To find whether a set of reduced density matrixes come from a common multi-party state is a hard and important problem. In this paper, (1) we introduce a method to find out some polytopes in one-party eigenvalue-space which are sufficient conditions of this problem. (2) We point out that there are some relations between the compatible conditions and the entanglement of pure states. And we show this idea more clearly in the three-qubit case. (3) We investigate the relations between the compatibility problem and the invariants of a matrix-set under some groups. Furthermore, we show that it is one of the reasons why the compatibility problems which involve the multi-party density matrixes are much more difficult than the one-party case.
It is a hard and important problem to find the criterion of a set of positive-definite matrices which can be written as reduced density operators of a multipartite global quantum state. This problem is closely related to the study of many-body quantum entanglement, which is one of the focuses of current quantum-information theory. We give several results on the necessary compatibility relations between a set of reduced and global density matrices, including (i) compatibility conditions for the one-party reduced density matrices of any $({N}_{A}\ifmmode\times\else\texttimes\fi{}{N}_{B})$-dimensional bipartite mixed quantum state, (ii) compatibility conditions for the one-party reduced matrices of any
M-partite quantum states with the dimension ${N}^{\ensuremath{\bigotimes}M}$, and (iii) compatibility conditions for the two-party reduced density matrixes of $(2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2)$-dimensional tripartite mixed quantum state.
To find the criterion of a set of positive-definite matrices which can be written as reduced density matrices of a multipartite matrix is a hard and important problem. When the problem is concerned with multiparty density matrices, it is much more significant for computational many-body physics and many-body quantum entanglement which is one of the focuses of current quantum information theory. We give several results on the necessary compatibility relations between a set of two-party reduced density matrices and a global state in Hilbert space ${N}_{A}\ifmmode\times\else\texttimes\fi{}{N}_{B}\ifmmode\times\else\texttimes\fi{}{N}_{C}$ where
NA,
NB, and
NC are arbitrary.
We present a necessary and sufficient condition for three qutrit density matrices to be the one-particle reduced density matrices of a pure three-qutrit quantum state. The condition consists of seven classes of inequalities satisfied by the eigenvalues of these matrices. One of them is a generalization of a known inequality for the qubit case. Some of these inequalities are proved algebraically whereas the proof of the others uses the fact that a continuous function of the state must have a minimum. Construction of states satisfying these inequalities relies on a representation of the convex set of the allowed set of eigenvalues in terms of corner points. We also present a result for a more general quantum system concerning the nature of the boundary surface of the set of the allowed set of eigenvalues.
We investigate some basic scenarios in which a given set of bipartite quantum states may consistently arise as the set of reduced states of a global
N-partite quantum state. Intuitively, we say that the multipartite state ``joins'' the underlying correlations. Determining whether, for a given set of states and a given joining structure, a compatible
N-partite quantum state exists is known as the quantum marginal problem. We restrict to bipartite reduced states that belong to the paradigmatic classes of Werner and isotropic states in
d dimensions and focus on two specific versions of the quantum marginal problem which we find to be tractable. The first is Alice-Bob, Alice-Charlie joining, with both pairs being in a Werner or isotropic state. The second is
m-
n sharability of a Werner state across
N subsystems, which may be seen as a variant of the
N-representability problem to the case where subsystems are partitioned into two groupings of
m and
n parties, respectively. By exploiting the symmetry properties that each class of states enjoys, we determine necessary and sufficient conditions for three-party joinability and 1-
n sharability for arbitrary
d. Our results explicitly show that although entanglement is required for sharing limitations to emerge, correlations beyond entanglement generally suffice to restrict joinability, and not all unentangled states necessarily obey the same limitations. The relationship between joinability and quantum cloning as well as implications for the joinability of arbitrary bipartite states are discussed.
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.