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arXiv:1306.0083 · Linear and Multilinear Algebra

Generalized Shemesh criterion, common invariant subspaces and irreducible completely positive superoperators

Andrzej Jamiołkowski, Grzegorz Pastuszak

Published Feb 5, 2014• 20 citations
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Abstract

Assume that are complex matrices. We give a computable criterion for existence of a common eigenvector of which generalize the result of D. Shemesh established for two matrices. We use this criterion to prove some necessary and sufficient condition for to have a common invariant subspace of dimension , , if every has pairwise different eigenvalues. Since the set of all matrices having multiple eigevalues has Lebesgue measure , the condition is sufficient in practical applications. Being motivated by quantum information theory, we give a flavour of such applications for irreducible completely positive superoperators.

DOI
10.1080/03081087.2013.865734

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Assume that are complex matrices. We give a computable criterion for existence of a common eigenvector of which generalize the result of D. Shemesh established for two matrices. We use this criterion to prove some necessary and sufficient condition for to have a common invariant subspace of dimension , , if every has pairwise different eigenvalues. Since the set of all matrices having multiple eigevalues has Lebesgue measure , the condition is sufficient in practical applications. Being motivated by quantum information theory, we give a flavour of such applications for irreducible completely positive superoperators.
Maps of the form Phi(X) =sum_{i=1}^s A_iXA^*, where A_1, . . . ,A_s are fixed complex n by n matrices and X is any complex n by n matrix are used in quantum information theory as representations of quantum channels. This article deals with computable conditions for the existence of decoherence--free subspaces for Phi. Since the definition of decoherence-free subspace for quantum channels relies only on the matrices A1, . . . ,As, the term of common reducing unitary subspace is used instead of the original one. Among the main results of the paper, there are computable conditions for the existence of common eigenvectors. These are related to common reducing unitary subspaces of dimension one. The new results on common eigenvectors provide new effective condition for the existence of common invariant subspaces of arbitrary dimensions.
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.
In this paper, our objective is to establish results that connect Shemesh theorem with quantum zero-error information theory. To achieve this, we will first introduce definitions and properties related to the zero-error capacity of a quantum channel, with particular emphasis on demonstrating that the eigenstates common to the Kraus operators representing the quantum channel are fixed points. This result has previously been employed to develop a test for determining whether a quantum channel possesses positive zero-error capacity. In this work, we demonstrate that the eigenstates common to the Kraus operators representing the channel reside within the Shemesh subspace. Consequently, the fixed points of the quantum channel act as a bridge between Shemesh theorem and quantum zero-error information theory, thereby enabling the derivation of new results on the zero-error capacity of quantum channels.
We introduce a simple sufficient criterion allowing to tell whether a subspace of a bipartite or multipartite Hilbert space is entangled. The main ingredient of our criterion is a bound on the minimal entanglement of a subspace in terms of entanglement of vectors spanning that subspace expressed for geometrical measures of entanglement. The criterion is applicable to both completely and genuinely entangled subspaces. We explore its applicability in several important scenarios. Further, an entanglement criterion for mixed states following directly from the condition is stated. As an auxiliary result we provide a formula for the generalized geometric measure of entanglement of the ddd--level Dicke states.