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arXiv:2104.06117

On a matrix equality involving partial transposition and its relation to the separability problem

Soni, Vaibhav, Rishone Deshwal, Garg, Aayush, Rohit Kumar, Satyabrata Adhikari

Published Apr 13, 2021• 0 citations
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Abstract

In matrix theory, a well established relation $(AB)^{T}=B^{T}A^{T}$ holds for any two matrices $A$ and $B$ for which the product $AB$ is defined. Here $T$ denote the usual transposition. In this work, we explore the possibility of deriving the matrix equality $(AB)^Γ=A^ΓB^Γ$ for any $4 \times 4$ matrices $A$ and $B$, where $Γ$ denote the partial transposition. We found that, in general, $(AB)^Γ\neq A^ΓB^Γ$ holds for $4 \times 4$ matrices $A$ and $B$ but there exist particular set of $4 \times 4$ matrices for which $(AB)^Γ= A^ΓB^Γ$ holds. We have exploited this matrix equality to investigate the separability problem. Since it is possible to decompose the density matrices $ρ$ into two positive semi-definite matrices $A$ and $B$ so we are able to derive the separability condition for $ρ$ when $ρ^Γ=(AB)^Γ=A^ΓB^Γ$ holds. Due to the non-uniqueness property of the decomposition of the density matrix into two positive semi-definte matrices $A$ and $B$, there is a possibility to generalise the matrix equality for density matrices lives in higher dimension. These results may help in studying the separability problem for higher dimensional and multipartite system.

DOI
10.48550/arxiv.2104.06117

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A quantum system consisting of two subsystems is separable if its density matrix can be written as $\ensuremath{\rho}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{A}^{}{w}_{A}{\ensuremath{\rho}}_{A}^{\ensuremath{'}}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{A}^{\ensuremath{'}\ensuremath{'}},$ where ${\ensuremath{\rho}}_{A}^{\ensuremath{'}}$ and ${\ensuremath{\rho}}_{A}^{\ensuremath{'}\ensuremath{'}}$ are density matrices for the two subsystems, and the positive weights wA{w}_{A}wA​ satisfy $\ensuremath{\Sigma}{w}_{A}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$. In this Letter, it is proved that a necessary condition for separability is that a matrix, obtained by partial transposition of \ensuremath{\rho}, has only non-negative eigenvalues. Some examples show that this criterion is more sensitive than Bell's inequality for detecting quantum inseparability.
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Recently it was proved that many results that are true for density matrices which are positive under partial transposition (or simply PPT), also hold for another class of matrices with a certain symmetry in their Hermitian Schmidt decompositions. These matrices were called symmetric with positive coefficients (or simply SPC). A natural question appeared: What is the connection between SPC matrices and PPT matrices? Is every SPC matrix PPT? Here we show that every SPC matrix is PPT in M2⊗M2M_2\otimes M_2M2​⊗M2​. This theorem is a consequence of the fact that every density matrix in M2⊗MmM_2\otimes M_mM2​⊗Mm​, with tensor rank smaller or equal to 3, is separable. Although, in M3⊗M3M_3\otimes M_3M3​⊗M3​, we present an example of SPC matrix with tensor rank 3 that is not PPT. We shall also provide a non trivial example of a family of matrices in Mk⊗MkM_k\otimes M_kMk​⊗Mk​, in which both, the SPC and PPT properties, are equivalent. Within this family, there exists a non trivial subfamily in which the SPC property is equivalent to separability.
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 MMM-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.
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