Entropic nonclassicality and quantum non-Gaussianity tests via beam splitting

We propose entropic nonclassicality criteria for quantum states of light that can be readily tested using homodyne detection with beam splitting operation. Our method draws on the fact that the entropy of quadrature distributions for a classical state is non-increasing under an arbitrary loss channel. We show that our test is strictly stronger than the variance-based squeezing condition and that it can also be extended to detect quantum non-Gaussianity in conjunction with phase randomization. Furthermore, we address how our criteria can be used to identify single-mode resource states to generate two-mode states demonstrating EPR paradox, i.e., quantum steering, via beam-splitter setting.


entropic nonclassicality and quantum non-Gaussianity tests via beam splitting
Jiyong park 1* , Jaehak Lee 2 & Hyunchul nha 3,2* We propose entropic nonclassicality criteria for quantum states of light that can be readily tested using homodyne detection with beam splitting operation. our method draws on the fact that the entropy of quadrature distributions for a classical state is non-increasing under an arbitrary loss channel. We show that our test is strictly stronger than the variance-based squeezing condition and that it can also be extended to detect quantum non-Gaussianity in conjunction with phase randomization. furthermore, we address how our criteria can be used to identify single-mode resource states to generate two-mode states demonstrating epR paradox, i.e., quantum steering, via beam-splitter setting.
Nonclassicality is a concept of fundamenatal and practical importance in quantum optics and in quantum information using continuous variables (CVs). A single-mode light field is called nonclasssical if it cannot be represented as a probabilistic mixture of coherent states. It is a characterization of quantum light enabling optical phenomena unattainable in classical domain, which can also be used for many applications in quantum technology. The nonclassicality of a single-mode state generally provides resource for creating quantum entanglement via beam splitter setting (BS) [1][2][3][4][5][6][7] . Nonclassical squeezed states can also be employed for quantum metrology, e.g. phase estimation with better sensitivity than classical schemes [8][9][10] . In addition, nonclassical states are necessary for testing quantum foundations 11,12 . Recently, beyond the notion of nonclassicality, there has also been a growing interest in addressing quantum non-Gaussianity, which are regarded as an essential ingredient for quantum information processing. This is because Gaussian states and Gaussian operations have limited capabilities in some crucial tasks, e.g. quantum computation 13,14 , entanglement distillation [15][16][17] and error correction 18 . A quantum state of light is called quantum non-Gaussian if it cannot be represented as a probabilistic mixture of Gaussian states. By its definition, the set of quantum non-Gaussian states belongs to the set of nonclassical states. The rigorous notion of quantum non-Gaussianity is introduced to detect higher-order optical processes that cannot be obtained by only Gaussian operations and their mixtures. Furthermore, the measures for quantum non-Gaussianity are developed by using a resource theoretical framework 19,20 .
We here propose experimentally feasible nonclassicality and quantum non-Gaussianity tests based on the entropy of quadrature distributions. Entropy, which quantifies the uncertainty about a random variable, is a key notion used in many branches of science including quantum foundation 21 , thermodynamics 22 , and information theory 23,24 . For continuous variable (CV) quantum information 25,26 , the entropy of quantum states has played a central role in establishing the capacities of Gaussian quantum channels [27][28][29][30] and the entanglement of formation for Gaussian states 31,32 . It has also been employed for measuring non-Gaussianity 33-35 and quantum non-Gaussianity 36 of quantum states. Furthermore, the entropy of quadrature distributions has been used for assessing the performance of CV communication protocols, e.g., CV quantum key distribution 37,38 , CV quantum dense-coding 39,40 and CV quantum communication in Gaussian regime [41][42][43] , and detecting quantum entanglement [44][45][46] and quantum steering 47,48 .
The entropy of a quadrature distribtion can be readily obtained using a highly efficient tool of homodyne detection 49 . While the homodyne detection can also be used to test usual quadrature squeezing, which is a prototypical nonclassical property, we show that our criterion is strictly stronger than the squeezing criterion. That is, our method detects more nonclassical states than squeezing test. Our approach relies on the fact that the entropy of a quadrature distribution is non-increasing via loss channel for the case of a classical state. In other words, given a quantum state of light, if the output entropy turns out to be larger than the input entropy for a certain quadrature distribution, it is nonclassical. This test requires the measurement of quadrature distributions for the input and the output fields, Results entropic nonclassicality criteria via beam splitting. For an input state ρ, the action of a loss channel can be described as where η B is a beam-splitting operation with transmittance η between system A and environment B. Similarly, its complementary channel, which represents a reflected state instead of a transmitted state, is described as We propose entropic criteria for nonclassicality as [ ] [ ] Here the differential entropy ρ H Q ( ) is obtained by in terms of quadrature distribution ) the Wigner function of ρ. That is, if the entropy of the output quadrature distribution is larger than that of the input distribution, the state is nonclassical ( Fig. 1 for illustration). Its proof goes as follows.
Proof: When an input state ρ is mixed with a vacuum state | 〉 0 at a beam splitter of transmissivity η = t and reflectivity η = − r 1 , the output two-mode Wigner function is given by W q p q p ( , , , ) 12 is the transmitted (reflected) output via a beam splitter. If the entropy of the output distribution turns out to be larger than that of the input distribution ρ M q ( ), the original state ρ is nonclassical.
with a non-negative P function, we obtain for a classical state. It is straightforward to see that the above analysis equally applies to an arbitrary quadrature amplitude θ θ ≡ − θˆq q p cos s in . Therefore, if the output entropy is larger than the input entropy for any quadrature distribution, the given state is confirmed to be nonclassical. comparison with other nonclassicality tests. In this section, we compare our entropic nonclassicality criteria with the usual squeezing criterion. Before doing so, we also present an entropic form of squeezing condition, which constitutes another simple nonclassicality test 52 .

Entropic squeezing criterion. It is given by
0 0 using the concavity of the differential entropy. That is, for a classical state, 2 0 0 cl Equation (8) tells that the state is nonclassical if its quadrature entropy is less than that of a vacuum state, which is = Remark: We note that the entropic test in Eq. (8) can be considered as a subset of our previously proposed criteria, i.e., the so-called demarginalization map (DM) approach 53 . In DM method, nonclassicality is confirmed by showing the unphysicality of a fictitious Wigner function, e.g., constructed as = ρ that of a vacuum state. If a given state ρ satisfies Eq. (8), we deduce Here Q and P represent the position and momentum quadratures, respectively, with their probability distributions ) obtained from the Wigner function ρ W q p ( , ), and the differential entropies 21 , which means that ρ DM is not a legitimate physical state confirming the nonclassicality of ρ.
Hierarchy of nonclassicality conditions. We here show that (i) the entropic squeezing condition in Eq. (8) is stronger than the usual variance-squeezing condition and that (ii) our main criteria in Eqs. (3) and (4) are stronger than the entropic squeezing condition in Eq. (8). Therefore, the hierarchy is given by {variance-squeezing} ⊂ {entropic-squeezing} ⊂ {entropy-nonclassicality via BS}.
(i) {variance-squeezing} ⊂ {entropic-squeezing} www.nature.com/scientificreports www.nature.com/scientificreports/ Suppose that a state, Gaussian or non-Gaussian, possesses a variance squeezing, i.e. < θ V 1 4 for a certain quadrature θ q . This condition can be expressed in terms of entropy as < is the entropy of a Gaussian distribution with variance V. Now using the fact that a Gaussian distribution takes a maximal entropy under the same variance constraint, we deduce , which is nothing but the condition in Eq. (8).
On the other hand, there are states that have entropic squeezing but no variance-squeezing. Example is given in the next subsection. Therefore, the entropic squeezing condition is strictly stronger than the variance-squeezing condition.
(ii) {entropic-squeezing} ⊂ {entropy-nonclassicality via BS} Now suppose that a state satisfies < . We here adopt an entropic inequality, i.e., 54,55 .  λ X Y means the addition of two random variables X and Y with fractions λ and λ − 1 , respectively. It is relevant to the action of a beamsplitter on the quadrature distributions. From the inequality, we deduce . On the other hand, there are states that satisfy our entropic criteria in Eqs. (3) and (4) but not Eq. (8). Example is again given in the next subsection. Therefore our main entropic criteria are stictly stronger than the entropic squeezing, and also the usual variance-squeezing by (i).
Examples. We here illustrate the usefulness of our criteria by examples. We are particularly interested in non-Gaussian states without variance-squeezing, which can nevertheless be detected by our criteria in Eqs. (3), (4) and (8).
Our first example is a photon-added thermal state, i.e. ρ = a thermal state of mean number n. Its quadrature distribution is given by The quadrature distribution after beam splitting operation can be obtained using Eq. (6). We would like emphasize that the photon-added thermal state has no entropic squeezing, i.e., > , and the entropic uncertainty relation, i.e., , which must be satisfied by all quantum states. In Fig. 2(a), we show the differential entropy of ρ path with = .
n 0 1 before and after a beam splitter of transmittance η. Black horizontal solid line represents the initial entropy . , respectively. We see that our entropic criteria detect nonclassicality by using a beam splitter of transmittance in the range  η . 0 667 and  η . 0 333, respectively, in view of Eqs. (3) and (4). We have numerically checked that the detectable range of η decreases with n and that our criteria detect ρ path for a thermal photon  ≤ . n 0 0203. Our second example is an odd cat state ψ γ γ | 〉 ∼ | 〉 − |− 〉 whose quadrature distribution is given by Here we look into the differential entropy of the momentum quadrature distribution, θ π = /2, as it manifests entropic squeezing. That is, the odd cat state, even though it does not have variance squeezing at all, satisfies entropic squeezing condition in Eq. (8) for γ ≥ . 0 891. When the coherent amplitude is smaller γ < . 0 891, our main criteria in Eqs. (3) and (4) can detect its nonclassicality.
In Fig. 2(b), we illustrate the cases with γ = . 0 5 (upper curves) and γ = 1 (lower curves). For γ = . 0 5, the input entropy . entropic quantum non-Gaussianity criterion. A phase randomization can be useful in enhancing the performance of quantum information protocols 56,57 . Here we further extend our entropic approach to detect quantum non-Gaussianity by using beam-splitting in conjunction with phase randomization. A complete phase randomization can be described as in in The quadrature distribution of a phase randomized state ρ [ ]  is written as   . The result in Eq. (16) indicates that the differential entropy of a phase-randomized Gaussian state always decreases after a loss channel. Using Jensen's inequality again, we readily see that the same argument applies to a statistical mixture of phase-randomized Gaussian states. Therefore, if a quantum state ρ satisfies the state is quantum non-Gaussian meaning that it cannot correspond to a probabilistic mixture of Gaussian states. Our criterion is particularly useful to detect quantum non-Gaussianity of a rotationally symmetric state in phase space, e.g. Fock states (Fig. 3).
Resource for quantum steerability. Now [ ] [ ] a two-mode quantum steerable state is produced by mixing ρ with vacuum at a beam-splitter. This can be seen by considering the entropic steering criterion proposed in ref. 47 as  . The condition in Eq. (19) means that the entropy of system B conditioned on the measurement outcome of system A beats this standard uncertainty relation due to quantum correlation, which evidences quantum steering 47 . On the other hand, one can also investigate B to A steering by interchanging indices ↔ A B in Eq. (19). Joint quadrature distribution of a quantum state

BS is given by
Therefore, if the condition in Eq. (18) is satisfied, the two-mode state after the beam splitter satisfies the entropic steering condition in Eq. (19). As a remark, our criteria in Eqs. (17) and (18)  Example. Here we illustrate the cases of Fock states | 〉 1 and | 〉 2 whose quantum non-Gaussianity and usefulness for quantum steering can be demonstrated by our criteria in Eqs. (17) and (18). These states are rotationally symmetric in phase space, which leads to identical probability distributions for all quadrature distributions. Therefore, all of our proposed criteria are satisfied in the same parameter regions. That is, the shaded regions in Fig. 3(a,b) represent the successful detection of nonclassicality, quantum non-Gaussianity, and usefulness for quantum steering simultaneously.
In particular, quantum non-Gaussianity of states ρ = | 〉〈 | 1 1 and ρ = | 〉〈 | 2 2 can be confirmed by Eq. (17) for  η . ≤ 0 383 1 and  η . ≤ 0 38 1 , respectively. One remark is in order. Beam-splitting operation is multiplicative, i.e. a BS with η 1 followed by another BS with η 2 correponds to a BS with η η η = 1 2 . Examining carefully the curves in Fig. 3, we see that not only a pure Fock state but also a noisy Fock state under loss can be detected by our analysis. For instance, let a single-photon state undergo a loss channel with η = .
0 9 1 (filled circle in Fig. 3). The nonclassicality of this noisy output state can be detected by injecting it at a beam splitter of transmittance, e.g., η = .
0 85 2 (hollow circle in Fig. 3). This is because the quadrature entropy increases with decreasing η from 1 to 0.728 in red curve of Fig. 3(a). The same argument can also be given for two-photon state, for which the entropy increases with decreasing η from 1 to 0.771 in red curve of Fig. 3(b).
For quantum steering, we identify the regions for one-way steering and two-way steering, respectively. Each colored region, red or blue, represents the A (transmitted field) to B (reflected field) steering or vice versa. The overlap region in purple represents the steering in both of the ways.
Comparison with other QNG criteria. It may be interesting to compare our QNG criterion with other existing criteria particularly in refs. [58][59][60] . Unlike the case of the nonclassicality criteria, we do not have a hierarchical relation among these existing QNG criteria and ours. That is, one criterion does not include another as a subset, but different criteria can be complementary to one another. For instance, while the criterion in ref. 60 is useful to detect QNG for a finite superposition of Gaussian states, e.g. generalized cat-states as discussed in ref. 60 , it is not suitable to address QNG of Fock states. On the other hand, the other criteria in refs. 58,59 and our criterion successfully detect noisy Fock states to some extent.
We further compare our criterion and the one in ref. 59 by showing how Fock-diagonal states can be detected via each method. Reference 59 introduced a QNG criterion in the form of + + P aP n k n k , , 1 , where P n k , represents a probability of firing k-detectors out of total n-detectors while a is a free parameter to optimize criterion. For each a, there exists a maximum value of + + P aP n k n k , , 1 achieved by a whole class of Gaussian mixture states so that the value above this bound becomes the signature of QNG 59 . In Fig. 4, we show the results for the state ρ = . | 〉〈 | + . | 〉〈 | + . | 〉〈 | 0 17 0 0 0 17 1 1 0 66 2 2 . As shown in Fig. 4(a), our criterion detect its QNG via the beam splitting of transmittance η > . 0 912. On the other hand, the criterion in ref. 59 does not detect QNG when the number of detectors is limited to two. That is, the value of + P aP 2,1 2 ,2 for the state ρ (brown dashed line) does not go above the Gaussian bound (blue solid line) for any a. On the other hand, it can be made successful by increasing the number of detectors to three, i.e. + P aP 3,2 3 ,3 for the state ρ (brown dashed line) beats the Gaussian bound (blue solid line) for a certain range of a.
The above example illustrates that the QNG criterion in ref. 59 requires an increasingly large number of detectors for higher Fock states, which may become less efficient with nonideal detector efficiency. However, ref. 59 also proposed a novel concept of QNG, i.e. genuine n-photon QNG, which certainly deserves a separate discussion elsewhere. In contrast, our criterion always uses the same experimental setup, i.e. homodyne detection known to be highly efficient, regardless of input states. From a fundamental point of view, it is also noteworthy that the criteria in refs. 58,59 consider the particle nature measuring photon-number distributions, whereas the one in ref. 60 and our criterion consider the wave nature measuring quadrature-amplitude distributions. experimental feasibility. To In quantum optics laboratory, homodyne detection is a well established, highly efficient, scheme to measure quadrature amplitudes constituting an integral part of state tomography 49 , thus our criterion is readily testable. On the other hand, there is one practical issue to consider for experimental feasibility. In a realistic homodyne detection, the measurement cannot discern the values of φ q within an interval of size σ, where σ represents the size of data binning. It leads to a coarse-grained probability distribution, instead of smooth continuous distribution 49,61 , as n with a step function x rect( ) This step-wise discontinuous distribution typically adds more entropy owing to information loss. To rigously address our criterion with coarse-graining σ, we come up with the entropic bound of nonclassicality due to σ in Methods. There we find the modified criterion of nonclassicality as , where > σ B 0 can be readily obtained numerically for each σ. That is, the condition of having a higher entropy after the loss channel still works with a nonzero adjustment B σ .
In Fig. 5, we show the results of our nonclassicality test with coarse-graining. We first numerically obtain the bound B σ = 0.0016639 and 0.0066226 for σ = 0.1 and σ = 0.2, respectively. As shown in the plots, our approach is still successful in detecting Fock states with finite binning. For a single photon state, we can detect it by using η .

Discussion
In this paper, we have proposed entropic nonclassicality criteria that look into the entropies of a quadrature amplitude before and after a loss channel. Our criteria can be readily tested using homodyne detection with a beam splitter. We have also shown the hierarchical relation among various nonclassicality tests, that is, our entropic tests are strictly stronger than the usual variance-squeezing test. We have illustrated the usefulness of our criteria with non-Gaussian states that do not possess quadrature squeezing but can be detected by our approach.
Furthermore, our approach has been extended to detect quantum non-Gaussianity in conjunction with phase randomization and to detect useful resource states demonstrating quantum steering. In future, we hope our approach here could be further developed to identify a full power of entropic analysis that can be a useful tool to investigate nonclassicality of CV states in general. For instance, our approach can be generalized to adopt Renyi entropies beyond Shannon entropy. It will also be interesting to examine how these entropic criteria can be useful for critical assessment of quantum tasks using continuous variables in relation to QNG 62 . www.nature.com/scientificreports www.nature.com/scientificreports/ Methods integral form of the log sum inequality. The log sum inequality is given by We may similarly construct the integral form of the log sum inequality as where      , the state is nonclassical. We numerically obtain B σ = 0.0016639 and 0.0066226 for σ = .