A Perron–Frobenius Type of Theorem for Quantum Operations

Abstract

Professor Wei-Shih Yang, Chair Quantum random walks are a generalization of classical Markovian random walks to a quantum mechanical or quantum computing setting. Quantum walks have promising applications but are complicated by quantum decoherence. We prove that the long-time limiting behavior of the class of quantum operations which are the convex combination of norm one operators is governed by the eigenvectors with norm one eigenvalues which are shared by the operators. This class includes all operations formed by a coherent operation with positive probability of orthogonal measurement at each step. We also prove that any operation that has range contained in a low enough dimension subspace of the space of density operators has limiting behavior isomorphic to an associated Markov chain. A particular class of such operations are coherent operations followed by an orthogonal measurement.

Time-inhomogeneous Quantum Markov Chains with Decoherence on Finite State Spaces

“…Note that for the special homogeneous case ζ = 0, the result is already known, and it was proved by Lagro et al [12] that the quantum Markov chain is convergent. [12] did not use the compound Markov chain representation. It used the spectral theory of the density operators.…”

“…Now we show our main result for the convergence of density operators. For ζ = 0, the following result has been obtained in [12] with more general conditions. Here we prove for the case 0 < ζ ≤ 1.…”

“…In this section we will prove convergence of density operators for ζ ≤ 1 and 0 < p ≤ 1. The homogeneous case ζ = 0 has been done in Lagro, Yang and Xiong [12] using spectral analysis of density operators. So we will just focus on the case when 0 < ζ ≤ 1.…”

“…For the time homogeneous case ζ = 0 and o < p ≤ 1, the limiting distribution has been obtained by Lagro, Yang and Xiong [12] under very general conditions similar to the Perron-Frobenius type of conditions. Our paper extends [12] to timeinhomogeneous case, but with stronger assumptions on the transition matrices.…”

The outer spectral radius and dynamics of completely positive maps

“…The condition that the coordinates generate the full algebra essentially corresponds to the condition of having all positive entries in the classical Perron-Frobenius theorem. Under an irreducibility type assumptions various Quantum Perron-Frobenius theorems which establish the existence of a simple real eigenvalue with maximum modulus have been obtained by Evans-Hoegh-Krohn [8], Schrader [26] and Lagro-Yang-Xiong [13]. We essentially gather some more detailed structure than the aforementioned works in the finite dimensional case by applying Theorem 1.3.…”