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Physical Review A

Compatibility relations between the reduced and global density matrices

Yong‐Jian Han, Y.-S. Zhang, Guang‐Can Guo

Published May 5, 2005• 13 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.71.052306

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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 (NA\ifmmode×\else\texttimes\fiNB)({N}_{A}\ifmmode\times\else\texttimes\fi{}{N}_{B})(NA​\ifmmode×\else\texttimes\fiNB​)-dimensional bipartite mixed quantum state, (ii) compatibility conditions for the one-party reduced matrices of any MMM-partite quantum states with the dimension N\ensuremath⨂M{N}^{\ensuremath{\bigotimes}M}N\ensuremath⨂M, and (iii) compatibility conditions for the two-party reduced density matrixes of (2\ifmmode×\else\texttimes\fi2\ifmmode×\else\texttimes\fi2)(2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2)(2\ifmmode×\else\texttimes\fi2\ifmmode×\else\texttimes\fi2)-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{N}_{A}NA​, NB{N}_{B}NB​, and NC{N}_{C}NC​ are arbitrary.
To find whether a set of reduced density matrices comes from a common multiparty state is a hard and important problem. In this paper, we do the following. (i) We introduce a method to find out some polytopes in one-party eigenvalue space which are sufficient conditions of this problem. (ii) 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. (iii) 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 multiparty density matrices are much more difficult than the one-party case.
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.
Subsystems of composite quantum systems are described by reduced density matrices, or quantum marginals. Important physical properties often do not depend on the whole wave function but rather only on the marginals. Not every collection of reduced density matrices can arise as the marginals of a quantum state. Instead, there are profound compatibility conditions -- such as Pauli's exclusion principle or the monogamy of quantum entanglement -- which fundamentally influence the physics of many-body quantum systems and the structure of quantum information. The aim of this thesis is a systematic and rigorous study of the general relation between multipartite quantum states, i.e., states of quantum systems that are composed of several subsystems, and their marginals. In the first part, we focus on the one-body marginals of multipartite quantum states; in the second part, we study general quantum marginals from the perspective of entropy.
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.