Revealing nonclassicality beyond Gaussian states via a single marginal distribution

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,

$${\mathcal{D}}_1:W_{\rho }(q,p)\mapsto M_{\rho }(x)M_{\rho }(y)\equiv W_{\rho }^{\text{DM1}}(x,y),$$ [1]

$${\mathcal{D}}_2:W_{\rho }(q,p)\mapsto M_{\rho }(x)M_{|0⟩⟨0|}(y)\equiv W_{\rho }^{\text{DM2}}(x,y),$$ [2]

where ${(x,y)}^T=\mathcal{R}(\theta ){(q,p)}^T$ is a pair of orthogonal quadratures rotated from position $q$ and momentum $p$ with $\mathcal{R}(\theta )=\left(\begin{array}{cc}\hfill \cos \theta \hfill & \hfill \sin \theta \hfill \\ \hfill -\sin \theta \hfill & \hfill \cos \theta \hfill \end{array}\right)$. $M_{\rho }(x)=\int dy{W}_{\rho }(x,y)$ is a marginal distribution of the Wigner function $W_{\rho }(q,p)=\frac{2}{\pi }\mathrm{tr}[\rho \widehat{D}(\alpha ){(-1)}^{\widehat{n}}{\widehat{D}}^{†}(\alpha )]$, where $\widehat{D}(\alpha )=e^{\alpha {\widehat{a}}^{†}-{\alpha }^{\ast }\widehat{a}}$ is a displacement operator with $\alpha =q+ip$ (43, 44).

Our DM methods proceed as follows. Given a state with its Wigner function $W_{\rho }(q,p)$, we measure a marginal distribution $M_{\rho }(x)$ along a certain axis, $x=q\cos \theta +p\sin \theta$. We then construct a fictitious, factorized, Wigner function $W_{\rho }^{\mathrm{DM}}(x,y)$ either by replicating the obtained distribution as $M_{\rho }(x)M_{\rho }(y)$ (DM1) or by multiplying the marginal distribution of a vacuum state as $M_{\rho }(x)M_{|0⟩⟨0|}(y)$ (DM2), with $M_{|0⟩⟨0|}(y)=\sqrt{\frac{2}{\pi }}{e}^{-2y^2}$ (Fig. 1). We test whether $W_{\rho }^{\mathrm{DM}}(x,y)$ is a legitimate Wigner function to represent a physical state.

Fig. 1.

DMs. (A) original Wigner function $W_{\rho }(x\text{,}y)$ of $|\mathrm{\Psi }⟩=\frac{1}{\sqrt{2}}\left(|0⟩+|2⟩\right)$, with its marginal $M_{\rho }(x)=\int dy{W}_{\rho }(x\text{,}y)$ in the backdrop. (B) A fictitious Wigner function $W^{\mathrm{DM1}}(x\text{,}y)\equiv M_{\rho }(x)M_{\rho }(y)$, with the same distribution $M_{\rho }(y)$ replicated along the orthogonal axis (red solid curve). (C) $W^{\mathrm{DM2}}(x\text{,}y)\equiv M_{\rho }(x)M_{|0⟩⟨0|}(y)$, with the vacuum-state distribution $M_{|0⟩⟨0|}(y)$ used (red solid curve). A–C, Bottom show the corresponding density matrix elements. $W^{\mathrm{DM1}}(x\text{,}y)$ and $W^{\mathrm{DM2}}(x\text{,}y)$ in B and C do not represent any physical states, confirming the nonclassicality of $|\mathrm{\Psi }⟩$.

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 $|\beta ⟩$ has a factorized Wigner function against all pairs of orthogonal quadratures, $W_{|\beta ⟩⟨\beta |}(x,y)=\frac{2}{\pi }{e}^{-2{(x-{\beta }_x)}^2}{e}^{-2{(y-{\beta }_y)}^2}$ (7). Owing to this factorizability, the maps ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ transform a classical state into another classical one. A mixture of coherent states has a Wigner function

$$W_{{\rho }_{\mathrm{cl}}}(x,y)=\int d^2\beta P({\beta }_x,{\beta }_y)W_{|\beta ⟩⟨\beta |}(x,y),$$ [3]

with the probability density $P({\beta }_x,{\beta }_y)$ for a coherent state $|\beta ⟩$ ($\beta ={\beta }_x+i{\beta }_y$). Applying each DM leads to

$${\mathcal{D}}_j[W_{{\rho }_{\mathrm{cl}}}(q,p)]=\int d^2\beta Q_1({\beta }_x)Q_j({\beta }_y)W_{|\beta ⟩⟨\beta |}(x,y)$$ [4]

$(j=1,2)$, where $Q_1({\beta }_x)=\int d{\beta }_{y}P({\beta }_x,{\beta }_y)$ and $Q_2({\beta }_x)=\delta ({\beta }_x)$ are nonnegative. The resulting distributions in Eq. 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 $\sigma$ that has a squeezed quadrature $\widehat{x}$ with $V_x\equiv {\mathrm{\Delta }}^2\widehat{x}<\frac{1}{4}$. Taking the squeezed marginal $M_{\sigma }(x)=\frac{1}{\sqrt{2\pi V_x}}{e}^{-\frac{{(x-⟨\widehat{x}⟩)}^2}{2V_x}}$ yields

$${\mathcal{D}}_1[W_{\sigma }(q,p)]=\frac{1}{2\pi V_x}{e}^{-\frac{{(x-⟨\widehat{x}⟩)}^2}{2V_x}}{e}^{-\frac{{(y-⟨\widehat{x}⟩)}^2}{2V_x}},{\mathcal{D}}_2[W_{\sigma }(q,p)]=\frac{1}{\pi \sqrt{V_x}}{e}^{-\frac{{(x-⟨\widehat{x}⟩)}^2}{2V_x}}{e}^{-2y^2},$$ [5]

both of which violate the uncertainty relation $\mathrm{\Delta }\widehat{x}\mathrm{\Delta }\widehat{y}\ge \frac{1}{4}$. Thus, the squeezed state turns into an unphysical state under our DMs. This method, of course, succeeds only when the observed marginal distribution is along a squeezed axis that generally extends to a finite range of angles, if not the whole range of angles (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, $\rho ={\sum }_{j,k=0}^N{\rho }_{jk}|j⟩⟨k|$, is nonclassical, because all coherent states (except vacuum), and their mixtures as well, have an extension to infinite Fock states. It is nontrivial to demonstrate the nonclassicality of FDS when one has access to limited information; for example, a noisy state $f|0⟩⟨0|+(1-f)|1⟩⟨1|$ for $f\ge \frac{1}{2}$ has no simple signatures of nonclassicality like squeezing and negativity of Wigner function. We prove that our DMs are able to detect all non-Gaussian states in finite dimension of any size, with details in 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 $|\mathrm{\Psi }⟩=\frac{1}{\sqrt{2}}\left(|0⟩+|2⟩\right)$, whose original Wigner function and matrix elements are displayed in Fig. 1A. Our DM methods yield matrix elements as shown in Fig. 1B and C. The nonpositivity of the density operator is then demonstrated by, for example, $⟨0|\rho |0⟩⟨8|\rho |8⟩-{|⟨0|\rho |8⟩|}^2<0$ under DM1 and $⟨0|\rho |0⟩⟨4|\rho |4⟩-{|⟨0|\rho |4⟩|}^2<0$ under DM2, respectively.

Nonclassicality Measure and Entanglement Potential.

We may define a measure of nonclassicality using our DMs as

$${\mathcal{N}}_{\mathrm{DM}}(\rho )\equiv \underset{\theta \in (0,\pi )}{\max }\frac{{||{\rho }_{\mathrm{DM}}^{\theta }||}_1-1}{2},$$ [6]

where $||\cdot |{|}_1$ is a trace norm and ${\rho }_{\mathrm{DM}}^{\theta }$ a density matrix under DM, using a marginal distribution at angle $\theta$. Our DM negativity possesses the following properties appropriate as a nonclassicality measure, with details in SI Appendix: (i) ${\mathcal{N}}_{\mathrm{DM}}=0$ for a classical state; (ii) convex, that is, nonincreasing via mixing states, ${\mathcal{N}}_{\mathrm{DM}}({\sum }_j{p}_j{\rho }_j)\le {\sum }_j{p}_j{\mathcal{N}}_{\mathrm{DM}}({\rho }_j)$; and (iii) invariant under a classicality-preserving unitary operation, ${\mathcal{N}}_{\mathrm{DM}}({\widehat{U}}_c\rho {\widehat{U}}_c^{†})={\mathcal{N}}_{\mathrm{DM}}(\rho )$, where ${\widehat{U}}_c$ refers to displacement or phase rotation. Combining ii and iii, we also deduce the property that (iv) ${\mathcal{N}}_{\mathrm{DM}}$ does not increase under generic classicality-preserving operations (mixture of unitary operations).

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, $W_{{\rho }_{12}}(q_1,p_1,q_2,p_2)\to W_{{\rho }_{12}}(q_1,p_1,q_2,-p_2)$. If the resulting Wigner function is unphysical, the state ${\rho }_{12}$ is NPT. We first show that all nonclassical states detected under our DMs can generate NPT entanglement via a BS setting.

We inject a single-mode state $\rho$ and its rotated version $\overline{\rho }=e^{i\frac{\pi }{2}\widehat{n}}\rho e^{-i\frac{\pi }{2}\widehat{n}}$ into a 50:50 BS, described as

$$W_{\rho }(q_1,p_1)W_{\overline{\rho }}(q_2,p_2)\stackrel{BS}{\to }{W}_{\rho }(\frac{q_1+q_2}{\sqrt{2}},\frac{p_1+p_2}{\sqrt{2}})W_{\overline{\rho }}(\frac{q_1-q_2}{\sqrt{2}},\frac{p_1-p_2}{\sqrt{2}}).$$ [7]

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

$$W_{\rho }(\frac{q_1+q_2}{\sqrt{2}},\frac{p_1-p_2}{\sqrt{2}})W_{\overline{\rho }}(\frac{q_1-q_2}{\sqrt{2}},\frac{p_1+p_2}{\sqrt{2}})\stackrel{BS}{\to }{W}_{\rho }(q_1,p_2)W_{\overline{\rho }}(q_2,p_1)=W_{\rho }(q_1,p_2)W_{\rho }(p_1,q_2).$$ [8]

Integrating over $q_2$ and $p_2$, the marginal Wigner function for mode 1 is given by $M_{\rho }(q_1)M_{\rho }(p_1)$, which is identical to DM1 of the state $\rho$ in Eq. 1. The other DM2 in Eq. 2 emerges when replacing the second input state $\overline{\rho }$ by a vacuum $\overline{\rho }=|0⟩⟨0|$. Therefore, if the original state $\rho$ is nonclassical under our DMs, the output entangled state via the BS scheme must be NPT.

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,

$${\mathcal{P}}_{\mathrm{ent}}[\rho ]\equiv \frac{{||{[{\widehat{U}}_{\mathrm{BS}}({\rho }_1\otimes |0⟩{⟨0|}_2){\widehat{U}}_{\mathrm{BS}}^{†}]}^{\mathrm{PT}}||}_1-1}{2},$$ [9]

where ${\widehat{U}}_{\mathrm{BS}}$ and ${[\cdot ]}^{\mathrm{PT}}$ represent 50:50 BS operation and partial transpose on mode 2, respectively. We then prove in SI Appendix that our DM2 measure provides a lower bound for the entanglement potential as

$${\mathcal{N}}_{\mathrm{DM2}}[\rho ]\le {\mathcal{P}}_{\mathrm{ent}}[\rho ].$$ [10]

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 ${}^{171}\text{Yb}^{+}$ ion. For the manipulation of motional state, the single phonon-mode $\widehat{a}$ along the $\mathrm{X}$ direction in 3D harmonic potential with trap frequencies $({\omega }_{\mathrm{X}},{\omega }_{\mathrm{Y}},{\omega }_{\mathrm{Z}})=2\pi (2.8,3.2,0.6)$ MHz is coupled to two internal levels of the $S_{1/2}$ ground-state manifold, $|F=1,m_F=0⟩\equiv |\uparrow ⟩$ and $|F=0,m_F=0⟩\equiv |\downarrow ⟩$ with transition frequency ${\omega }_{\mathrm{HF}}=(2\pi )\mathrm{\hspace{0.25em}12.642821}$ GHz. We implement the anti-Jaynes–Cummings interaction $H_{\mathrm{aJC}}=\frac{\eta \mathrm{\Omega }}{2}{\widehat{a}}^{†}{\widehat{\sigma }}_{+}+\mathrm{h}.\mathrm{c}.$ and the Jaynes–Cummings interaction $H_{\mathrm{JC}}=\frac{\eta \mathrm{\Omega }}{2}\widehat{a}{\widehat{\sigma }}_{+}+\mathrm{h}.\mathrm{c}.$ with ${\sigma }_{+}=|\uparrow ⟩⟨\downarrow |$. $H_{\mathrm{aJC}}$ is realized by two counterpropagating laser beams with beat frequency near ${\omega }_{\mathrm{HF}}+{\omega }_{\mathrm{X}}$ and $H_{\mathrm{JC}}$ with frequency near ${\omega }_{\mathrm{HF}}-{\omega }_{\mathrm{X}}$ (7). $\eta =\mathrm{\Delta }k\sqrt{\mathrm{\hslash }/2M{\omega }_{\mathrm{X}}}$ is the Lamb–Dicke parameter, $\mathrm{\Omega }$ the Rabi frequency of internal transition, $\mathrm{\Delta }k$ the net wave vector of the Raman laser beams, and $M$ the ion mass.

For our test, we generate the Fock states $|n=1⟩$ and $|n=2⟩$, together with the ground state $|n=0⟩$. First, we prepare the ground state by applying the standard Doppler cooling and the Raman sideband cooling. Then we produce the Fock states by a successive application of the $\pi$ pulse of $H_{\mathrm{aJC}}$, transferring the state $|\downarrow ,n⟩$ to $|\uparrow ,n+1⟩$, and the $\pi$ pulse for internal state transition $|\uparrow ,n+1⟩$ to $|\downarrow ,n+1⟩$. We also generate a superposition state $\frac{1}{\sqrt{2}}(|0⟩+|2⟩)$ by applying the $\pi /2$ pulse of $H_{\mathrm{aJC}}$ and then the $\pi$ pulse of $H_{\mathrm{JC}}$.

Nonclassicality Test.

We measure a characteristic function $C_{\rho }(k_{\theta })\equiv ⟨e^{-2ik{\widehat{x}}_{\theta }}⟩$ with ${\widehat{x}}_{\theta }=\widehat{x}\cos \theta +\widehat{p}\sin \theta$, by first making the evolution $\widehat{U}=e^{-ik{\widehat{x}}_{\theta }{\widehat{\sigma }}_x}$ (simultaneously applying $H_{\mathrm{aJC}}$ and $H_{\mathrm{JC}}$ with proper phases) and then measuring internal state ${\widehat{\sigma }}_z=|\uparrow ⟩⟨\uparrow |-|\downarrow ⟩⟨\downarrow |$ at times $t_i$ ($k=\eta \mathrm{\Omega }t$) (45, 46). Using ${\widehat{U}}^{†}{\widehat{\sigma }}_z\widehat{U}=\cos (2k{\widehat{x}}_{\theta }){\widehat{\sigma }}_z+\sin (2k{\widehat{x}}_{\theta }){\widehat{\sigma }}_y$, we obtain $⟨\cos (2k{\widehat{x}}_{\theta })⟩$ and $⟨\sin (2k{\widehat{x}}_{\theta })⟩$, with the internal state initially prepared in the eigenstates ${|+⟩}_z$ and ${|+⟩}_y$ of ${\sigma }_z$ and ${\sigma }_y$, respectively. The Fourier transform of $C_{\rho }(k_{\theta })$ gives the marginal distribution of ${\widehat{x}}_{\theta }$ (45, 46). In contrast, we directly use it without the Fourier transform, for which our DMs work equally well as for the Wigner function. We test $C^{\mathrm{DM1}}\equiv C_{\rho }(k_x)C_{\rho }(k_y)$ or $C^{\mathrm{DM2}}\equiv C_{\rho }(k_x)C_{|0⟩⟨0|}(k_y)=C_{\rho }(k_x)e^{-\frac{1}{2}{\lambda }_{k_y}^2}$, with its density operator $\rho =\frac{1}{\pi }\int d{k}_{x}d{k}_{y}C(k_x,k_y){\widehat{D}}^{†}(k_x,k_y)$ unphysical for a nonclassical state.

To set a benchmark (noise level) for classical states, we prepared the motional ground state $|n=0⟩$ and obtained its marginal distributions along six axes with 1,000 repetitions for each time $t_i$. It yielded the negativity ${\mathcal{N}}_{\mathrm{DM2}}=0.019\pm 0.02$ represented by gray shading in Fig. 2 (SI Appendix). On the other hand, the Fock states $|n=1⟩$ and $|n=2⟩$ clearly manifest nonclassicality for each marginal distribution taken at three different angles $\theta$ in Fig. 2A, 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 $\frac{1}{\sqrt{2}}(|0⟩+|2⟩)$ not rotationally symmetric in phase space. As shown in Fig. 2B, its nonclassicality is well demonstrated for all measured angles individually whereas the degree of negativity varies with the measured axis.

Fig. 2.

(A and B) DM2 negativity against $\theta$ of the measured distribution $⟨e^{-2ik{\widehat{x}}_{\theta }}⟩$ for (A) Fock states $|1⟩$ and $|2⟩$ and (B) a superposition $\frac{1}{\sqrt{2}}(|0⟩+|2⟩)$. Dashed lines, theoretical value; bullets with error bars, experiment; gray shading, noise level for classical states. (C and D) Negativity under deconvolution test for Fock states $|1⟩$ (blue), $|2⟩$ (brown), and $\frac{1}{\sqrt{2}}(|0⟩+|2⟩)$ (red), together with $|0⟩$ (gray), using a $5\times 5$-moment matrix.

Compared with our DM, one might look into nonclassicality directly via deconvolution, that is, examine whether a marginal distribution $P(x)$ can be written as a sum of coherent-state distributions as $P(x)=\sqrt{\frac{2}{\pi }}\int d\stackrel{\sim }{x}\stackrel{\sim }{P}(\stackrel{\sim }{x})e^{-2{(x-\stackrel{\sim }{x})}^2}$, where $\stackrel{\sim }{P}(\stackrel{\sim }{x})$ must be positive definite for classical states. $\stackrel{\sim }{P}(\stackrel{\sim }{x})$ is nothing but the marginal of Glauber–Sudarshan P function and thus typically ill-behaved. One can test the positivity of $\stackrel{\sim }{P}(\stackrel{\sim }{x})$ alternatively using an $n\times n$-moment matrix with elements $M_{ij}\equiv ⟨{\stackrel{\sim }{x}}^{i+j}⟩$ $(i,j=0,\mathrm{\dots },n-1)$ (47). Fig. 2 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 $C_{\rho }(\xi )\equiv \mathrm{tr}[\rho \widehat{D}(\xi )]$ for a legitimate quantum state must yield an $n\times n$ positive matrix $\mathcal{M}>0$ with matrix elements

$${\mathcal{M}}_{jk}=C({\xi }_j-{\xi }_k)e^{\frac{1}{2}({\xi }_j{\xi }_k^{\ast }-{\xi }_j^{\ast }{\xi }_k)},$$ [11]

for an arbitrary set of complex variables $\{{\xi }_1,{\xi }_2,\mathrm{\dots },{\xi }_n\}$. In our case, we test the positivity of a matrix ($n=9$) constructed using $3\times 3$ points of rectangular lattice of size $d$ for the characteristic function under DM2 (SI Appendix). As shown in Fig. 3A, the ground state $|0⟩$ shows nonnegativity (and thus the mixture of coherent states as well due to convexity of our method) for all values of $d$, whereas a nonclassical state $|\mathrm{\Psi }⟩=\frac{1}{\sqrt{2}}\left(|0⟩+|2⟩\right)$ manifests negativity in a certain range of $d$, confirming nonclassicality for each measured distribution at $\theta =\frac{\pi }{3},\frac{\pi }{2}$, and $\frac{2\pi }{3}$ (red solid curves) in Fig. 3 B–D, respectively. Furthermore, note that a mixture of the vacuum and the nonclassical state, $f|0⟩⟨0|+(1-f)|\mathrm{\Psi }⟩⟨\mathrm{\Psi }|$, possesses a positive-definite Wigner function for $f\ge 0.66$, so even a full tomography may not directly show its nonclassicality via negativity. In contrast, our simple method manifests nonclassicality for $f=0.66$, shown by blue dashed curves in Fig. 3 B–D. For Fock states, we consider the matrix test using $5\times 5$ lattice points, which confirms negativity at the mixing $f=0.5$ with vacuum giving a nonnegative Wigner function for both states $|1⟩$ and $|2⟩$ in Fig. 3 E and F.

Fig. 3.

KLM test under DM2 using a matrix of $3\times 3$ (A–D) and $5\times 5$ (E and F) lattice points, respectively, with ${\lambda }_{\min }$ the lowest eigenvalue for each lattice size $d$ (SI Appendix). Negative ${\lambda }_{\min }$ manifests the nonclassicality of the considered state. Solid (pure-state $|\mathrm{\Psi }⟩$) and dashed (mixed-state $f|0⟩⟨0|+(1-f)|\mathrm{\Psi }⟩⟨\mathrm{\Psi }|$) curves represent theoretical predictions and bullets with error bars represent experimental data. (A) motional ground state $|0⟩$. (B–D) $|\mathrm{\Psi }⟩=\frac{1}{\sqrt{2}}\left(|0⟩+|2⟩\right)$ for the measured angles at $\theta =\frac{\pi }{3}$, $\frac{\pi }{2}$, and $\frac{2\pi }{3}$, respectively. (E) Fock state $|1⟩$. (F) Fock state $|2⟩$. For mixed states (dashed curves), we use $f=0.66$ in B–D and $f=0.5$ in E and F, respectively.

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, $\rho =\sum p_n|n⟩⟨n|$, and thus rotationally symmetric in phase space. For this class, one may detect nonclassicality using photon-number moments (18), which can be experimentally addressed efficiently by phase-averaged quadrature measurements (64, 65). Lvovsky and Shapiro experimentally demonstrated the nonclassicality of a noisy single-photon state $f|0⟩⟨0|+(1-f)|1⟩⟨1|$ for an arbitrary $f$ (66), using the Vogel criterion (67). In contrast, we look into the genuine non-Gaussianity of non-Gaussian states as follows.

For a Gaussian state ${\rho }_{\mathrm{G}}$, the phase randomization gives $\sigma \equiv \frac{1}{N}{\sum }_{k=0}^{N-1}{e}^{-i{\theta }_k\widehat{n}}{\rho }_{\mathrm{G}}{e}^{i{\theta }_k\widehat{n}}$ with ${\theta }_k\equiv \frac{k}{N}\pi$. As the number $N$ of phase rotations grows, the DM negativity of Gaussian states decreases. With $N\to \mathrm{\infty }$ (full phase randomization), we obtain the Gaussian bound ${\mathcal{B}}_G\approx 0.0887$ (SI Appendix). Thus, if a state manifests a larger DM negativity as $\mathcal{N}>{\mathcal{B}}_{\mathrm{G}}$, it confirms genuine non-Gaussianity. We plot the Gaussian bounds for finite rotations $N=6$ and $N=12$ with ${\mathcal{B}}_{\mathrm{G}}\approx 0.0887$ against energy $n$ in Fig. 4. Our data for the state $|2⟩$, which shows negativity insensitive to measured angles in Fig. 2, indicate genuine non-Gaussianity for the mixed states $f|0⟩⟨0|+(1-f)|2⟩⟨2|$ with $f=1-\frac{n}{2}$. For example, the $N=12$ case (Fig. 4, brown dotted-dashed line) as well as the full phase randomization (Fig. 4, black dashed horizontal line) confirms quantum non-Gaussianity at $f=\frac{1}{2}$, corresponding to a positive Wigner function.

Fig. 4.

DM negativity of $f|0⟩⟨0|+(1-f)|2⟩⟨2|$ (bullets with error bars) from experimental data. Maximum Gaussian negativity under $N=6$ (blue dotted line), $N=12$ (brown dotted-dashed line), and $N\to \mathrm{\infty }$ (black dashed horizontal line) phase rotations is given against energy $n$, the negativity above which confirms genuine non-Gaussianity.

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.