In a celebrated paper ([Phys. Rev. A 58, 883 (1998)]), K. Zyczkowski, P. Horodecki, A. Sanpera,and M. Lewenstein proved for the frst time a very interesting theorem that the volume of separable quantum states is nonzero. Inspired by their ideas, we obtain a general analytical lower bound of the volume of separable states (VOSS) for arbitrary N-dimensional system. Our results give quite simple and computable suffcient conditions for separability. Moreover, for bipartite system, an upper bound of the VOSS is also presented.
The (complex) two-qubit systems comprise a 15-dimensional convex set and the real two-qubit systems, a 9-dimensional convex set. While formulas for the Hilbert-Schmidt volumes of these two sets are known -- owing to recent important work of Sommers and Zyczkowski (J. Phys. A {36}, 10115 [2003]) -- formulas have not been so far obtained for the volumes of the separable subsets. We reduce these two problems to the determination of certain functions of a single variable.
Paralleling our recent computationally-intensive work for the case N=4 (quant-ph/0308037), we undertake the task for N=6 of computing to high numerical accuracy, the formulas of Sommers and Zyczkowski (quant-ph/0304041) for the (N^2-1)-dimensional volume and (N^2-2)-dimensional hyperarea of the (separable and nonseparable) N x N density matrices, based on the Bures (minimal monotone) metric. At the same time, we estimate the unknown volumes and hyperareas based on a number of other monotone metrics of interest. Additionally, we estimate -- but with perhaps unavoidably diminished accuracy -- al…
Every choice of an orthonormal frame in the d-dimensional Hilbert space of a system corresponds to one set of all mutually commuting density matrices or, equivalently, a classical statistical state space of the system; the quantum state space itself can thus be profitably viewed as an SU(d) orbit of classical state spaces, one for each orthonormal frame. We exploit this connection to study the relative volume of separable states of a bipartite quantum system. While the two-qubit case is studied in considerable analytic detail, for higher dimensional systems we fall back on Monte Carlo. Several…
We study the stability of NPT property of an arbitrary pure entangled state under the mixture of arbitrary pure separable states. For bipartite pure states with Schmidt number $n$ $(n>1)$ which is NPT, we show that this state is still NPT when it is mixed with no more than $\frac{n(n-1)}{2}-1$ arbitrary pure separable states. This result is generalized to multipartite case.
After introducing the partially separable concept, we proved the equivalence between the partial separability of a given $m$-partite subsystem with $m$ qubits and the purity of states of this $m$-partite subsystem for a pure state in multipartite systems with arbitrary finite $n(>m)$ qubits. Furthermore, we give out the operational realizations (corollaries) of our theorem, which are the sufficient and necessary criterions of partial separability of states and can be used to classification of states. Our results are helpful to understand and describe quantum entanglement in multipartite system…
We investigate the joint (separable) numerical range of multiple measurements, i.e., the regions of expectation values accessible with (separable) quantum states for given observables. This not only enables efficient entanglement detection, but also sheds light on the geometry of the set of quantum states. More precisely, in an experiment, if the confidence region for the obtained data and the separable numerical range are disjoint, entanglement is reliably detected. Generically, the success of such an experiment is more likely the smaller the separable numerical range is compared to the stand…
Extensive numerical integration results lead us to conjecture that the silver mean, that is, s = \sqrt{2}-1 = .414214 plays a fundamental role in certain geometries (those given by monotone metrics) imposable on the 15-dimensional convex set of two-qubit systems. For example, we hypothesize that the volume of separable two-qubit states, as measured in terms of (four times) the minimal monotone or Bures metric is s/3, and 10s in terms of (four times) the Kubo-Mori monotone metric. Also, we conjecture, in terms of (four times) the Bures metric, that that part of the 14-dimensional boundary of se…
We investigate the possibility of deriving analytical formulas for the 15-dimensional separable volumes, in terms of any of a number of metrics of interest (Hilbert-Schmidt [HS], Bures,...), of the two-qubit (four-level) systems. This would appear to require 15-fold symbolic integrations over a complicated convex body (defined by both separability and feasibility constraints). The associated 15-dimensional integrands -- in terms of the Tilma-Byrd-Sudarshan Euler-angle-based parameterization of the 4 x 4 density matrices ρ(math-ph/0202002) -- would be the products of 12-dimensional Haar measure…
In this note a very crude but simple approximation to the set of separable states in an arbitrary simplex of commutative states is given using the fact that on the lines connecting the maximally mixed state and an arbitrary pure state the positivity of the partial transpose is both necessary and sufficient condition for separability of states. The lower limit to the volume of separable states in a simplex is slightly improved.
We investigate optimal separable approximations (decompositions) of states rho of bipartite quantum systems A and B of arbitrary dimensions MxN following the lines of Ref. [M. Lewenstein and A. Sanpera, Phys. Rev. Lett. 80, 2261 (1998)]. Such approximations allow to represent in an optimal way any density operator as a sum of a separable state and an entangled state of a certain form. For two qubit systems (M=N=2) the best separable approximation has a form of a mixture of a separable state and a projector onto a pure entangled state. We formulate a necessary condition that the pure state in t…
Paralleling our recent computationally-intensive (quasi-Monte Carlo) work for the case N=4 (quant-ph/0308037), we undertake the task for N=6 of computing to high numerical accuracy, the formulas of Sommers and Zyczkowski (quant-ph/0304041) for the (N^2-1)-dimensional volume and (N^2-2)-dimensional hyperarea of the (separable and nonseparable) N x N density matrices, based on the Bures (minimal monotone) metric -- and also their analogous formulas (quant-ph/0302197) for the (non-monotone) Hilbert-Schmidt metric. With the same seven billion well-distributed (``low-discrepancy'') sample points, w…
In this note we give sharp estimates on the volume of the set of separable states on N qubits. In particular, the magnitude of the "effective radius" of that set in the sense of volume is determined up to a factor which is a (small) power of N, and thus precisely on the scale of powers of its dimension. Additionally, one of the appendices contains sharp estimates (by known methods) for the expected values of norms of the GUE random matrices. We employ standard tools of classical convexity, high-dimensional probability and geometry of Banach spaces.
The problem of of how many entangled or, respectively, separable states there are in the set of all quantum states is investigated. We study to what extent the choice of a measure in the space of density matrices describing N--dimensional quantum systems affects the results obtained. We demonstrate that the link between the purity of the mixed states and the probability of entanglement is not sensitive to the measure chosen. Since the criterion of partial transposition is not sufficient to distinguish all separable states for N > 6, we develop an efficient algorithm to calculate numerically th…
We obtain two sided estimates for the Bures volume of an arbitrary subset of the set of $N\times N$ density matrices, in terms of the Hilbert-Schmidt volume of that subset. For general subsets, our results are essentially optimal (for large $N$). As applications, we derive in particular nontrivial lower and upper bounds for the Bures volume of sets of separable states and for sets of states with positive partial transpose. PACS numbers: 02.40.Ft, 03.65.Db, 03.65.Ud, 03.67.Mn
This note deals with estimating the volume of the set of separable mixed quantum states when the dimension of the state space grows to infinity. This has been studied recently for qubits; here we consider larger particles and conclude that, in all cases, the proportion of the states that are separable is super-exponentially small in the dimension of the set. We also show that the partial transpose criterion becomes imprecise when the dimension increases, and that the lower bound $6^{-N/2}$ on the (Hilbert-Schmidt) inradius of the set of separable states on N qubits obtained recently by Gurvits…
In this paper we show an explicit construction of multipartite class of entangled states with the PPT (Positive Partial Transposition) property in every cut. We investigate properties of this class of states focusing on the trace distance from the set of separable states. We provide an explicit sub-class of the multipartite entangled PPT states which are arbitrary far from the set of separable states. We argue, that in the multipartite case the mentioned distance increases with dimension of the local Hilbert space. In our construction is we do not have to use many copies of initial state livin…
In this paper, we show that an arbitrary separable state can be the output of a certain entanglement-breaking channel corresponding exactly to the input of a maximally entangled state. A necessary and sufficient separability criterion and some sufficient separability criteria from entanglement-breaking channels are given.
We introduce an operational procedure to determine, with arbitrary probability and accuracy, optimal entanglement witness for every multipartite entangled state. This method provides an operational criterion for separability which is asymptotically necessary and sufficient. Our results are also generalized to detect all different types of multipartite entanglement.
A natural measure in the space of density matrices describing N-dimensional quantum systems is proposed. We study the probability P that a quantum state chosen randomly with respect to the natural measure is not entangled (is separable). We find analytical lower and upper bounds for this quantity. Numerical calculations give P = 0.632 for N=4 and P=0.384 for N=6, and indicate that P decreases exponentially with N. Analysis of a conditional measure of separability under the condition of fixed purity shows a clear dualism between purity and separability: entanglement is typical for pure states, …
We use the linear programming algorithm introduced by Akulin et al. [V. M. Akulin, G. A. Kabatiansky, and A. Mandilara, Phys. Rev. A 92, 042322 (2015)] to perform best separable approximation on two-qutrit random density matrices. We combine the numerical results with theoretical methods in order to generate random representative families of positive partial transposed bound entangled (BE) states and analyze their properties. Our results are disclosing that for the two-qutrit system the BE states have negligible volume and that these form tiny `islands' sporadically distributed over the surfac…
A lower bound on the amount of noise that must be added to a GHZ-like entangled state to make it separable (also called the random robustness) is found using the transposition condition. The bound is applicable to arbitrary numbers of subsystems, and dimensions of Hilbert space, and is shown to be exact for qubits. The new bound is compared to previous such bounds on this quantity, and found to be stronger in all cases. It implies that increasing the number of subsystems, rather than increasing their Hilbert space dimension is a more effective way of increasing entanglement. An explicit decomp…
In a previous study (quant-ph/9911058), several remarkably simple exact results were found, in certain specialized m-dimensional scenarios (m<5), for the a priori probability that a pair of qubits is unentangled/separable. The measure used was the volume element of the Bures metric (identically one-fourth the statistical distinguishability [SD] metric). Here, making use of a newly-developed (Euler angle) parameterization of the 4 x 4 density matrices (math-ph/0202002), we extend the analysis to the complete 15-dimensional convex set (C) of arbitrarily paired qubits -- the total SD volume of wh…
We pursue a number of analytical directions, motivated to some extent initially by the possibility of developing a methodology for formally proving or disproving a certain conjecture of quantum-theoretical relevance (quant-ph/0308037). It asserts that the 15-dimensional volume occupied by the separable two-qubit density matrices is (\sqrt{2}-1)/3, as measured in terms of the statistical distinguishability metric (four times the Bures or minimal monotone metric). Somewhat disappointingly, however, the several various analyses that we report, though we hope of independent/autonomous interest, ap…
In this paper, we treat an asymptotic hypothesis testing (or state discrimination with asymmetric treatment of errors) between an arbitrary fixed bipartite pure state and the completely mixed state by one-way LOCC, two-way LOCC, and separable POVMs. As a result, we derive single-letterized formulas for the Stein's lemma type of optimal error exponents under one-way LOCC, two-way LOCC and separable POVMs, the Chernoff bounds under one-way LOCC POVMs and separable POVMs, and the Hoeffding bounds under one-way LOCC POVMs in the whole region of a parameter and under separable POVMs on a restricted…
We compare the classification as entangled or separable of Bell diagonal bipartite qudits with positive partial transposition (PPT) and their properties for different dimensions. For dimension $d \geq 3$, a form of entanglement exists that is hard to detect and called bound entanglement due to the fact that such entangled states cannot be used for entanglement distillation. Up to this date, no efficient solution is known to differentiate bound entangled from separable states. We address and compare this problem named separability problem for a family of bipartite Bell diagonal qudits with spec…
Universal two-particle entanglement processes are analyzed in arbitrary dimensional Hilbert spaces. On the basis of this analysis the class of possible optimal universal entanglement processes is determined whose resulting output states do not contain any separable states. It is shown that these processes form a one-parameter family. For all Hilbert space dimensions larger than two the resulting optimally entangled output states are mixtures of anti-symmetric states which are freely entangled and which also preserve information about input states. Within this one-parameter family there is only…
We derive Bohm's trajectories from Bell's beables for arbitrary bipartite systems composed by dissipative noninteracting harmonic oscillators at finite temperature. As an application of our result, we calculate the Bohmian trajectories of particles described by a generalized Werner state, comparing the trajectories when the sate is either separable or entangled. We show that qualitative differences appear in the trajectories for entangled states as compared with those for separable states.
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 …
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…
Employing the volume of quantum steering ellipsoids (QSEs) as a measure on the fifteen-dimensional convex set of two-qubit states, we estimate the ratio of the integral of the measure over the separable states to its integral over all (separable and entangled) states to be 0.0288. This can be contrasted with the considerably larger separability ratios (probabilities) of $\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424$ and $\frac{25}{341}=\frac{5^2}{11 \cdot 31} \approx 0.0733138$ that various forms of evidence point to with the use of the prominent Hilbert-Schmidt and Bures measures, r…
This paper aims to study the $\a$-volume of $\cK$, an arbitrary subset of the set of $N\times N$ density matrices. The $\a$-volume is a generalization of the Hilbert-Schmidt volume and the volume induced by partial trace. We obtain two-side estimates for the $\a$-volume of $\cK$ in terms of its Hilbert-Schmidt volume. The analogous estimates between the Bures volume and the $\a$-volume are also established. We employ our results to obtain bounds for the $\a$-volume of the sets of separable quantum states and of states with positive partial transpose (PPT). Hence, our asymptotic results provide…
