We explore separability of bipartite divisions of mixed Gaussian states based on the positivity of the Abe-Rajagopal (AR) q-conditional entropy. The AR q-conditional entropic characterization provide more stringent restrictions on separability (in the limit q tending to infinity) than that obtained from the corresponding von Neumann conditional entropy (q = 1 case)--similar to the situation in finite dimensional states. Effectiveness of this approach, in relation to the results obtained by partial transpose criterion, is explicitly analyzed in three illustrative examples of two-mode Gaussian states of physical significance.
We discuss the entropic criterion for separability of compound quantum systems for general non-additive entropic forms based on arbitrary concave functions $f$. For any separable state, the generalized entropy of the whole system is shown to be not smaller than that of the subsystems, for any choice of $f$, providing thus a necessary criterion for separability. Nevertheless, the criterion is not sufficient and examples of entangled states with the same property are provided. This entails, in particular, that the conjecture about the positivity of the conditional Tsallis entropy for all $q$, a …
Beyond the standard entropic inequalities: stronger scalar separability criteria and their applications
最近，它表明，如果一个给定的状态满足约化准则，它也必须满足已知的熵不等式。现在的问题是，假设满足基于正映射但不完全正映射的更强准则，是否有可能导出一些比熵不等式更强的标量不等式。在本文中，我们证明了在某些假设下，扩展的约化准则[H.- P. Breuer，Phys. Rev. Lett 97，080501（2006）; W. Hall，J.Phys.A40，6183（2007）]导致了一些类似熵的不等式，这些不等式比它们的熵对应物强得多。与其他可分离性判据的比较表明，这种方法可能会导致强标量判据检测可分离纠缠和束缚纠缠。特别地，在SO（3）不变态的情况下，它表明，目前的不等式检测纠缠的区域中的纠缠证人的基础上扩展的约简映射失败。还应该强调的是，在$2\otimes N$状态的情况下，导出的不等式有效地检测纠缠，而扩展的约简映射在作用于量子比特子系统时是无用的。此外，有一个自然的方式来构建一个多副本纠缠证人的基础上导出的不等式，所以，在原则上，有实验实现的可能性。一些开放的问题和进一步研究的可能性进行了概述。
Separability conditions for a bipartite quantum system of finite-dimensional subsystems are formulated in terms of Rényi and Tsallis entropies. Entropic uncertainty relations often lead to entanglement criteria. We propose new approach based on the convolution of discrete probability distributions. Measurements on a total system are constructed of local ones according to the convolution scheme. Separability conditions are derived on the base of uncertainty relations of the Maassen-Uffink type as well as majorization relations. On each of subsystems, we use a pair of sets of subnormalized vecto…
In this paper, we give Separability criterion for the multi-mode Gaussian states using the Marchenko-Pastur theorem. We show that the Marchenko-Pastur theorem from random matrix theory as necessary and sufficient condition for separability of multimode Gaussian states.
We investigate the separability of the two-mode Gaussian states by using the variances of a pair of Einstein-Podolsky-Rosen (EPR)-like observables. Our starting point is inspired by the general necessary condition of separability introduced by Duan {\em et al.} [Phys. Rev. Lett. {\bf 84}, 2722 (2000)]. We evaluate the minima of the normalized forms of both the product and sum of such variances, as well as that of a regularized sum. Making use of Simon's separability criterion, which is based on the condition of positivity of the partial transpose (PPT) of the density matrix [Phys. Rev. Lett. {…
Currently available separability criteria for continuous-variable states are generally based on the covariance matrix of quadrature operators. The well-known separability criterion of Duan et al. [Phys. Rev. Lett. 84, 2722 (2000)] and Simon [Phys. Rev. Lett. 84, 2726 (2000)] , for example, gives a necessary and sufficient condition for a two-mode Gaussian state to be separable, but leaves many entangled non-Gaussian states undetected. Here, we introduce an improvement of this criterion that enables a stronger entanglement detection. The improved condition is based on the knowledge of an additi…
Gaussian distribution of a quantum state with continuous spectrum is known to maximize the Shannon entropy at a fixed variance. Applying it to a pair of canonically conjugate quantum observables $\hat x$ and $\hat p$, quantum entropic uncertainty relation can take a suggestive form, where the standard deviations $σ_x$ and $σ_p$ are featured explicitly. From the construction, it follows in a transparent manner that: (i) the entropic uncertainty relation implies the Kennard-Robertson uncertainty relation in a modifed form, $σ_xσ_p\geq\hbar e^{\cal N}/2$; (ii) the additional factor ${\cal N}$ qua…
We present a novel approach to the separability problem for Gaussian quantum states of bosonic continuous variable systems. We derive a simplified necessary and sufficient separability criterion for arbitrary Gaussian states of $m$ vs $n$ modes, which relies on convex optimisation over marginal covariance matrices on one subsystem only. We further revisit the currently known results stating the equivalence between separability and positive partial transposition (PPT) for specific classes of Gaussian states. Using matrix analysis techniques such as Schur complements and matrix means, we provide…
We prove, using symplectic methods and The Wigner formalism, a refinement of a criterion due to Werner and Wolf for the separability of bipartite Gaussian mixed states in an arbitrary number of dimensions. We use our result to show that one can characterize separability by comparing these states with separable pure Gaussian states.
The notion of partial trace of a density operator is essential for the understanding of the entanglement and separability properties of quantum states. In this paper we investigate these notions putting an emphasis on the geometrical properties of the covariance ellipsoids of the reduced states. We thereafter focus on Gaussian states and we give new and easily numerically implementable sufficient conditions for the separability of all Gaussian states. Unlike the positive partial transposition criterion, none of these conditions is however necessary.
The Tsallis relative entropy $S_q (\hatρ,\hatσ)$ measures the distance between two arbitrary density matrices $\hatρ$ and $\hatσ$. In this work the approximation to this quantity when $q=1+δ$ ($δ\ll 1$) is obtained. It is shown that the resulting series is equal to the von Neumann relative entropy when $δ=0$. Analyzing the von Neumann relative entropy for arbitrary $\hatρ$ and a thermal equilibrium state $\hatσ=e^{- β\hat{H}}/{\rm Tr}(e^{- β\hat{H}})$ is possible to define a new inequality relating the energy, the entropy, and the partition function of the system. From this inequality, a param…
Motivated by the notions of $k$-extendability and complete extendability of the state of a finite level quantum system as described by Doherty et al (Phys. Rev. A, 69:022308), we introduce parallel definitions in the context of Gaussian states and using only properties of their covariance matrices derive necessary and sufficient conditions for their complete extendability. It turns out that the complete extendability property is equivalent to the separability property of a bipartite Gaussian state. Following the proof of quantum de Finetti theorem as outlined in Hudson and Moody (Z. Wahrschein…
We present a numerical method for solving the separability problem of Gaussian quantum states in continuous-variable quantum systems. We show that the separability problem can be cast as an equivalent problem of determining the feasibility of a set of linear matrix inequalities. Thus, it can be efficiently solved using existent numerical solvers. We apply this method to the identification of bound entangled Gaussian states. We show that the proposed method can be used to identify bound entangled Gaussian states that could be simple enough to be producible in quantum optics.
We have derived a general separability criterion for a class of two mode non-Gaussian continuous variable systems, obtained earlier using PPT, violation of which provides sufficient condition for entanglement. It has been obtained by utilizing the Cauchy-Schwarz inequality and from the basic definition of separable states. This criterion coincides with the work of Agarwal and Biswas [4] which involved inequality involving higher order correlation, for testing entanglement in non-Gaussian states.
We show that a proper expression of the uncertainty relation for a pair of canonically-conjugate continuous variables relies on entropy power, a standard notion in Shannon information theory for real-valued signals. The resulting entropy-power uncertainty relation is equivalent to the entropic formulation of the uncertainty relation due to Bialynicki-Birula and Mycielski, but can be further extended to rotated variables. Hence, based on a reasonable assumption, we give a partial proof of a tighter form of the entropy-power uncertainty relation taking correlations into account and provide exten…
We derive a hierarchy of separability criteria for multi-mode continuous variable systems. They permit to study in a unified way the k-partite entanglement of broad classes of Gaussian and non- Gaussian states. With specific examples we demonstrate the strength of the criteria, and, we discuss their assessment based on data obtained from Gaussian measurements.
Quantumness and separability criteria for continuous variable systems are discussed for the case of a noncommutative (NC) phase-space. In particular, the quantum nature and the entanglement configuration of NC two-mode Gaussian states are examined. Two families of covariance matrices describing standard quantum mechanics (QM) separable states are deformed into a NC QM configuration and then investigated through the positive partial transpose criterium for identifying quantum entanglement. It is shown that the entanglement of Gaussian states may be exclusively induced by switching on the NC def…
We survey the state of the art for the proof of the quantum Gaussian optimizer conjectures of quantum information theory. These fundamental conjectures state that quantum Gaussian input states are the solution to several optimization problems involving quantum Gaussian channels. These problems are the quantum counterpart of three fundamental results of functional analysis and probability: the Entropy Power Inequality, the sharp Young's inequality for convolutions, and the theorem "Gaussian kernels have only Gaussian maximizers." Quantum Gaussian channels play a key role in quantum communicatio…
