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ON THE SEPARABILITY OF PURE STATES

Horia Scutaru

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

It is well know that the problem of the description of separable states of finite dimensional quantum system is central for the quantum information theory. The aim of the present paper is to point out that the conjecture of An Min Wang [1] concerning the separability of pure states is valid for pure states of bipartite quantum systems. A general condition for separability of a quantum state could be in principle obtained from the measure of entanglement. For a bipartite pure state the natural measure of entanglement is given by the von Neumann entropy of the reduced density matrices. Then, a bipartite state is separable if and only if this measure of entanglement is equal to zero, i.e., if and only if the reduced states are pure states. The purity of any state, described by a density matrix, can be verified directly using the Bloch vector, associated with that state. One of the restrictions is on the value of the squared Euclidean norm of the Bloch vector. Exactly on this restriction is based the An Min Wang criterion for the separability of a bipartite quantum state. The other restrictions concern the symmetric product of the Bloch vector with itself. We shall prove that the equations of the second set of restrictions follows from the An Min Wang restrictions and from the equations fulfilled by the Fano parameters of any pure bipartite state. In order to prove the An Min Wang conjecture we shall use three different parametrizations for the pure states of bipartite quantum systems: the generalized Bloch vector parametrization [2-10]; the Fano parametrization [3-10]; the Schmidt parametrization [11,12], and the relations between them.

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It is well know that the problem of the description of separable states of finite dimensional quantum system is central for the quantum information theory. The aim of the present paper is to point out that the conjecture of An Min Wang [1] concerning the separability of pure states is valid for pure states of bipartite quantum systems. A general condition for separability of a quantum state could be in principle obtained from the measure of entanglement. For a bipartite pure state the natural measure of entanglement is given by the von Neumann entropy of the reduced density matrices. Then, a bipartite state is separable if and only if this measure of entanglement is equal to zero, i.e., if and only if the reduced states are pure states. The purity of any state, described by a density matrix, can be verified directly using the Bloch vector, associated with that state. One of the restrictions is on the value of the squared Euclidean norm of the Bloch vector. Exactly on this restriction is based the An Min Wang criterion for the separability of a bipartite quantum state. The other restrictions concern the symmetric product of the Bloch vector with itself. We shall prove that the equations of the second set of restrictions follows from the An Min Wang restrictions and from the equations fulfilled by the Fano parameters of any pure bipartite state. In order to prove the An Min Wang conjecture we shall use three different parametrizations for the pure states of bipartite quantum systems: the generalized Bloch vector parametrization [2-10]; the Fano parametrization [3-10]; the Schmidt parametrization [11,12], and the relations between them.
The study of the entanglement properties of systems of N fermions has attracted considerable interest during the last few years. Various separability criteria for pure states of N identical fermions have been recently discussed but, except for the case of two-fermions systems, these criteria are difficult to implement and are of limited value from the practical point of view. Here we advance simple necessary and sufficient separability criteria for pure states of N identical fermions. We found that to be identified as separable, a state has to comply with one single identity involving either the purity or the von Neumann entropy of the single-particle reduced density matrix. These criteria, based on the verification of only one identity, are drastically simpler than the criteria discussed in the recent literature. We also derive two inequalities verified, respectively, by the purity and the entropy of the single-particle, reduced density matrix, which lead to natural entanglement measures for N -fermion pure states. Our present considerations are related to some classical results from the Hartree-Fock theory, which are here discussed from a different point of view in order to clarify some important points concerning the separability of fermionic pure states.
We study the separability of bipartite quantum systems in arbitrary dimensions using the Bloch representation of their density matrix. This approach enables us to find an alternative characterization of the separability problem, from which we derive a necessary condition and sufficient conditions for separability. For a certain class of states the necessary condition and a sufficient condition turn out to be equivalent, therefore yielding a necessary and sufficient condition. The proofs of the sufficient conditions are constructive, thus providing decompositions in pure product states for the states that satisfy them. We provide examples that show the ability of these conditions to detect entanglement. In particular, the necessary condition is proved to be strong enough to detect bound entangled states.
The entanglement of a pure bipartite state is uniquely measured by the von-Neumann entropy of its reduced density matrices. Though it cannot specify all the non-local characteristics of pure entangled states. It was proven that for every possible value of entanglement of a bipartite system, there exists an infinite number of equally entangled pure states, not comparable(satisfies Nielsen’s criteria) to each other. In this work, we investigate other correlation measures of pure bipartite states that are able to differentiate the quantum correlations of the states with entropy of entanglement. In Schmidt rank 3, we consider the whole set of states having same entanglement and compare how minutely such states can be distinguished by other correlation measures. Then for different values of entanglement we compare the sets of states belonging to the same entanglement and also investigate the graphs of different correlation measures. We extend our search to Schmidt rank 4 and 5 also.
Quantum entanglement has been regarded as one of the key physical resources in quantum information sciences. However, the determination of whether a mixed state is entangled or not is generally a hard issue, even for the bipartite system. In this work we propose an operational necessary and sufficient criterion for the separability of an arbitrary bipartite mixed state, by virtue of the multiplicative Horn's problem. The work follows the work initiated by Horodecki et al. and uses the Bloch vector representation introduced to the separability problem by J. De Vicente. In our criterion, a complete and finite set of inequalities to determine the separability of compound system is obtained, which may be viewed as trade-off relations between the quantumness of subsystems. We apply the obtained result to explicit examples, e.g. the separable decomposition of arbitrary dimension Werner state and isotropic state.
In this paper, we obtain an efficient separability criterion for bipartite quantum pure state systems, which is based on the two-order minors of the coefficient matrix corresponding to quantum state. Then, we generalize this criterion to multipartite arbitrarily dimensional pure states. Our criterion is directly built upon coefficient matrices, but not density matrices or observables, so it has the advantage of being computed easily. Indeed, to judge separability for an arbitrary n-partite pure state in a d-dimensional Hilbert space, it only needs at most O(d) times operations of multiplication and comparison. Our criterion can be extended to mixed states. Compared with Yu's criteria, our methods are faster, and can be applied to any quantum state.
Abstract The relation between the distillability of entanglement of three bipartite reduced density matrices from a tripartite pure state has been studied in Hayashi and Chen [2011 Phys. Rev. A 84 012325]. We extend this result to the tripartite mixed state of rank at most three. In particular we show that if the state has two bipartite reduced density operators with rank two, then the third bipartite reduced density operator additionally having non positive partial transpose (non-PPT) is distillable. In contrast, we show that the tripartite PPT state with two reduced density operators of rank two is a three-qubit fully separable state. We obtain these facts by proving a conjectured matrix inequality for the bipartite matrix M with Schmidt rank at most three. This is one of the main results of this paper. We also prove it for some M with arbitrary Schmidt rank.