† Corresponding author. E-mail:
Project supported by the Natural Science Foundation of Hunan Province, China (Grant No. 2017JJ2214) and the Key Project Foundation of the Education Department of Hunan Province, China (Grant No. 14A114.
We provide a measure to characterize the non-Gaussianity of phase-space function of bosonic quantum states based on the cumulant theory. We study the non-Gaussianity dynamics of two-mode squeezed number states by analyzing the phase-averaged kurtosis for two different models of decoherence: amplitude damping model and phase damping model. For the amplitude damping model, the non-Gaussianity is very fragile and completely vanishes at a finite time. For the phase damping model, such states exhibit rich non-Gaussian characters. In particular, we obtain a transition time that such states can transform from sub-Gaussianity into super-Gaussianity during the evolution. Finally, we compare our measure with the existing measures of non-Gaussianity under the independent dephasing environment.
The non-Gaussianity of a quantum state has been widely acknowledged to be a useful resource in implementing relevant quantum-information tasks and in understanding quantum mechanics since it can enhance the fidelity in continuous-variable quantum teleportation,[1,2] construct a test platform for optical links suitable for quantum key distribution,[3] reveal the non-classicality of quantum states,[4] and measure the robustness of real single photons against optical losses.[5] To characterize this property in a quantitative manner, different measures have been proposed in recent years. Genoni et al. first used the Hilbert–Schmidt distance to quantify the non-Gaussian character of a bosonic quantum state and evaluated the non-Gaussianity of some relevant states.[6] Subsequently, they developed the entropic measure of non-Gaussianity based on the quantum relative entropy,[7] by which they investigated the performance of conditional Gaussification toward twin-beam and de-Gaussification processes driven by Kerr interaction. These measures have recently been used in the studies of the non-Gaussianity of other quantum states. For instance, Tang et al. have investigated the non-Gaussianity of the single-photon-added and -subtracted coherent (SPASC) superposition states. It was found that squeezing the input field can enhance the robustness of non-Gaussianity.[8] Taghiabadi et al. have analyzed the non-Gaussianity of two two-mode continuous-variable separable states with the same marginal states and shown that the difference in behavior of their non-Gaussianity is the same as the difference between negativity of their Wigner functions.[9] Park et al. have examined the non-Gaussianity for the correlation of a bipartite quantum states and found that Gaussian extremality holds for none of these measures.[10] However, we emphasize that computing those measures is a formidable task for multimode continuous-variable (CV) non-Gaussian entangled states because one requires to look for the corresponding reference Gaussian counterparts.
The present work aims at overcoming this roadblock by introducing the cumulants. As is well known, the cumulants were first introduced in the late 1800s by Thiele under the name of semi-invariants,[11] but entered the wider scene of statistics only with Fisher’s fundamental paper under the name of cumulative moment functions.[12] Up to now it has been widely accepted that the cumulants are useful quantities for measuring the statistical properties of probability distributions. In particular, the cumulants of order greater than two serves as a quantitative statement of the departure of the shape of probability distribution from Gaussian.[13,14] For a univariate case, the third-order (or called skewness) is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean, while the fourth-order cumulant (or called kurtosis) provides a measure of the distance to Gaussianity. Positive kurtosis indicates heavy tails and peakedness relative to the Gaussian distribution, whereas negative kurtosis indicates light tails and flatness. Therefore, the larger the kurtosis is different from zero, the more highly the distribution is non-Gaussian. The multivariate skewness and kurtosis measures were also developed to assess departure from multivariate normality.[15–18] On the other hand, the characteristic function to any quantum operator including the density matrix operator can be expanded by its cumulants.[19] A quantum operator is Gaussian if all the cumulants of greater two orders of its operator’s characteristic function vanish identically; otherwise, it is non-Gaussian. Recently, an ongoing research effort has been devoted to the study of the non-Gaussianity of quantum operators on quantum Bosonic systems. Dubost et al. have studied theoretically and experimentally the quantification of non-Gaussian spin distributions using theory of cumulants and demonstrated that cumulant-based estimation is very efficient.[20] In the same spirit, Moreno–Cardoner et al. have analyzed the behavior of the full distribution of collective observables in quantum spin chains.[21] It has been shown that the non-Gaussianity in the critical phase can be evident from the analysis of the Binder cumulant and quantum fluctuations at criticality leads to highly non-Gaussian distribution. Olsen et al.[22] investigated the non-Gaussian statistic of the Kerr-squeezed state by calculating higher-order cumulants of quadrature variables. It was found that the nonlinear interaction can skew the distribution of the quadrature variables, giving rise to large third- and fourth-order cumulants for sufficiently long interaction times. However, unlike the proposed measures in Refs.[20] and [22], we use the phase-averaged technique to estimate the non-Gaussianity of quadrature distribution of a collective quadrature operator of bosonic quantum system, which belongs to a single-variable estimation of non-Gaussianity, thus greatly removing the complexity of cumulant computation. Additionally, we note that resource theories of quantum non-Gaussianity have been developed very recently for non-Gaussian operations[23] and for non-Gaussian states.[24,25] It is of great help to guide the deeply understanding of non-Gaussian resources and the development of quantum information processing thereof.
Two-mode squeezed number states are a broad and meaningful class of continuous-variable quantum bipartite states. The set of them contains mostly non-Gaussian entangled states but includes the known two-mode squeezed vacuum state. Such states were originally introduced by Chizhow where the photon number statistics and the phase properties of these states are investigated.[26] It is shown that the photon number distribution and the Pegg–Barnett phase distribution for such states have a similar (N + 1)-peak structure for a nonzero value of the difference in the numbers of photons between the modes. The analysis of their inseparabilities has been carried out via the Neumann entropy and the Shchukin-Vogel positive partial transposition (PPT) criterion.[27] However, to the best of our knowledge, non-Gaussianity dynamics of such states was never studied as yet.
In the present paper, we mainly study the non-Gaussianity of two-mode squeezed Bell states, which can be generated by operating with two-mode squeezer on Bell states, thus allowing us to discuss the influence of Gaussian operation on non-Gaussian entangled states. The remainder of this paper is organized as follows. In Section
As is well known, it is convenient to describe a continuous-variable quantum state by the s-order characteristic function, which is defined as[28,29]
![](cpb_27_10_100305/cpb_27_10_100305_ieqn1.gif)
![](cpb_27_10_100305/cpb_27_10_100305_ieqn2.gif)
![](cpb_27_10_100305/cpb_27_10_100305_ieqn3.gif)
![](cpb_27_10_100305/cpb_27_10_100305_ieqn4.gif)
Without loss of generality, the characteristic function of any continuous-variable quantum state can be expressed as in the following form:
![](cpb_27_10_100305/cpb_27_10_100305_ieqn5.gif)
![](cpb_27_10_100305/cpb_27_10_100305_ieqn6.gif)
![](cpb_27_10_100305/cpb_27_10_100305_ieqn7.gif)
![](cpb_27_10_100305/cpb_27_10_100305_ieqn8.gif)
Another equivalent description of a probability distribution is through the cumulant generating function, which is the logarithm of the moment-generation function
![](cpb_27_10_100305/cpb_27_10_100305_ieqn9.gif)
Any multi-mode Gaussian state is fully determined in the phase space by the first and second moments of the quadrature operators.[33,34] Thus, in terms of the definition of cumulant, one can easily show that the first-order cumulants of Gaussian states are given by
We see that the function f(
![](cpb_27_10_100305/cpb_27_10_100305_ieqn10.gif)
![](cpb_27_10_100305/cpb_27_10_100305_ieqn11.gif)
Among these higher-order cumulants, the fourth-order cumulant or the kurtosis can be regarded as the classical measure of non-Gaussianity. The kurtosis of a zero-mean random variable y is classically defined by
Two light modes are expressed by the annihilations a1 and a2 and are mixed on a beam splitter with reflectivity η. After beam splitter, we have
![](cpb_27_10_100305/cpb_27_10_100305_ieqn14.gif)
![](cpb_27_10_100305/cpb_27_10_100305_ieqn15.gif)
![](cpb_27_10_100305/cpb_27_10_100305_ieqn16.gif)
It is well known that any quantum system is unavoidably influenced by the surrounding environment, thus giving rise to deteriorate the degree of non-classicality of quantum states. Let us consider a simple case where two modes are independently coupled to their own thermal environment with the same coupling constant γ, so the dynamics of the system is described by the following master equation in the interaction picture
![](cpb_27_10_100305/cpb_27_10_100305_ieqn17.gif)
Using standard operator correspondences we can transform the mater equation (
![](cpb_27_10_100305/cpb_27_10_100305_ieqn18.gif)
The characteristic function of the state (
![](cpb_27_10_100305/cpb_27_10_100305_ieqn19.gif)
In the light of the discussion above and using Eqs. (
Let us first study a case where the system does not interact with the environment. As seen from Eq. (
Let us return to another decoherence model: phase damping (PD), whose mater equation is given by
![](cpb_27_10_100305/cpb_27_10_100305_ieqn24.gif)
Using the integration
Furthermore, we can obtain the phase-averaged kurtosis of two-mode quadrature operator
![](cpb_27_10_100305/cpb_27_10_100305_ieqn26.gif)
![](cpb_27_10_100305/cpb_27_10_100305_ieqn27.gif)
We end this section with the following remarks. First, quantum homodyne tomographies are an important topic in the field of quantum optics and quantum information science. Especially, it is a desirable research how to reconstruct quantum state using by a simple experimental arrangement. In our two-mode measurement scheme, it is consist of two balanced lossless beam splitters, two photo-detectors and a strong coherent local oscillator (LO), thus greatly simplifying the complexity of the experiment. Second, we propose an one-parameter measure to characterize the non-Gaussian characters of a multi-mode continuous-variable system: the phase-averaged kurtosis. Working with our method can have an advantage over the method in Ref. [39] where one needs a lots of the joint cumulants to completely characterize the non-Gaussianity of a bosonic quantum state. It was found that among these cumulants, some are equivalent and redundant. But we are able to circumvent this obstruction in our work and more importantly, one removes the terms with the phase of local oscillator in Eq. (
As stated in Refs. [6], [7], and [37], the geometric and entropic measures of non-Gaussianity have the same basic properties and also share the same qualitative behavior under Gaussification and de-Gaussification processes. So in this section, we only compare our phase-averaged kurtosis with the Hilbert-Schmidt measure of non-Gaussianity introduced in Ref. [6], where the non-Gaussianity of a state ρ is defined as
To simplify our analysis, we focus our attention on the second noise model. For the matrix elements of a two-mode state: ρnn;mm = ⟨m,m|ρ|n,n⟩, we can obtain the solution of Eq. (
It is seen that these two measures display the similar asymptotic behaviors of of non-Gaussianity. Namely, there exists a nonzero asymptotic value of the non-Gaussianity in the long-time limit, which is in very good agreement with the results obtained in Ref. [7] in the phase-diffusion evolution. However, they have a striking difference. One sees that the Hilbert–Schmidt degree of non-Gaussianities monotonically decreases with the evolution time, implying that the phase-damping noise has a merely detrimental effect on the δHS, in accordance with the corresponding property in the presence of the thermal noise reported in Ref. [37]. But this is not the case with our introduced measure. It is a complicated problem. We see that the kurtosis begins to decrease until it reaches zero, then increases and eventually approaches to the asymptotic value (see the red dashed line). This difference is due to the fact that the non-Gaussianity measure of Genoni et al. is based on the distance between the state ρ and the reference Gaussian state. So the key step is to choose the correct reference Gaussian state; if not, it will lead to misleading results in assessing which the quantum state is the most non-Gaussian. It is challenging task. On the contrary, our measure rests on the shape of the phase-space distribution of quantum state itself and hence can quantify the non-Gaussianity in a safe manner. Of course, how to quantify non-Gaussianity for quantum state by using a justified measure is our work in the future research.
In conclusion, we have proposed a method to characterize non-Gaussianity of multi-mode continuous-variable quantum state based on the cumulants. Unlike the other geometric and entropic distances, our measure only need to focus on the non-Gaussian components of the characteristic function of quantum states, thus greatly simplifying the process of calculation. Using the proposed measure, we have study the non-Gaussianity evolution of two-mode squeezed number states subject to two sources of decoherence: i) the amplitude damping and ii) the phase damping due to the coupling to the thermal environment. Our analysis shows that in the amplitude damping model, the non-Gaussianity can rapidly vanish at a finite time, while in the phase damping model, such a quantum state can transform from sub-Gaussianity into super-Gaussianity during the evolution. We hope that our work will be helpful for deeply understanding the non-Gaussianity of other quantum states and may be have potential applications in quantum entanglement distillation and quantum information processing.
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