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International Journal of Geometric Methods in Modern Physics

EFFECTIVE METHODS IN INVESTIGATION OF IRREDUCIBLE QUANTUM OPERATIONS

Andrzej Jamiołkowski

Published Mar 1, 2012• 1 citation
Publisher

Abstract

In this paper we discuss some constructive procedures which can be used in characterization of irreducible quantum operations and quantum channels. Our method makes use of an explicit form of a fixed operation in its Kraus representation. A generalization of the Shemesh criterion is discussed.
DOI
10.1142/s0219887812600146

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In this paper, we discuss some constructive procedures which can be used in characterizations of linear transformations which preserve the set of states of a fixed quantum system. Our methods are based on analyzing an explicit form of a linear positive map in its Kraus representation. In particular, we discuss the so-called partial commutativity of operators and its applications to investigation of decoherence-free subspaces. These subspaces can also be considered as a special class of quantum error correcting codes. Using the concept of standard polynomials and Amitsur–Levitzki theorem and other ideas from the so-called polynomial identity algebras (PI-algebras) we discuss some effective algorithms for analyzing properties of quantum channels.
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Understanding the limits of quantum communication is a central problem in quantum information theory. Structural properties of quantum channels provide tools for analyzing these limits. In particular, approximate degradability links quantum-capacity bounds to the accuracy with which a channel's environment can be reconstructed from its output. We study this reconstruction problem for noisy Landau-Streater channels across finite-spin representations. For spin jjj and noise probability ppp, we construct an explicit degrading map with diamond-norm error at most [2+8/(j(j+1))]p2≤(38/3)p2[2+8/(j(j+1))]p^2\le(38/3)p^2[2+8/(j(j+1))]p2≤(38/3)p2 on the common interval 0≤p≤1/80\le p\le1/80≤p≤1/8. A lower bound establishes quadratic order for this construction at each fixed spin. Floating-point semidefinite optimization places its error within 2.48%2.48\%2.48% of the numerical optimum on the sampled grid for four spins. Covariance and conditional-entropy concavity yield explicit two-sided quantum-capacity bounds for this family with a spin-uniform remainder. Building on the low-noise Pauli cancellation strategy, we formulate a sufficient condition based on a common scalar adjoint action on fixed, possibly nonunitary noise Kraus operators. The resulting reconstruction-error bound is uniform in system dimension and Kraus count. The criterion applies to irreducible SU(N)SU(N)SU(N) generator channels and noisy Werner-Holevo channels, with the SU(3)SU(3)SU(3) adjoint representation providing an example outside the mixed-unitary class.