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QP-PQ, quantum probability and white noise analysis

QUANTUM MARKOV CHAINS : THE RECURRENCE PROBLEM

Luigi Accardi, Dmitri Koroliouk

Published Oct 1, 1991• 17 citations
Publisher
DOI
10.1142/9789814360203_0003

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A notion of entangled Markov chain was introduced by Accardi and Fidaleo in the context of quantum random walk. They proved that, in the finite-dimensional case, the corresponding states have vanishing entropy density, but they did not prove that they are entangled. In this note this entropy result is extended to the infinite-dimensional case under the assumption of finite speed of hopping. Then the entanglement problem is discussed for spin-1/2, entangled Markov chains generated by a binary symmetric channel with hopping probability 1-q. The von Neumann entropy of these states, restricted on a sublattice is explicitly calculated and shown to be independent of the size of the sublattice. This is a new, purely quantum, phenomenon. Finally the entanglement property between the sublattices [Formula: see text] and [Formula: see text] is investigated using the PPT criterium. It turns out that, for q≠ 0, 1, ½ the states are non-separable, thus truly entangled, while for q = 0, 1, ½, they are separable.

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EQUIVALENCE CLASSES AND LOCAL ASYMPTOTIC NORMALITY IN SYSTEM IDENTIFICATION FOR QUANTUM MARKOV CHAINS

Mădălin Guţǎ, Jukka Kiukas

Abstract. We consider the problems of identifying and estimating dynamical parameters of an ergodic quantum Markov chain, when only the stationary output is accessible for measurements. On the identifiability question, we show that the knowledge of the output state completely fixes the dynamics up to a ‘coordinate transformation’ consisting of a multiplication by a phase and a unitary conjugation of the Kraus operators. When the dynamics depends on an unknown parameter, we show that the latter can be estimated at the ‘standard ’ rate n−1/2, and give an explicit expression of the (asymptotic) quantum Fisher information of the output, which is proportional to the Markov variance of a certain ‘generator’. More generally, we show that the output is locally asymptotically normal, i.e. it can be approximated by a simple quantum Gaussian model consisting of a coherent state whose mean is related to the unknown parameter. As a consistency check we prove that a parameter related to the ‘coordinate transformation ’ unitaries, has zero quantum Fisher information. 1.

•Aug 17, 2016•23 citations
We introduce the notion of Markov states and chains on the Canonical Anticommutation Relations algebra over ℤ, emphasizing some remarkable differences with the infinite tensor product case. We describe the structure of the Markov states on this algebra and show that, contrarily to the infinite tensor product case, not all these states are diagonalizable. A general method to construct nontrivial quantum Markov chains on the CAR algebra is also proposed and illustrated by some pivotal examples. This analysis provides a further step for a satisfactory theory of quantum Markov processes.
A notion of entangled Markov chain was introduced by Accardi and Fidaleo in the context of quantum random walk. They proved that, in the finite-dimensional case, the corresponding states have vanishing entropy density, but they did not prove that they are entangled. In this note this entropy result is extended to the infinite-dimensional case under the assumption of finite speed of hopping. Then the entanglement problem is discussed for spin-1/2, entangled Markov chains generated by a binary symmetric channel with hopping probability 1-q. The von Neumann entropy of these states, restricted on a sublattice is explicitly calculated and shown to be independent of the size of the sublattice. This is a new, purely quantum, phenomenon. Finally the entanglement property between the sublattices [Formula: see text] and [Formula: see text] is investigated using the PPT criterium. It turns out that, for q≠ 0, 1, ½ the states are non-separable, thus truly entangled, while for q = 0, 1, ½, they are separable.
Abstract. We consider the problems of identifying and estimating dynamical parameters of an ergodic quantum Markov chain, when only the stationary output is accessible for measurements. On the identifiability question, we show that the knowledge of the output state completely fixes the dynamics up to a ‘coordinate transformation’ consisting of a multiplication by a phase and a unitary conjugation of the Kraus operators. When the dynamics depends on an unknown parameter, we show that the latter can be estimated at the ‘standard ’ rate n−1/2, and give an explicit expression of the (asymptotic) quantum Fisher information of the output, which is proportional to the Markov variance of a certain ‘generator’. More generally, we show that the output is locally asymptotically normal, i.e. it can be approximated by a simple quantum Gaussian model consisting of a coherent state whose mean is related to the unknown parameter. As a consistency check we prove that a parameter related to the ‘coordinate transformation ’ unitaries, has zero quantum Fisher information. 1.