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arXiv:1412.3860 · IEEE Transactions on Information Theory

Completely Reducible Maps in Quantum Information Theory

Daniel Cariello

Published Feb 26, 2016• 10 citations
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Abstract

In order to compute the Schmidt decomposition of A ∈ Mk⊗ Mm, we must consider an associated self-adjoint map. Here, we show that if A is positive under partial transposition (PPT) or symmetric with positive coefficients (SPCs) or invariant under realignment, then its associated self-adjoint map is completely reducible. We give applications of this fact in quantum information theory. We recover some theorems (recently proved for PPT and SPC matrices), and we prove them for matrices invariant under realignment using theorems of the Perron-Frobenius theory. We also provide a new proof of the fact that if Ckcontains k mutually unbiased bases, then there exists another orthonormal basis which is mutually unbiased with these k bases. We study other types of matrices that could have the same property. We consider a collection of linear transformations acting on Mk⊗ Mk, which contains the partial transpositions and the realignment map. For each linear transformation, we consider the set of matrices in Mk⊗ Mk≃ M(k2) that are positive and remain positive, or invariant, under the action of this linear transformation. Within this family of sets, we have the set of PPT matrices, the set of SPC matrices and the set of matrices invariant under realignment. We show that these three sets are the only sets of this family, such that the associated self-adjoint map of each matrix is completely reducible. We also show that every matrix invariant under realignment is PPT in M2⊗ M2and we present a counterexample in Mk⊗ Mkand k ≥ 3.

DOI
10.1109/tit.2016.2535176

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In order to compute the Schmidt decomposition of A ∈ Mk⊗ Mm, we must consider an associated self-adjoint map. Here, we show that if A is positive under partial transposition (PPT) or symmetric with positive coefficients (SPCs) or invariant under realignment, then its associated self-adjoint map is completely reducible. We give applications of this fact in quantum information theory. We recover some theorems (recently proved for PPT and SPC matrices), and we prove them for matrices invariant under realignment using theorems of the Perron-Frobenius theory. We also provide a new proof of the fact that if Ckcontains k mutually unbiased bases, then there exists another orthonormal basis which is mutually unbiased with these k bases. We study other types of matrices that could have the same property. We consider a collection of linear transformations acting on Mk⊗ Mk, which contains the partial transpositions and the realignment map. For each linear transformation, we consider the set of matrices in Mk⊗ Mk≃ M(k2) that are positive and remain positive, or invariant, under the action of this linear transformation. Within this family of sets, we have the set of PPT matrices, the set of SPC matrices and the set of matrices invariant under realignment. We show that these three sets are the only sets of this family, such that the associated self-adjoint map of each matrix is completely reducible. We also show that every matrix invariant under realignment is PPT in M2⊗ M2and we present a counterexample in Mk⊗ Mkand k ≥ 3.
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.
This paper is devoted to the study of the separability problem in the field of Quantum information theory. We focus on the bipartite finite dimensional case and on two types of matrices: SPC and PPT matrices (see definitions 32 and 33). We prove that many results hold for both types. If these matrices have specific Hermitian Schmidt decompositions then they are separable in a very strong sense (see theorem 38 and corollary 39). We prove that both types have what we call \textbf{split decompositions} (see theorems 41 and 42). We also define the notion of weakly irreducible matrix (see definition 43), based on the concept of irreducible state defined recently in \cite{chen1}, \cite{chen} and \cite{chen2}.}{These split decomposition theorems imply that every SPC (((PPT))) matrix can be decomposed into a sum of s+1s+1s+1 SPC (((PPT))) matrices of which the first sss are weakly irreducible, by theorem 48, and the last one has a further split decomposition of lower tensor rank, by corollary 49. Thus the SPC (((PPT))) matrix is decomposed in a finite number of steps into a sum of weakly irreducible matrices. Different components of this sum have support on orthogonal local Hilbert spaces, therefore the matrix is separable if and only if each component is separable. This reduces the separability problem for SPC (((PPT))) matrices to the case of weakly irreducible SPC (((PPT))) matrices. We also provide a complete description of weakly irreducible matrices of both types (see theorem 46).}{Using the fact that every positive semidefinite Hermitian matrix with tensor rank 2 is separable (see theorem 58), we found sharp inequalites providing separability for both types (see theorems 61 and 62).
This paper is devoted to the study of the separability problem in the field of Quantum information theory. We deal mainly with the bipartite finite dimensional case and with two types of matrices, one of them being the PPT matrices. We proved that many results holds for both types. If these matrices have specific Hermitian Schmidt decompositions then the matrices are separable in a very strong sense. We proved that both types have what we call split decompositions. We defined the notion of weak irreducible matrix, based on the concept of irreducible state defined recently. These split decomposition theorems together with the notion of weak irreducible matrix, imply that these matrices are weak irreducible or a sum of weak irreducible matrices of the same type. The separability problem for these types of matrices can be reduced to the set of weak irreducible matrices of the same type. We also provided a complete description of weak irreducible matrices of both types. Using the fact that every positive semidefinite Hermitian matrix with tensor rank 2 is separable, we found sharp inequalites providing separability for both types.
Genuine high-dimensional entanglement, i.e., the property of having a high Schmidt number, constitutes an instrumental resource in quantum communication, overcoming limitations of low-dimensional systems. States with a positive partial transpose (PPT) are generally considered weakly entangled, as they can never be distilled into pure entangled states. This naturally raises the question of whether high Schmidt numbers are possible for PPT states. Volume estimates suggest that optimal, i.e., linear, scaling in the local dimension should be possible, albeit without providing insight into the possible slope. We provide the first explicit construction of a family of PPT states that achieves linear scaling in the local dimension and we prove that random PPT states typically share this feature. Our construction also allows us to prove a recent conjecture of Chen et al. on the existence of PPT states whose Schmidt number increases by an arbitrarily large amount upon partial transposition. Finally, we link the Schmidt number to entangled sub-block matrices of a quantum state. We use this connection to prove that quantum states that are either (i) invariant under partial transposition on the smallest of their two subsystems, or (ii) absolutely PPT cannot have a maximal Schmidt number. Overall, our findings shed new light on some fundamental problems in entanglement theory.
We construct a large class of quantum d⊗d states which are positive under partial transposition (so called PPT states). The construction is based on certain direct sum decomposition of the total Hilbert space which is governed by by cyclic permutation from the symmetric group Sd-i. It turns out that partial transposition maps any such decomposition into another one corresponding to 'complementary' permutation. This class contains many well known examples of PPT states from the literature and gives rise to a huge family of completely new states.
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.