A New Bound on Quantum Wielandt Inequality

Abstract

A new bound on quantum version of Wielandt inequality for positive (not necessarily completely positive) maps has been established. Also bounds for entanglement breaking and PPT channels are put forward which are better bound than the previous bounds known. We prove that a primitive positive map E acting on M d that satisfies the Schwarz inequality becomes strictly positive after at most 2(d − 1) 2 iterations. This is to say, that after 2(d − 1) 2 iterations, such a map sends every positive semidefinite matrix to a positive definite one. This finding does not depend on the number of Kraus operators as the map may not admit any Kraus decomposition. The motivation of this work is to provide an answer to a question raised in the article [17] by Sanz-García-Wolf and Cirac.

Mathematical open problems in projected entangled pair states

“…In [39] it is shown that p(T ) ≤ (D 2 − d + 1)D 2 . This result has been recently improved [36] to p(T ) ≤ 2(D − 1) 2 . The order O(D 2 ) is optimal just by invoking the optimality of the classical Wielandt inequality.…”

Entanglement Breaking Channels, Stochastic Matrices, and Primitivity

“…Motivated by this, the concept of a primitive quantum channel was introduced in [15] as follows (see also [14] for more recent work). Definition 4.4.…”

“…In this paper, we contribute to this study with an investigation of an important class of quantum operations, which are given mathematically by completely positive trace-preserving maps on matrices, called entanglement breaking channels [4,7]. We bring two key notions from matrix theory to their study by identifying stochastic matrix representations for the channels, and, built on this, we conduct an analysis of the primitivity [14,15] of such channels based on corresponding matrix primitivity [6,11]. More specifically, we show how every so-called Holevo form of an entanglement breaking channel induces a certain stochastic matrix representation that has the same non-zero spectrum as the channel.…”

Quantum Version of Wielandt’s Inequality Revisited

“…In particular, we confirm that the exponent conjectured in [PGVWC07] is indeed equal to two. After finishing the articles the authors learned that a related topic has been recently studied by Rahaman in [Rah18]. Under additional positivity assumptions the author proves an O(D 2 ) bound for the index of primitivity.…”