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Pairwise completely positive matrices with applications in Quantum Information Theory

Olivia R. MacLean

Published Apr 8, 2019• 0 citations
OpenAlex

Abstract

This thesis introduces a generalization of the set of completely positive matrices that we call “pairwise completely positive” (PCP) matrices. PCP matrix pairs are defined so that one matrix in the pair is necessarily positive semidefinite while the other is necessarily entrywise non-negative. After PCP matrices are defined we explore their basic properties and establish numerous necessary and sufficient conditions that can help test whether or not a pair meets our definition of PCP. We then relate these matrix pairs to the separability of conjugate local diagonal unitary invariant (CLDUI) quantum states. In particular, we show that determining whether or not a pair of matrices is pairwise completely positive is equivalent to determining whether or not a corresponding CLDUI state is separable.

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This thesis introduces a generalization of the set of completely positive matrices that we call “pairwise completely positive” (PCP) matrices. PCP matrix pairs are defined so that one matrix in the pair is necessarily positive semidefinite while the other is necessarily entrywise non-negative. After PCP matrices are defined we explore their basic properties and establish numerous necessary and sufficient conditions that can help test whether or not a pair meets our definition of PCP. We then relate these matrix pairs to the separability of conjugate local diagonal unitary invariant (CLDUI) quantum states. In particular, we show that determining whether or not a pair of matrices is pairwise completely positive is equivalent to determining whether or not a corresponding CLDUI state is separable.
A generalization of the set of completely positive matrices called pairwise completely positive (PCP) matrices is introduced. These are pairs of matrices that share a joint decomposition so that one of them is necessarily positive semidefinite while the other one is necessarily entrywise non-negative. Basic properties of these matrix pairs are explored and several testable necessary and sufficient conditions are developed to help determine whether or not a pair is PCP. A connection with quantum entanglement is established by showing that determining whether or not a pair of matrices is pairwise completely positive is equivalent to determining whether or not a certain type of quantum state, called a conjugate local diagonal unitary invariant state, is separable. Many of the most important quantum states in entanglement theory are of this type, including isotropic states, mixed Dicke states (up to partial transposition), and maximally correlated states. As a specific application of these results, a wide family of states that have absolutely positive partial transpose are shown to in fact be separable.
A generalization of the set of completely positive matrices called pairwise completely positive (PCP) matrices is introduced. These are pairs of matrices that share a joint decomposition so that one of them is necessarily positive semidefinite while the other one is necessarily entrywise non-negative. Basic properties of these matrix pairs are explored and several testable necessary and sufficient conditions are developed to help determine whether or not a pair is PCP. A connection with quantum entanglement is established by showing that determining whether or not a pair of matrices is pairwise completely positive is equivalent to determining whether or not a certain type of quantum state, called a conjugate local diagonal unitary invariant state, is separable. Many of the most important quantum states in entanglement theory are of this type, including isotropic states, mixed Dicke states (up to partial transposition), and maximally correlated states. As a specific application of these results, a wide family of states that have absolutely positive partial transpose are shown to in fact be separable.
We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the literature, which includes notable entries like the Diagonal Symmetric states and the Choi-type maps, we show that this class of matrices (and maps) encompasses a wide variety of scenarios, thereby unifying their study. We examine their linear algebraic structure and investigate different notions of positivity through their convex conic manifestations. In particular, we generalize the well-known cone of completely positive matrices to that of triplewise completely positive matrices and connect it to the separability of the relevant invariant states (or the entanglement breaking property of the corresponding quantum channels). For linear maps, we provide explicit characterizations of the stated covariance in terms of their Kraus, Stinespring, and Choi representations, and systematically analyze the usual properties of positivity, decomposability, complete positivity, and the like. We also describe the invariant subspaces of these maps and use their structure to provide necessary and sufficient conditions for separability of the associated invariant bipartite states.
For any nnn-partite state ρA1A2⋅⋅⋅Anρ_{A_{1}A_{2}\cdot\cdot\cdot A_{n}}ρA1​A2​⋅⋅⋅An​​, we define its quantum mutual information matrix as an nnn by nnn matrix whose (i,j)(i,j)(i,j)-entry is given by quantum mutual information I(ρAiAj)I(ρ_{A_{i}A_{j}})I(ρAi​Aj​​). Although each entry of quantum mutual information matrix, like its classical counterpart, is also used to measure bipartite correlations, the similarity ends here: quantum mutual information matrices are not always positive semidefinite even for collections of up to 3-partite states. In this work, we obtain necessary and sufficient conditions for the positive semidefinite quantum mutual information matrix. We further define the \emph{genuine} nnn-partite mutual information which can be easily calculated. This definition is symmetric, nonnegative, bounded and more accurate for measuring multipartite states.
Separability from the spectrum is an important and ongoing research topic in quantum entanglement. In this study, we investigate properties related to absolute separability from the spectrum in qudit-qudit states in the bipartite state space Hmn=Hm⊗Hn. First, we propose the necessary and sufficient conditions for absolute positive partial transpose states in the Hilbert space H4n. These conditions are equivalent to the positive semidefiniteness of twelve matrices resulting from the symmetric matricizations of eigenvalues. Furthermore, we demonstrate that these sufficient conditions can be extended to the general Hmn case, improving existing conclusions in the literature. These sufficient conditions depend only on the first few leading and last few leading eigenvalues, significantly reducing the complexity of determining absolute separable states. On the other hand, we also introduce additional sufficient conditions for determining that states in Hmn are not absolutely separable. These conditions only depend on 2m − 1 eigenvalues of the mixed states. Our sufficient conditions are not only simple and easy to implement, but also effective in practice. As applications, we derive distance bounds for eigenvalues and purity bounds for general absolutely separable states.