Xiao-Hong Wang, Shao-Ming Fei
By using the "subtracting projectors" method in proving the separability of PPT states on multiple quantum spaces, we derive a canonical form of PPT states in ${\Cb}^{K_1} \otimes {\Cb}^{K_2} \otimes ... \otimes {\Cb}^{K_m} \otimes {\Cb}^N$ composite quantum systems with rank $N$, from which a sufficient separability condition for these states is presented.
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We investigate the canonical forms of positive partial transposition (PPT) density matrices in ${\cal C}^2 \otimes {\cal C}^M \otimes {\cal C}^N$ composite quantum systems with rank $N$. A general expression for these PPT states are explicitly obtained. From this canonical form a sufficient separability condition is presented.
We give a canonical form of PPT states in ${\cal C}^2 \otimes {\cal C}^2\otimes {\cal C}^2 \otimes {\cal C}^N$ with rank=$N$. From this canonical form a necessary separable condition for these states is presented.
We study the separability problem in mixtures of Dicke states i.e., the separability of the so-called Diagonal Symmetric (DS) states. First, we show that separability in the case of DS in $C^d\otimes C^d$ (symmetric qudits) can be reformulated as a quadratic conic optimization problem. This connection allows us to exchange concepts and ideas between quantum information and this field of mathematics. For instance, copositive matrices can be understood as indecomposable entanglement witnesses for DS states. As a consequence, we show that positivity of the partial transposition (PPT) is sufficien…
Positivity of the density operator reflects itself in terms of sequences of inequalities on observable moments. Uncertainty relations for non-commuting observables form a subset of these inequalities. In addition, criterion of positivity under partial transposition (PPT) imposes distinct bounds on moments, violations of which signal entanglement. We present bounds on some novel sets of composite moments, consequent to positive partial transposition of the density operator and report their violation by entangled multiqubit states. In particular, we derive separability bounds on a multiqubit mom…
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. {…
Nonequivalence between absolute separability and positive partial transposition in the symmetric subspace
本研究针对量子纠缠理论中绝对可分态与绝对PPT态等价性的开放问题，探讨对称多量子比特态的情况，通过构造纠缠的五量子比特及更大奇数个量子比特的SAPPT态反例，证明SAPPT态并非总是对称绝对可分。
We study quantum states for which the PPT criterion is both sufficient and necessary for separability. We present a class of 3x3 bipartite mixed states and show that these states are separable if and only if they are PPT.
In a series of papers with Kossakowski, the first author has examined properties of densities for which the positive partial transpositrionm (PPT) property can be readily checked. These densities were also investigated from a different perspective by Baumgartner, Hiesmayr and Narnhofer. In this paper we show how the support of such densities can be expressed in terms of lines in a finite geometry and how that same structure lends itself to checking the necessary PPT condition and to a novel sufficient condition for separability.
We study the general representations of positive partial transpose (PPT) states in ${\cal C}^K \otimes {\cal C}^M \otimes {\cal C}^N$. For the PPT states with rank-$N$ a canonical form is obtained, from which a sufficient separability condition is presented.
We consider low rank density operators $\varrho$ supported on a $M\times N$ Hilbert space for arbitrary $M$ and $N$ ($M\leq N$) and with a positive partial transpose (PPT) $\varrho^{T_A}\ge 0$. For rank $r(\varrho) \leq N$ we prove that having a PPT is necessary and sufficient for $\varrho$ to be separable; in this case we also provide its minimal decomposition in terms of pure product states. It follows from this result that there is no rank 3 bound entangled states having a PPT. We also present a necessary and sufficient condition for the separability of generic density matrices for which th…
With a probability of success of $95 \%$ we solve the separability problem for Bell diagonal qutrit states with positive partial transposition (PPT). The separability problem, i.e. distinguishing separable and entangled states, generally lacks an efficient solution due to the existence of bound entangled states. In contrast to free entangled states that can be used for entanglement distillation via local operations and classical communication, these states cannot be detected by the Peres-Horodecki criterion or PPT criterion. We analyze a large family of bipartite qutrit states that can be sepa…
The computable cross norm (CCN) criterion is a new powerful analytical and computable separability criterion for bipartite quantum states, that is also known to systematically detect bound entanglement. In certain aspects this criterion complements the well-known Peres positive partial transpose (PPT) criterion. In the present paper we study important analytical properties of the CCN criterion. We show that in contrast to the PPT criterion it is not sufficient in dimension 2 x 2. In higher dimensions we prove theorems connecting the fidelity of a quantum state with the CCN criterion. We also a…
Jakobczyk and Siennicki studied two-dimensional sections of a set of (generalized) Bloch vectors corresponding to n x n density matrices of two-qubit systems (that is, the case n = 4). They found essentially five different types of (nontrivial) separability regimes. We compute the Euclidean/Hilbert-Schmidt (HS) separability probabilities assigned to these regimes, and conduct parallel two-dimensional sectional analyses for the higher-level cases n=6,8,9 and 10. Making use of the newly-introduced capability for integration over implicitly defined regions of version 5.1 of Mathematica -- also fr…
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…
In this paper, we present new progress on the study of the symmetric extension criterion for separability. First, we show that a perturbation of order O(1/N) is sufficient and, in general, necessary to destroy the entanglement of any state admitting an N Bose symmetric extension. On the other hand, the minimum amount of local noise necessary to induce separability on states arising from N Bose symmetric extensions with Positive Partial Transpose (PPT) decreases at least as fast as O(1/N^2). From these results, we derive upper bounds on the time and space complexity of the weak membership probl…
We study certain quantum states for which the PPT criterion is both sufficient and necessary for separability. A class of $n\times n$ bipartite mixed states is presented and the conditions of PPT for these states are derived. The separable pure state decompositions of these states are explicitly constructed when they are PPT.
We generalize the definition of strong positive partial transpose (SPPT) to the multipartite system. The tripartite case was first considered by X.-Y. Yu and H. Zhao [ Int. J. Theor. Phys.,54, 292, (2015)]. In this extension, unfortunately, desired properties such as the PPT of SPPT states the separability of super and pure SPPT states are not preserved. In contrast, this paper provides an alternative generalization to multipartite cases with these properties preserved. We also provide sufficient conditions for the separability of SPPT states.
Entanglement as a vital resource for information processing can be described by special properties of the quantum state. Using the well-known Weyl basis we propose a new Bloch decomposition of the quantum state and study its separability problem. This decomposition enables us to find an alternative characterization of the separability based on the correlation matrix. We shaw that the criterion is effective in detecting entanglement for the isotropic states, Bell-diagonal states and some PPT entangled states. We also use the Weyl operators to construct an detecting operator for quantum teleport…
One of the most important problems in quantum information is the separability problem, which asks whether a given quantum state is separable. We investigate multipartite states of rank at most four which are PPT (i.e., all their partial transposes are positive semidefinite). We show that any PPT state of rank two or three is separable and has length at most four. For separable states of rank four, we show that they have length at most six. It is six only for some qubit-qutrit or multiqubit states. It turns out that any PPT entangled state of rank four is necessarily supported on a 3x3 or a 2x2…
Milz and Strunz ({\it J. Phys. A}: {\bf{48}} [2015] 035306) recently studied the probabilities that two-qubit and qubit-qutrit states, randomly generated with respect to Hilbert-Schmidt (Euclidean/flat) measure, are separable. They concluded that in both cases, the separability probabilities (apparently exactly $\frac{8}{33}$ in the two-qubit scenario) hold {\it constant} over the Bloch radii ($r$) of the single-qubit subsystems, jumping to 1 at the pure state boundaries ($r=1$). Here, firstly, we present evidence that in the qubit-qutrit case, the separability probability is uniformly distrib…
We employ a quasirandom methodology, recently developed by Martin Roberts, to estimate the separability probabilities, with respect to the Bures (minimal monotone/statistical distinguishability) measure, of generic two-qubit and two-rebit states. This procedure, based on generalized properties of the golden ratio, yielded, in the course of almost seventeen billion iterations (recorded at intervals of five million), two-qubit estimates repeatedly close to nine decimal places to $\frac{25}{341} =\frac{5^2}{11 \cdot 31} \approx 0.073313783$. Howeer, despite the use of over twenty-three billion it…
An important variable in the 2017 analysis of Lovas and Andai, formally establishing the Hilbert-Schmidt separability probability conjectured by Slater of $\frac{29}{64}$ for the 9-dimensional convex set of two-rebit density matrices, was the ratio ($\varepsilon =\frac{σ_2}{σ_1}$) of the two singular values ($σ_1 \geq σ_2 \geq 0$) of $D_2^{\frac{1}{2}} D_1^{-\frac{1}{2}}$. There, $D_1$ and $D_2$ were the diagonal $2 \times 2$ blocks of a $4 \times 4$ two-rebit density matrix $ρ$. Working within the Lovas-Andai "separability function" ($\tildeχ_d(\varepsilon)$) framework, Slater was able to ver…
We begin by seeking the qubit-qutrit and rebit-retrit counterparts to the now well-established Hilbert-Schmidt separability probabilities for (the 15-dimensional convex set of) two-qubits of $\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424$ and (the 9-dimensional) two-rebits of $\frac{29}{64} =\frac{29}{2^6} \approx 0.453125$. Based in part on extensive numerical computations, we advance the possibilities of a qubit-qutrit value of $\frac{27}{1000} = (\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = 0.027$ and a rebit-retrit one of $\frac{860}{6561} =\frac{2^2 \cdot 5 \cdot 43}{3^8} \approx…
Internal Boundary between Entanglement and Separability Within a Quantum State
该研究提出量子态的纠缠/可分离性由态内阈值决定，引入最优可分离态概念，证明两体态可分解为最优纠缠态与最优可分离态的凸混合，并给出求解BSA等分解的可操作算法，解决了高维BSA计算的开放问题。
We propose an interpretation of quantum separability based on a physical principle: local time reversal. It immediately leads to a simple characterization of separable quantum states that reproduces results known to hold for binary composite systems and which thereby is complete for low dimensions. We then describe a constructive algorithm for finding the canonical decomposition of separable and non separable mixed states of dimensions 2x2 and 2x3.
We give a new separability criterion, a necessary condition for separability of $N$-partite quantum states. The criterion is based on the Bloch representation of a $N$-partite quantum state and makes use of multilinear algebra, in particular, the matrization of tensors. Our criterion applies to {\it arbitrary} $N$-partite quantum states in $\mathcal{H}=\mathcal{H}^{d_1}\otimes \mathcal{H}^{d_2} \otimes ... \otimes \mathcal{H}^{d_N}.$ The criterion can test whether a $N$-partite state is entangled and can be applied to different partitions of the $N$-partite system. We provide examples that sho…
The problem of determining whether a given quantum state is entangled lies at the heart of quantum information processing, which is known to be an NP-hard problem in general. Despite the proposed many methods such as the positive partial transpose (PPT) criterion and the k-symmetric extendibility criterion to tackle this problem in practice, none of them enables a general, effective solution to the problem even for small dimensions. Explicitly, separable states form a high-dimensional convex set, which exhibits a vastly complicated structure. In this work, we build a new separability-entanglem…
Sufficient criteria for absolute separability in arbitrary dimensions via linear map inverses
本文利用线性映射及其逆提出任意维数量子态绝对可分性的新充分判据与极点，并结合凸几何优化及绝对PPT分析，将结果推广至多体系统。
In this work, new types of EWs are introduced. They are constructed by using real skew-symmetric operators defined on a single party subsystem of a bipartite dxd system and a maximal entangled state in that system. A canonical form for these witnesses is proposed which is called canonical EW in corresponding to canonical real skew-symmetric operator. Also for each possible partition of the canonical real skew-symmetric operator corresponding EW is obtained. The method used for dxd case is extended to d1xd2 systems. It is shown that there exist Cd2xd1 distinct possibilities to construct EWs for…
We analyze a class of positive partial transpose states (PPT) such that the positivity of its partial transposition is recognized with respect to canonical factorization of the original density operator (Cholesky block decomposition). We call such PPT states strong PPT states (SPPT). This property, contrary to PPT, is basis dependent. It is shown that there exists a proper subset of SPPT states which are separable and provide a separable decomposition for any of these states.
We address an open question about the existence of entangled continuous-variable (CV) Werner states with positive partial transpose (PPT). We prove that no such state exists by showing that all PPT CV Werner states are separable. The separability follows by observing that these CV Werner states can be approximated by truncating the states into a finite-dimensional convex mixture of product states. In addition, the constituents of the product states comprise a generalized non-Gaussian measurement which gives, rather surprisingly, a strictly tighter upper bound on quantum discord than photon cou…
It is known that he bipartite quantum states, with rank strictly smaller than the maximum of the ranks of its two reduced states, are distillable by local operations and classical communication. Our first main result is that this is also true for NPT states with rank equal to this maximum. (A state is PPT if the partial transpose of its density matrix is positive semidefinite, and otherwise it is NPT.) This was conjectured first in 1999 in the special case when the two local ranks are equal. Our second main result provides a complete solution of the separability problem for bipartite states of…
Bipartite states with vanishing quantum discord are necessarily separable and hence positive partial transpose (PPT). We show that 2 x N states satisfy additional property: the positivity of their partial transposition is recognized with respect to the canonical factorization of the original density operator. We call such states SPPT (for strong PPT). Therefore, we provide a natural witness for a quantum discord: if a 2 x N state is not SPPT it must contain nonclassical correlations measured by quantum discord. It is an analog of the celebrated Peres-Horodecki criterion: if a state is not PPT …
Recently, in [1], the author proved that many results that are true for PPT matrices also hold for another class of matrices with a certain symmetry in their Hermitian Schmidt decompositions. These matrices were called SPC in [1] (definition 1.1). Before that, in [9], Tóth and Gühne proved that if a state is symmetric then it is PPT if and only if it is SPC. A natural question appeared: What is the connection between SPC matrices and PPT matrices? Is every SPC matrix PPT? Here we show that every SPC matrix is PPT in $M_2\otimes M_2$ (theorem 4.3). This theorem is a consequence of the fact that…
We construct a parameterized family of $n\otimes n$ PPT (positive partial transpose) states of corank one for each $n\ge 3$. With a suitable choice of parameters, we show that they are $n\otimes n$ PPT entangled edge states of corank one for $3\le n\le 1000$. They violate the range criterion for separability in the most extreme way. Note that corank one is the smallest possible corank for such states. The corank of the partial transpose is given by $2n-3$, which is also the smallest possible corank for the partial transposes of PPT entangled edge states of corank one. They provide the first ex…
We construct a large class of bipartite M x N quantum states which defines a proper subset of states with positive partial transposes (PPT). Any state from this class is PPT but the positivity of its partial transposition is recognized with respect to canonical factorization of the original density operator. We propose to call elements from this class states with strong positive partial transposes (SPPT). We conjecture that all SPPT states are separable.
Entanglement in cyclic sign invariant quantum states
研究了循环符号不变量子态的纠缠特性，展示了其在不同维度下的分离性和PPT性质，构建了新的纠缠见证并推广了相关理论。
We investigate the structure of SO(3)-invariant quantum systems which are composed of two particles with spins j_1 and j_2. The states of the composite spin system are represented by means of two complete sets of rotationally invariant operators, namely by the projections P_J onto the eigenspaces of the total angular momentum J, and by certain invariant operators Q_K which are built out of spherical tensor operators of rank K. It is shown that these representations are connected by an orthogonal matrix whose elements are expressible in terms of Wigner's 6-j symbols. The operation of the partia…
We study entanglement-related properties of random quantum states which are unitarily invariant, in the sense that their distribution is left unchanged by conjugation with arbitrary unitary operators. In the large matrix size limit, the distribution of these random quantum states is characterized by their limiting spectrum, a compactly supported probability distribution. We prove several results characterizing entanglement and the PPT property of random bipartite unitarily invariant quantum states in terms of the limiting spectral distribution, in the unbalanced asymptotical regime where one o…
Let E' denote the set of non-normalized two-qutrit entangled states of rank four having positive partial transpose (PPT). We show that the set of SLOCC equivalence classes of states in E', equipped with the quotient topology, is homeomorphic to the quotient R/A_5 of the open rectangular box R in the Euclidean space R^4 by an action of the alternating group A_5. We construct an explicit map omega: Omega -> E', where Omega is the open positive orthant in R^4, whose image meets every SLOCC equivalence class E containeed in E'. Although the intersection of the image of omega and E is not necessari…
Power symmetric matrices defned and studied by R. Sinkhorn (1981) and their generalization by R.B. Bapat, S.K. Jain and K. Manjunatha Prasad (1999) have been utilized to give positive block matrices with trace one possessing positive partial transpose, the so-called PPT states. Another method to construct such PPT states is given, it uses the form of a matrix unitarily equivalent to its transpose obtained by S.R. Garcia and J.E. Tener (2012). Evolvement or suppression of separability or entanglement of various levels for a quantum dynamical semigroup of completely positive maps has been studie…
General physical background of Peres-Horodecki positive partial transpose (ppt-) separability criterion is revealed. Especially, the physical sense of partial transpose operation is shown to be equivalent to the "local causality reversal" (LCR-) procedure for all separable quantum systems or to the uncertainty in a global time arrow direction in all entangled cases. Using these universal causal considerations the heuristic causal separability criterion has been proposed for arbitrary $ D^{N} \times D^{N}$ density matrices acting in $ \mathcal{H}_{D}^{\otimes N} $ Hilbert spaces which describe …
Let K, K' be convex cones residing in finite-dimensional real vector spaces E, E'. An element in the tensor product E \otimes E' is K \otimes K'-separable if it can be represented as finite sum \sum_l x_l \otimes x'_l with x_l \in K and x_l' \in K' for all l. Let S(n), H(n), Q(n) be the spaces of n x n real symmetric, complex hermitian and quaternionic hermitian matrices, respectively. Let further S_+(n), H_+(n), Q_+(n) be the cones of positive semidefinite matrices in these spaces. If a matrix in H(mn) = H(m) \otimes H(n) is H_+(m) \otimes H_+(n)-separable, then it fulfills also the so-called…
A hierarchy of multimode uncertainty relations on the second moments of n pairs of canonical operators is derived in terms of quantities invariant under linear canonical (i.e. symplectic) transformations. Conditions for the separability of multimode continuous variable states are derived from the uncertainty relations, generalizing the inequalities obtained in [Phys. Rev. Lett. 96, 110402 (2006)] to states with some transposed symplectic eigenvalues equal to 1. Finally, to illustrate the methodology proposed for the detection of continuous variable entanglement, the separability of multimode n…
On the entanglement induced by the deformation of phase-space
本文研究非对易相空间形变对纠缠的影响，将可分离性PPT判据推广到Bopp平移，发现形变参数诱导纠缠，并应用于各向异性二维谐振子，提出验证相空间非对易性的思想实验。
Tuning the separability in noncommutative space
本文通过 Bopp 移位与 Lewis-Riesenfeld 不变量方法，将含时非对易两体高斯系统转化为对易空间形式，并利用 Simon 条件证明 NC 形变及振子参数可调控态的可分性。
We present a description of entanglement in composite quantum systems in terms of symplectic geometry. We provide a symplectic characterization of sets of equally entangled states as orbits of group actions in the space of states. In particular, using Kostant-Sternberg theorem, we show that separable states form a unique Kaehler orbit, whereas orbits of entanglement states are characterized by different degrees of degeneracy of the canonical symplectic form on the complex projective space. The degree of degeneracy may be thus used as a new geometric measure of entanglement and we show how to c…
Spacetime quantum and classical mechanics with dynamical foliation
本研究通过将勒让德变换的时间选择动力学化、扩展泊松括号为时空对称形式，构建时空相空间，发展出带量子叶状结构的协变经典与量子力学框架，并通过时空关联子建立其与传统量子力学的对应，提出因果结构从系统-环境量子关联中涌现的诠释。
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