CiteULike: kevina's separability

Theoretical investigations of separability and entanglement of bipartite quantum systems

P Rungta — 2007-05-24T07:52:26-00:00

Negativity as a distance from a separable state

M Khasin — 2007-05-22T16:17:51-00:00

Physical Review A (Atomic, Molecular, and Optical Physics), Vol. 75, No. 5. (2007)

The computable measure of the mixed-state entanglement, the negativity, is shown to admit a clear geometrical interpretation, when applied to Schmidt-correlated (SC) states: the negativity of a SC state equals a distance of the state from a pertinent separable state. As a consequence, the Peres-Horodecki criterion of separability is both necessary and sufficient for SC states. Another remarkable consequence is that the negativity of a SC can be estimated “at a glance” on the density matrix. These results are generalized to mixtures of SC states, which emerge in bipartite evolutions with additive integrals of motion.

"All versus Nothing" Inseparability for Two Observers

Adán Cabello — 2007-04-21T05:43:27-00:00

Physical Review Letters, Vol. 87, No. 1. (18 June 2001), 010403.

A recent proof of Bell’s theorem without inequalities [A. Cabello; Phys. Rev. Lett. 86 ; 1911 (2001)] is formulated as a Greenberger-Horne-Zeilinger–type proof involving just two observers. On one hand; this new approach allows us to derive an experimentally testable Bell inequality which is violated by quantum mechanics. On the other hand; it leads to a new state-independent proof of the Kochen-Specker theorem and provides a wider perspective on the relations between the major proofs of no hidden variables.

Separability criterion and inseparable mixed states with positive partial transposition

Pawel Horodecki — 2006-06-19T13:35:44-00:00

Physics Letters A, Vol. 232, No. 5. (4 August 1997), pp. 333-339.

It is shown that any separable state on the Hilbert space = 1 [circle times operator] 2 can be written as a convex combination of N pure product states with N 2. Then a new separability criterion for mixed states in terms of the range of the density matrix is obtained. It is used in the construction of inseparable mixed states with positive partial transposition in the case of 3 x 3 and 2 x 4 systems. The states represent an entanglement which is hidden in a more subtle way than known so far.

Separability of mixed states: necessary and sufficient conditions

Michal Horodecki — 2007-04-21T04:59:01-00:00

Physics Letters A, Vol. 223, No. 1-2. (25 November 1996), pp. 1-8.

We provide necessary and sufficient conditions for the separability of mixed states. As a result we obtain a simple criterion of the separability for 2 x 2 and 2 x 3 systems. Here, the positivity of the partial transposition of a state is necessary and sufficient for its separability. However, this is not the case in general. Some examples of mixtures which demonstrate the utility of the criterion are considered.

Volume of the set of separable states

Karol Życzkowski — 2007-04-21T04:16:34-00:00

Physical Review A, Vol. 58, No. 2. (1998), 883.

The question of how many entangled or; respectively; separable states there are in the set of all quantum states is considered. We propose a natural measure in the space of density matrices ϱ describing N -dimensional quantum systems. We prove that; under this measure; the set of separable states possesses a nonzero volume. Analytical lower and upper bounds of this volume are also derived for N =2×2 and N =2×3 cases. Finally; numerical Monte Carlo calculations allow us to estimate the volume of separable states; providing numerical evidence that it decreases exponentially with the dimension of the composite system. We have also analyzed a conditional measure of separability under the condition of fixed purity. Our results display a clear dualism between purity and separability: entanglement is typical of pure states; while separability is connected with quantum mixtures. In particular; states of sufficiently low purity are necessarily separable.

Bipartite-mixed-states of infinite-dimensional systems are generically nonseparable

Rob Clifton — 2007-03-29T01:09:19-00:00

Physical Review A, Vol. 61, No. 1. (14 December 1999), 012108.

Given a bipartite quantum system represented by a Hilbert space H 1 ⊗ H 2 ; we give an elementary argument to show that if either dim H 1 =∞ or dim H 2 =∞; then the set of nonseparable density operators on H 1 ⊗ H 2 is trace-norm dense in the set of all density operators (and the separable density operators nowhere dense). This result complements recent detailed investigations of separability; which show that when dim H i < ∞ for i =1;2; there is a separable neighborhood (perhaps very small for large dimensions) of the maximally mixed state.

Complete separability and Fourier representations of n-qubit states

Arthur Pittenger — 2007-03-28T00:12:45-00:00

Physical Review A, Vol. 62, No. 4. (2000), 042306.

Necessary conditions for separability are most easily expressed in the computational basis; while sufficient conditions are most conveniently expressed in the spin basis. We use the Hadamard matrix to define the relationship between these two bases and to emphasize its interpretation as a Fourier transform. We then prove a general sufficient condition for complete separability in terms of the spin coefficients and give necessary and sufficient conditions for the complete separability of a class of generalized Werner densities. As a further application of the theory; we give necessary and sufficient conditions for full separability for a particular set of n -qubit states whose densities all satisfy the Peres condition.

Further results on the cross norm criterion for separability

Oliver Rudolph — 2007-03-21T20:48:46-00:00

(21 Feb 2002)

In the present paper the cross norm criterion for separability of density matrices is studied. In the first part of the paper we determine the value of the greatest cross norm for Werner states, for isotropic states and for Bell diagonal states. In the second part we show that the greatest cross norm criterion induces a novel computable separability criterion for bipartite systems. This new criterion is a necessary but in general not a sufficient criterion for separability. It is shown, however, that for all pure states, for Bell diagonal states, for Werner states in dimension d=2 and for isotropic states in arbitrary dimensions the new criterion is necessary and sufficient. Moreover, it is shown that for Werner states in higher dimensions (d greater than 2), the new criterion is only necessary.

Separability properties of tripartite states with UxUxU-symmetry

T Eggeling — 2007-01-22T18:18:36-00:00

(27 Oct 2000)

We study separability properties in a 5-dimensional set of states of quantum systems composed of three subsystems of equal but arbitrary finite Hilbert space dimension d. These are the states, which can be written as linear combinations of permutation operators, or, equivalently, commute with unitaries of the form UxUxU. We compute explicitly the following subsets and their extreme points: (1) triseparable states, which are convex combinations of triple tensor products, (2) biseparable states, which are separable for a twofold partition of the system, and (3) states with positive partial transpose with respect to such a partition. Tripartite entanglement is investigated in terms of the relative entropy of tripartite entanglement and of the trace norm.

Covariance matrices and the separability problem

O Gühne — 2006-12-19T20:43:06-00:00

(29 Nov 2006)

We propose a unified approach to the separability problem which uses a representation of a quantum state by a covariance matrix of suitable observables. From the practical point of view, our approach leads to entanglement criteria that allow to detect the entanglement of many bound entangled states in higher dimensions and which are at the same time necessary and sufficient for two qubits. From a fundamental point of view, our approach leads to insights into the relations between several known entanglement criteria as well as their limitations.

Separable States Are More Disordered Globally than Locally

MA Nielsen — 2006-08-21T23:58:14-00:00

Physical Review Letters, Vol. 86, No. 22. (28 May 2001), 5184.

A remarkable feature of quantum entanglement is that an entangled state of two parties; Alice ( A ) and Bob ( B ); may be more disordered locally than globally. That is; S ( A ) > S ( A ; B ); where S (Ì) is the von Neumann entropy. It is known that satisfaction of this inequality implies that a state is nonseparable. In this paper we prove the stronger result that for separable states the vector of eigenvalues of the density matrix of system AB is majorized by the vector of eigenvalues of the density matrix of system A alone. This gives a strong sense in which a separable state is more disordered globally than locally and a new necessary condition for separability of bipartite states in arbitrary dimensions.

Separability Criterion for Density Matrices

Asher Peres — 2006-08-03T01:32:21-00:00

Physical Review Letters, Vol. 77, No. 8. (1996), 1413.

A quantum system consisting of two subsystems is separable if its density matrix can be written as ÏÂ =Â Î£ A w A Ï A â² âÏ A â²â² ; where Ï A â² and Ï A â²â² are density matrices for the two subsystems; and the positive weights w A satisfy Î£ w A Â =Â 1. In this Letter; it is proved that a necessary condition for separability is that a matrix; obtained by partial transposition of Ï; has only non-negative eigenvalues. Some examples show that this criterion is more sensitive than Bell's inequality for detecting quantum inseparability.

Local description of quantum inseparability

Anna Sanpera — 2006-07-27T20:36:27-00:00

Physical Review A, Vol. 58, No. 2. (1998), 826.

We show how to decompose any density matrix of the simplest binary composite systems; whether separable or not; in terms of only product vectors. We determine for all cases the minimal number of product vectors needed for such a decomposition. Separable states correspond to mixing from one to four pure product states. Inseparable states can be described as pseudomixtures of four or five pure product states; and can be made separable by mixing them with one or two pure product states.

Separability criterion of tripartite qubit systems

Chang Yu — 2006-07-27T18:13:19-00:00

Physical Review A (Atomic, Molecular, and Optical Physics), Vol. 72, No. 2. (2005)

In this paper, we present a method to construct full separability criterion for tripartite systems of qubits. The spirit of our approach is that a tripartite pure state can be regarded as a three-order tensor that provides an intuitionistic mathematical formulation for the full separability of pure states. We extend the definition to mixed states and give out the corresponding full separability criterion. As applications, we discuss the separability of several bound entangled states, which shows that our criterion is feasible.

The Necessary and Sufficient Conditions of Separability for Multipartite Pure States

D Li — 2006-07-27T18:11:26-00:00

(20 Apr 2006)

In this paper we present the necessary and sufficient conditions of separability for multipartite pure states. These conditions are very simple, and they don't require Schmidt decomposition or tracing out operations. We also give a necessary condition for a local unitary equivalence class for a bipartite system in terms of the determinant of the matrix of amplitudes and explore a variance as a measure of entanglement for multipartite pure states.

Separability and Distillability of Multiparticle Quantum Systems

W Dur — 2006-07-19T03:03:37-00:00

Physical Review Letters, Vol. 83, No. 17. (25 October 1999), 3562.

We present a family of 3-qubit states to which any arbitrary state can be depolarized. We fully classify those states with respect to their separability and distillability properties. This provides a sufficient condition for nonseparability and distillability for arbitrary states. We generalize our results to N -particle states.

Classification of multiqubit mixed states: Separability and distillability properties

W Dur — 2006-07-18T20:22:48-00:00

Physical Review A, Vol. 61, No. 4. (17 March 2000), 042314.

We give a complete; hierarchic classification for arbitrary multiqubit mixed states based on the separability properties of certain partitions. We introduce a family of N -qubit states to which any arbitrary state can be depolarized. This family can be viewed as the generalization of Werner states to multiqubit systems. We fully classify those states with respect to their separability and distillability properties. This provides sufficient conditions for nonseparability and distillability for arbitrary states.