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Xiang Shaohua, Zhu Xixiang, Song Kehui. Non-Gaussianity dynamics of two-mode squeezed number states subject to different types of noise based on cumulant theory. Chinese Physics B, 2018, 27(10): 100305  
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Non-Gaussianity dynamics of two-mode squeezed number states subject to different types of noise based on cumulant theory
Xiang Shaohua †, Zhu Xixiang, Song Kehui
School of Mechanical, Optoelectronics and Physics, Huaihua University, Huaihua 418008, China

 

† Corresponding author. E-mail: shxiang97@163.com

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.

Abstract

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.

PACS: 03.65.Yz;03.67.Mn;42.50.Dv;42.50.Lc
Keyword:non-Gaussianity;cumulants;squeezed number states;decoherence
1. Introduction

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 2, we define the cumulants of a quantum state in the characteristic function representation and provide an explicit formula for cumulants of any order. We illustrate in Section 3 how two-mode squeezer effects the non-Gaussianity of quantum state using two examples of two-mode squeezed Bell states. In Section 4, we compare our measure with the Hilbert–Schmidt degree of non-Gaussianity for a case of the two modes subjected to their own phase-diffusive noise. We conclude with a summary and outlook in Section 5.

2. Higher-order cumulants of quantum non-Gaussian states

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]

where , ρ is the density operator, and ξl is the complex variable corresponding to the operator . The values s = −1,0,1 correspond, respectively, to the Q function,[30] the Wigner function,[31] and the P function.[32] Here we make use of the Wigner characteristic function, i.e., s = 0. Therefore, once these functions are known, the expectation values of any products of creation operators and annihilations quantum system can be computed.

Without loss of generality, the characteristic function of any continuous-variable quantum state can be expressed as in the following form:

where χG(ξ) is equivalently given by an N−mode Gaussian state as follows:

where the average value is given by and a matrix element of 2N × 2N covariance matrix V is given by , in which for convenience to write, we collect the N-pair quadrature operators into a vector of operators , , and {·,·} denotes the anti-commutator. We see from Eq. (3) that if the function f(ξ) is a constant, then a state described by Eq. (2) is a Gaussian one; otherwise, it is a non-Gaussian state.

Another equivalent description of a probability distribution is through the cumulant generating function, which is the logarithm of the moment-generation function

Once χ[ρ](ξ) is known, the k-th order cumulant can be generated by evaluating the l-th derivative of Γ(ξ) at ξ = 0 and is given by

where the superscript NG (G) denotes the non-Gaussian (Gaussian) component, , and l = l1 + l2 + … +l2N. So we see that the cumulant of a CV quantum state is a sum of the cumulants of its Gaussian and the resulting non-Gaussian parts.

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

and the corresponding second-order cumulants are the vectors of the covariance matrix, i.e.,

where ξl denotes the l-th component of vector ξ.

We see that the function f(ξ) is of high relevance in deciding whether a quantum state is Gaussian or not. So we return to consider the non-Gaussianity of such a function, namely, its higher order cumulants. In order to calculate these cumulants, we can write the function f(ξ) as a formal Taylor expansion

where are the expansion coefficients of Eq. (8) and are in fact given by

where denotes the expectation value.

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

Thus, kurtosis is zero for a Gaussian random variable. For non-Gaussian random variables, kurtosis is nonzero. Kurtosis can be both positive or negative. Random variables that have a negative kurtosis are called sub-Gaussian, and those with positive kurtosis are called super-Gaussian. On the other hand, it is possible to determinate quasi-probability distribution in terms of probability distribution for the rotated quadrature phase.[35,36] Such a quadrature operator is defined as

For two-mode case, we propose a scheme for measuring quadrature distribution via two-mode homodyne tomography with a single local oscillator, as shown in Fig. 1.

Fig. 1. (color online) Scheme of two-mode quadrature measurement. Mode interacts with mode through a beam splitter with reflectivity η ∈ [0,1]. After such a beam splitter, a probability distribution of a linear superposition of two single-mode quadratures is measured by means of standard single-mode homodyne detection. BS: beam splitter; LO: local oscillator; PD: photon-detectors.

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

Mode c and a local-oscillator (LO) pulse are mixed at a beam splitter reflectivity η. Subsequently, the distribution of quadrature X(θ) is measured by means of standard single-mode homodyne detection. If we choose , we have

where θ is the phase determined by a local oscillator. Thus a complete set of homodyne measurements over all phase angles θ ∈ [0,2π] can be used to construct the density matrix ρ or equivalently its phase-space function. That is, the measured quadrature distribution can contain the complete statistics information on a quantum state. Therefore, we introduce a phase-averaged kurtosis of two-mode quadrature as the non-Gaussianity measure of quantum state, which is defined as

where the nth moment of operator is given by

where : : denotes the normal ordering and the terms ⟨: :⟩ can be obtained from Eq. (9).

3. Non-Gaussianity evolution of two-mode squeezed number states
3.1. Independent amplitude damping

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

where stands for the average photon number of the reservoir and the Lindblad superoperator is defined as

This model refers as the amplitude damping (AD).

Using standard operator correspondences we can transform the mater equation (16) into the Fokker–Planck equation:

The interaction with reservoir leads to the time development

where χ12(ξ1,ξ2; 0) is the characteristic function at time t = 0 and χΓ(ξ1,ξ2) is given by

Assume that the system is initially in a two-mode squeezed number state, which is given by

where the two-mode squeezing operator is given by with r being the squeezing degree and where |n,m⟩ denotes the Fock state of two modes.

The characteristic function of the state (20) can be written as

where Ln(z) denotes a Laguerre polynomial, , l,m = 1,2, l ≠ m, and we have used the formula

From Eq. (18), we find that the characteristic function of two modes at time t becomes

where the function f(ξ1,ξ2; t) is given by

where

and where the Newton binomial

In the light of the discussion above and using Eqs. (8), (9), and (15), we can obtain the phase-averaged kurtosis for operator at time t as

Let us first study a case where the system does not interact with the environment. As seen from Eq. (25) that if the system is initially in a two-mode number state, we can calculate the phase-averaged kurtosis of operator as κ4(X) = −3n(n + 1), that is, it is a sum of kurtosis for two single-mode Fock states obtained in Ref. [35]. When two modes are initially in two-mode squeezed vacuum state, i.e., n = 0, this phase-averaged kurtosis naturally vanishes, as expected. We now focus on the dynamics of this phase-averaged kurtosis of two-mode squeezed number state evolving in the independent amplitude damping, as shown in Fig. 2. We see that under the amplitude damping, the phase-averaged kurtosis for two-mode operator shows an exponential law decay behavior and ultimately vanishes completely in a finite interaction time, meaning that the non-Gaussian correlations in two-mode squeezed number state disappear. To further see this point, we can calculate all other higher-order cumulants of such an operator at any time as κm(X;t) = e−mγt/2κm(X), in which κm(X) is the m-th order cumulant of the two-mode phase-averaged quadrature operator at the initial time. It tells again that the cumulant-based non-Gaussianity is very fragile against the local amplitude damping decoherence, which can drive non-Gaussian states into Gaussian ones and makes the states more Gaussian. This process is called a Gaussification one.[37] Additionally, we observe from Fig. 2 that increasing the squeezing degree and the photon numbers of two modes have the same dynamical behavior of the κ4(X) and can lead to higher non-Gaussian correlations of quantum states. For instance, for a case of r = 0.8 and n = 4, one gets κ4(X)|τ = 0.5 = −372.53, while for r = 1.2 and n = 10, κ4(X)|τ = 0.5 = − 1.134 × 104 where τ = 1−e−2γt, which is 0 for t = 0 and 1 for t = ∞. That is to say, they can improve the robustness of non-Gaussianity.

Fig. 2. (color online) The phase-averaged kurtosis of as a function of interaction time γt for (a) various squeezing parameters r at n = 2 and for (b) different photon numbers n at r = 0.4 when two modes are initially in a squeezed number state under the independent amplitude damping model.
3.2. Independent phase damping

Let us return to another decoherence model: phase damping (PD), whose mater equation is given by

Repeating the same procedure as earlier, we can obtain the characteristic function of the evolved state as[38]

where and d{x} ≡ dx1 dx2.

Using the integration

we can easily obtain the characteristic function of the state (16) subject to PD decoherence as

where the function fII(ξ1,ξ2; t) is given by

where h = 2q2 + 2k2 + g1 − g2 − l1 − l2.

Furthermore, we can obtain the phase-averaged kurtosis of two-mode quadrature operator as

As seen from Eq. (31), when the two modes are initially in a Fock state, the averaged-phase kurtosis remains unchanged and equals to a sum of that of two individual Fock states, i.e., κ4(X) = −3n(n + 1), implying that there exists the possibility of non trivial phenomenon of time-invariant non-Gaussianity for two-mode quantum system in initially a number state. For the case of the system in being squeezed number state, we see that in the long-time limit, the κ4(X) does not vanish and approaches asymptotically to a nonzero steady value, which is determined by the squeezing parameters and photon numbers of the initial quantum state. Especially we have κ4(X)|t → ∞ = (3/2)sinh2(2r) in the long-time limit when the initial condition is the known two-mode squeezed vacuum state of the system. We plot the dynamics of the averaged-phase kurtosis of two-mode quadrature operator when the system is initially in two-mode squeezed number state evolving in the local dephasing environment, as shown in Fig. 3. It can be seen that if the mean initial intensity is enough small, the presence of dephasing environment almost have no great influence to the evolution of the κ4(X); on the contrary, one sees that the bigger the mean initial intensity is, the more pronounced the variation of non-Gaussianity becomes. Or more accurately the κ4(X) has a linear-dynamics behavior in the short-time limit, while at other evolution times, it exhibits an exponential-law behavior. For example, with Eq. (31), we can obtain the time-dependent phase-averaged kurtosis as κ4(X) = (3/2) sinh2(2r)(1 − e−2γt) when taking into account two-mode squeezed vacuum state as initial state. Additionally, it is interesting to see that the independent phase damping can lead to flip the sign of the κ4(X) of the evolved state, namely, the transition from sub-Gaussian state to super-Gaussian one. We can obtain easily this transition time

in such a way that the κ4(X) will be positive if the evolution time t > tf; otherwise, it is negative.

Fig. 3. (color online) The dynamics of κ4(X) for two-mode squeezed number state as a function of (a) the squeezing parameter at n = 10 and (b) photon numbers at r = 1.2 under the independent phase damping, where time τ = 1−e−2γt. We also present the crossover lines for the negative and positive regimes of the phase-averaged kurtosis.

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. (15) thanks to the phase-averaged technique, i.e., greatly simplifying the complexity of the complication. Third, unlike existing measures of non-Gaussianity,[6,7] the cumulant-based non-Gaussianity is aim at quantify the degree of peak of phase-space distribution for quantum state, so avoiding the selection of the exact reference Gaussian state. Finally, we analysis the dynamical difference of the phase-averaged kurtosis in two different decoherence models. As is well known, the amplitude damping channel is a quantum operation that describes the energy dissipation effects due to the loss of energy in a quantum system.[40] Such a channel is viewed as a Gaussian noise that can drive a quantum state into Gaussian thermal state in the long-time limit. Thus, it is valid for the case of two-mode squeezed number input state. While the phase damping channel describes a quantum map with loss of quantum phase information without loss of energy. It is regarded as a non-Gaussian noise, whose statistics may be generally expected away from thermal equilibrium. As a result, the dephase damping makes the system evolving towards a non-Gaussian state in the long-time limit.

4. Comparison with other measures

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

where μ[ρ] is the purity of the state ρ, κ[ρ,ρG] denotes the overlap of ρ and ρG, and ρG is a reference Gaussian state with the same vector and same covariance matrix as ρ under investigation, namely, X[ρ] = X[ρG] and σ[ρ] = σ[ρG].

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. (26) as follows:

To evaluate the non-Gaussianity of the damped state, we can write the state (20) in the two-mode Fock basis as[26]

where S(r) is a two-mode squeezing operator as before and the expansion coefficients bl are given by

According to Eqs. (33) and (34), we can calculate the purity of the damped state

and the purity of the reference Gaussian state

where

The overlap of the state under examination and the reference Gaussian state is evaluated in the phase space by means of the characteristic functions

where the characteristic functions χ[ρ](ξ1, ξ2;t) is given by Eq. (29) and since the vector is zero, the χ[ρG](ξ1,ξ2;t) of the reference Gaussian state can be written as

We have used Eqs. (35)–(40) to evaluate numerically two measures of non-Gaussianity we are interested in δHS and κ4(X) via the formulas (31) and (32) in the specific case of two-mode squeezed number subject to the independent phase damping evolution. The results are shown in Fig. 4.

Fig. 4. (color online) Time evolution of the phase-averaged kurtosis (red line) and the Hilbert-Schmidt degree of non-Gaussianity (blue line) under the independent phase damping model. The input state is a two-mode squeezed number state. Solid line: N = 2, r = 0.1; dotted line: N = 1, r = 0.6585. Throughout, the phase-averaged kurtosis is divided by a hundred for clarity.

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.

5. Conclusions

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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