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# Revealing nonclassicality beyond Gaussian states via a single marginal distribution

Edited by Luiz Davidovich, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil, and approved December 13, 2016 (received for review October 25, 2016)

## Significance

Quantum states possess nonclassical properties inaccessible from classical physics, providing a profound basis of quantum physics and a crucial resource for quantum information technology. We propose a general framework to manifest nonclassicality via single marginal distributions, unlike quantum-state tomography using many marginal distributions, applicable to a broad range of quantum systems. Our approach provides a fundamentally unique insight showing how partial information on a quantum state can be sufficient to confirm nonclassicality and a practical efficiency, yielding conclusive evidence of nonclassicality by directly analyzing experimental data without numerical optimization. Remarkably, our method works regardless of measurement axis for all non-Gaussian states of finite dimension. We also experimentally demonstrate our framework, using motional states of a trapped ion.

## Abstract

A standard method to obtain information on a quantum state is to measure marginal distributions along many different axes in phase space, which forms a basis of quantum-state tomography. We theoretically propose and experimentally demonstrate a general framework to manifest nonclassicality by observing a single marginal distribution only, which provides a unique insight into nonclassicality and a practical applicability to various quantum systems. Our approach maps the 1D marginal distribution into a factorized 2D distribution by multiplying the measured distribution or the vacuum-state distribution along an orthogonal axis. The resulting fictitious Wigner function becomes unphysical only for a nonclassical state; thus the negativity of the corresponding density operator provides evidence of nonclassicality. Furthermore, the negativity measured this way yields a lower bound for entanglement potential—a measure of entanglement generated using a nonclassical state with a beam-splitter setting that is a prototypical model to produce continuous-variable (CV) entangled states. Our approach detects both Gaussian and non-Gaussian nonclassical states in a reliable and efficient manner. Remarkably, it works regardless of measurement axis for all non-Gaussian states in finite-dimensional Fock space of any size, also extending to infinite-dimensional states of experimental relevance for CV quantum informatics. We experimentally illustrate the power of our criterion for motional states of a trapped ion, confirming their nonclassicality in a measurement-axis–independent manner. We also address an extension of our approach combined with phase-shift operations, which leads to a stronger test of nonclassicality, that is, detection of genuine non-Gaussianity under a CV measurement.

Nonclassicality is a fundamentally profound concept to identify quantum phenomena inaccessible from classical physics. It also provides a practically useful resource, for example, entanglement, making possible a lot of applications in quantum information processing beyond classical counterparts (1⇓–3). A wide range of quantum systems, for example, field amplitudes of light, collective spins of atomic ensembles, motional modes of trapped ions, and Bose–Einstein condensate and mechanical oscillators, can be used for quantum information processing based on continuous variables (CVs) (2). It is of crucial importance to establish efficient and reliable criteria of nonclassicality for CV systems, desirably testable with fewer experimental resources, for example, fewer measurement settings (4⇓⇓⇓–8) and with the capability of detecting a broad class of nonclassical states. In this paper, in view of the Glauber–Sudarshan P function (9, 10), those states that cannot be represented as a convex mixture of coherent states are referred to as nonclassical.

A standard method to obtain information on a CV quantum state is to measure marginal distributions along many different axes in phase space constituting quantum-state tomography (11). This tomographic reconstruction may reveal nonclassicality to some extent, for example, negativity of Wigner function making only a subset of whole nonclassicality conditions. However, it typically suffers from a legitimacy problem; that is, the measured distributions do not yield a physical state when directly used due to finite data and finite binning size (11, 12). Much efforts was made to use estimation methods finding a most probable quantum state closest to the obtained data (13⇓⇓–16). There were also numerous studies to directly detect nonclassicality, for example, an increasingly large number of hierarchical conditions (4) requiring information on two or more marginal distributions or measurement of many higher-order moments (17⇓–19). An exception would be the case of Gaussian states, with its nonclassical squeezing demonstrated by the variance of distribution along a squeezed axis.

Here we theoretically propose and experimentally demonstrate a simple, powerful, method to directly manifest nonclassicality by observing a single marginal distribution applicable to a wide range of nonclassical states. Our approach makes use of a phase-space map that transforms the marginal distribution (obtained from measurement) to a factorized Wigner distribution by multiplying the same distribution or the vacuum-state distribution along an orthogonal axis. We refer to those mathematical procedures as demarginalization maps (DMs), because a one-dimensional marginal distribution is converted to a fictitious 2D Wigner function. The same method can be applied equally to the characteristic function as well as the Wigner function. We show that a classical state, that is, a mixture of coherent states, must yield a physical state under our DMs. That is, the unphysicality emerging under DMs is a clear signature of nonclassicality. Remarkably, for all non-Gaussian states in finite-dimensional space, our test works for an arbitrary single marginal distribution thus experimentally favorable. (For Gaussian states, our method, if directly applied, works only for the squeezed axes not covering the whole range of quadrature axis. As we show in *SI Appendix*, however, a phase randomization, which does not create nonclassicality, modifies a Gaussian state to a non-Gaussian state for which nonclassicality can be detected regardless of quadrature axis.) It also extends to non-Gaussian states in infinite dimension, particularly those without squeezing effect. We introduce a quantitative measure of nonclassicality using our DMs, which provides a lower bound of entanglement potential (20)—an entanglement measure under a beam-splitter setting versatile for CV entanglement generation (20⇓⇓–23). Along this way, our method makes a rigorous connection between single-mode nonclassicality and negative partial transpose (NPT) entanglement (24⇓⇓⇓⇓⇓⇓⇓⇓–33), which bears on entanglement distillation (34) and nonlocality (35⇓⇓⇓⇓⇓⇓–42).

As the measurement of a marginal distribution is highly efficient in various quantum systems, for example, homodyne detection in quantum optics, our proposed approach can provide a practically useful and reliable tool in a wide range of investigations for CV quantum physics. We here experimentally illustrate the power of our approach by manifesting nonclassicality of motional states in a trapped-ion system. Specifically, we confirm the nonclassicality regardless of measured quadrature axis by introducing a simple faithful test using only a subset of data points, not requiring data manipulation under numerical methods, unlike the case of state reconstruction. We also extend our approach combined with phase randomization to obtain a criterion on genuine non-Gaussianity.

## DMs and Nonclassicality Measure

### Nonclassicality Test via DMs.

We first introduce our main tools, that is, DMs,

Our DM methods proceed as follows. Given a state with its Wigner function

#### Nonclassicality criteria.

The constructed functions in Eqs. **1** and **2** are both in factorized forms, so judging their legitimacy is related to the problem of what quantum states can possess a factorized Wigner function. (Note also that a factorized Wigner function must be everywhere nonnegative as each term in it represents its marginal distribution so is nonnegative.) Every coherent state **4** also represent a certain mixture of coherent states and hence a physical state. Therefore, if an unphysical Wigner function emerges under our DMs, the input state must be nonclassical.

#### Gaussian states.

Let us first consider a Gaussian state *SI Appendix*). We can further make the test successful regardless of quadrature axis by introducing a random phase rotation on a Gaussian state (*SI Appendix*). Note that a mixture of phase rotations, which transforms a Gaussian to a non-Gaussian state, does not create nonclassicality, so the nonclassicality detected after phase rotations is attributed to that of the original state.

#### Non-Gaussian states.

More importantly, we now address non-Gaussian states. Every finite-dimensional state (FDS) in Fock basis, that is, *SI Appendix*, section S4. The essence of our proof is that there always exists a submatrix of the density operator corresponding to DMs, which is not positive definite. Remarkably, this nonpositivity emerges for a marginal distribution along an arbitrary direction, which means that the nonclassicality of FDS is confirmed regardless of the quadrature axis measured, just like the phase-randomized Gaussian states introduced in *SI Appendix*. This makes our DM test experimentally favorable, whereas the degree of negativity may well depend on the quadrature axis except in rotationally symmetric states. Our criteria can further be extended to non-Gaussian states in infinite dimension, particularly those without squeezing effect (*SI Appendix*).

As an illustration, we show the case of a FDS *A*. Our DM methods yield matrix elements as shown in Fig. 1*B* and *C*. The nonpositivity of the density operator is then demonstrated by, for example,

### Nonclassicality Measure and Entanglement Potential.

We may define a measure of nonclassicality using our DMs as*SI Appendix*: (*i*) *ii*) convex, that is, nonincreasing via mixing states, *iii*) invariant under a classicality-preserving unitary operation, *ii* and *iii*, we also deduce the property that (*iv*)

Our nonclassicality measure also makes a significant connection to entanglement potential as follows. A prototypical scheme to generate a CV entangled state is to inject a single-mode nonclassical state into a beam splitter (BS) (20⇓⇓–23). It is important to know the property of those entangled states under partial transposition (PT), which bears on the distillibility of the output to achieve higher entanglement. Our formalism makes a connection between nonclassicality of single-mode resources and NPT of output entangled states. The effect of PT in phase space is to change the sign of momentum,

We inject a single-mode state

Applying PT on mode 2 and injecting the state again into a 50:50 BS, we have

Integrating over **1**. The other DM2 in Eq. **2** emerges when replacing the second input state

In ref. 20, single-mode nonclassicality is characterized by entanglement potential via a BS setting, where a vacuum is used as an ancillary input to BS to generate entanglement. We may take negativity, instead of logarithmic negativity in ref. 20, as a measure of entanglement potential; that is,*SI Appendix* that our DM2 measure provides a lower bound for the entanglement potential as

Thus, the nonclassicality measured under our framework indicates the degree of entanglement achievable via BS setting.

## Experiment

We experimentally illustrate the power of our approach by detecting nonclassicality of several motional states of a trapped

For our test, we generate the Fock states

### Nonclassicality Test.

We measure a characteristic function

To set a benchmark (noise level) for classical states, we prepared the motional ground state *SI Appendix*). On the other hand, the Fock states *A*, at much higher negativity with error bars considering finite data of 1,000. To further show that our method works regardless of measured axis, we also tested a superposition state *B*, its nonclassicality is well demonstrated for all measured angles individually whereas the degree of negativity varies with the measured axis.

Compared with our DM, one might look into nonclassicality directly via deconvolution, that is, examine whether a marginal distribution *C* and *D* shows the results under deconvolution, using the same experimental data as in Fig. 2 *A* and *B*. To confirm nonclassicality, the degree of negativity must be large enough to beat that of the vacuum state, including the statistical errors. Although those states produce negativity under deconvolution, their statistical errors substantially overlap with that of the vacuum state, providing much weaker evidence of nonclassicality than our DM. Full details are given in *SI Appendix*.

Instead of using an entire characteristic function, we can also test our criterion by examining a subset of data using the Kastler–Loupias–Miracle–Sole (KLM) condition (48⇓⇓–51). This simple test provides clear evidence of nonclassicality against experimental imperfections, for example, coarse graining and finite-data acquisition in other experimental platforms as well. The KLM condition states that the characteristic function *SI Appendix*). As shown in Fig. 3*A*, the ground state *B*–*D*, respectively. Furthermore, note that a mixture of the vacuum and the nonclassical state, *B*–*D*. For Fock states, we consider the matrix test using *E* and *F*.

### Genuine Non-Gaussianity.

We further extend our approach combined with phase randomization to derive a criterion on genuine non-Gaussianity. Notably there exist quantum tasks that cannot be achieved by Gaussian resources, for example, universal quantum computation (52), CV nonlocality test (53, 54), entanglement distillation (55⇓–57), and error correction (58). It is a topic of growing interest to detect genuine non-Gaussianity that cannot be addressed by a mixture of Gaussian states. Previous approaches all address particle nature like the photon-number distribution (59⇓–61) and the number parity in phase space (7, 62, 63) for this purpose. Here we propose a method to examine genuine CV characteristics of marginal distributions. Our criterion can be particularly useful to test a class of non-Gaussian states diagonal in the Fock basis,

For a Gaussian state *SI Appendix*). Thus, if a state manifests a larger DM negativity as

## Conclusion and Remarks

Measuring marginal distributions along different axes in phase space forms a basis of quantum-state tomography with a wide range of applications. A marginal distribution is readily obtained in many different experimental platforms, for example, by an efficient homodyne detection in quantum optical systems (11, 68⇓⇓–71) and by other quadrature measurements in trapped-ion (45, 46, 72), atomic ensembles (73), optomechanics (74, 75), and circuit quantum electrodynamics (QED) systems (76, 77). We here demonstrated that only a single marginal distribution can manifest nonclassicality by using our DMs. Our DM methods are powerful to detect a wide range of nonclassical states, particularly non-Gaussian states. They provide a practical merit with less experimental effort and make a stronger test of nonclassicality by analyzing data without numerical manipulation unlike state tomography.

Remarkably, nonclassicality can be demonstrated regardless of measured quadrature axis for all FDSs, which was also experimentally confirmed using a trapped-ion system. We clearly showed that the proposed method provides a reliable nonclassicality test by directly using a finite number of data, which can be further extended to other CV systems. In addition to the KLM test used here, we can manifest nonclassicality by looking into single marginal distributions under other forms, for example, functional (33) and entropic (78, 79) inequalities. We also extended our approach to introduce a criterion on genuine non-Gaussianity, using marginal distributions combined with a phase-randomization process. Our nonclassicality and non-Gaussianity tests were experimentally shown to successfully detect non-Gaussian states even with positive-definite Wigner functions whose nonclassicality is thus not immediately evident by the tomographic construction of Wigner function. As a remark for those nonclassical states with positive Wigner functions, one may use generalized quasi-probability distributions like a filtered P function (80⇓–82). For example, the experiment in ref. 83 introduced a nonclassicality filter to construct a generalized P function that yields a regularized distribution with negativity as a signature of nonclassicality for the case of photon-added thermal states. On the other hand, our DM method does not require a tomographic construction and provides a faithful test that is reliable against experimental imperfections like finite data and coarse graining.

Moreover, we established the connection between single-mode nonclassicality and NPT entanglement via a BS setting—a prototypical model of producing CV entanglement. The negativity under our DM framework provides a quantitative measure of a useful resource by identifying the minimum level of entanglement achievable in Eq. **10** (as shown in *SI Appendix*, the relation in Eq. **10** holds regardless of the measured axis). Nonclassicality and non-Gaussianity are important resources, making a lot of quantum tasks possible far beyond their classical counterparts. We thus hope our proposed method could provide a valuable experimental tool and a fundamental insight for future studies of CV quantum physics by critically addressing them.

## Acknowledgments

M.S.Z. and H.N. were supported by National Priorities Research Program Grant 8-751-1-157 from the Qatar National Research Fund and K.K. by the National Key Research and Development Program of China under Grants 2016YFA0301900 and 2016YFA0301901 and the National Natural Science Foundation of China under Grants 11374178 and 11574002.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: hyunchul.nha{at}qatar.tamu.edu.

Author contributions: J.P. and H.N. designed research; J.P., J.L., M.S.Z., and H.N. developed theory; Y.L., Y.S., K.Z., S.Z., and K.K. performed the experiment; J.P. and H.N. analyzed experimental data; J.P., K.K., and H.N. wrote the paper; and H.N. supervised the project.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1617621114/-/DCSupplemental.

Freely available online through the PNAS open access option.

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