, due to the projection onto , the equivalent observables for do not in general commute. Nevertheless, their linear combination given by still defines an observable with as its unique null-eigenvalue eigenstate. These observables constitute, to our knowledge42,52,53, the first examples of nullifiers for non-Gaussian states.

Additional information

How to cite this article: Aolita, L. et al. Reliable quantum certification of photonic state preparations. Nat. Commun. 6:8498 doi: 10.1038/ncomms9498 (2015).