Irreducible Decompositions and Stationary States of Quantum Channels

Carbone, Raffaella; Pautrat, Yan

doi:10.1016/s0034-4877(16)30032-5

Cited by 24 publications

(36 citation statements)

References 26 publications

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“…Specifically, using a generalized notion of quantum sufficient statistic [24][25][26][27], we show that a local operation on part of a system is efficient if and only if it unitarily preserves the minimal sufficient statistic of the part for arXiv:2001.02258v3 [quant-ph] 1 Feb 2020 the whole. Our geometric interpretation of this also draws on recent progress on fixed points of quantum channels [28][29][30][31].…”

Section: Introductionmentioning

confidence: 99%

Thermodynamically-efficient local computation and the inefficiency of quantum memory compression

Loomis

Crutchfield

2020

Phys. Rev. Research

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Modularity dissipation identifies how locally-implemented computation entails costs beyond those required by Landauer's bound on thermodynamic computing. We establish a general theorem for efficient local computation, giving the necessary and sufficient conditions for a local operation to have zero modularity cost. Applied to thermodynamically-generating stochastic processes it confirms a conjecture that classical generators are efficient if and only if they satisfy retrodiction, which places minimal memory requirements on the generator. This extends immediately to quantum computation: Any quantum simulator that employs quantum memory compression cannot be thermodynamically efficient.CB is given by W min = k B T ln 2 (H[ρ AB ] − H[ρ CB ]).

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Section: Introductionmentioning

confidence: 99%

Thermodynamically-efficient local computation and the inefficiency of quantum memory compression

Loomis

Crutchfield

2020

Phys. Rev. Research

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“…A necessary and sufficient algebraic condition for uniqueness of this invariant state is (see e.g. [7,20,5])…”

Section: Introductionmentioning

confidence: 99%

Invariant measure for quantum trajectories

Benoist

Fraas

Pautrat

et al. 2018

Probab. Theory Relat. Fields

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We study a class of Markov chains that model the evolution of a quantum system subject to repeated measurements. Each Markov chain in this class is defined by a measure on the space of matrices. It is then given by a random product of correlated matrices taken from the support of the defining measure. We give natural conditions on this support that imply that the Markov chain admits a unique invariant probability measure. We moreover prove the geometric convergence towards this invariant measure in the Wasserstein metric. Standard techniques from the theory of products of random matrices cannot be applied under our assumptions, and new techniques are developed, such as maximum likelihood-type estimations.

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“…Each spin is L-dimensional and indexed by the physical index . Let us consider a faithful channel E = {E } L =1 (with E N × N matrices for some bond dimension N ) and write its corresponding MPS |Φ (47) where the N × N boundary matrix B provides the ability to pin the boundary of the chain to various states [98]. Physically meaningful boundaries are either B = I (the identity) for translationally invariant MPS's or B = |r l| for some states |r , |l quantifying the effect of the boundary on the right and left ends of the chain.…”

Section: Mps From Faithful Channelsmentioning

confidence: 99%

Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states

Albert

2019

Quantum

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This work derives an analytical formula for the asymptotic state-the quantum state resulting from an infinite number of applications of a general quantum channel on some initial state. For channels admitting multiple fixed or rotating points, conserved quantities-the left fixed/rotating points of the channeldetermine the dependence of the asymptotic state on the initial state. The formula stems from a Noether-like theorem stating that, for any channel admitting a full-rank fixed point, conserved quantities commute with that channel's Kraus operators up to a phase. The formula is applied to adiabatic transport of the fixed-point space of channels, revealing cases where the dissipative/spectral gap can close during any segment of the adiabatic path. The formula is also applied to calculate expectation values of noninjective matrix product states (MPS) in the thermodynamic limit, revealing that those expectation values can also be calculated using an MPS with reduced bond dimension and a modified boundary.

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Irreducible Decompositions and Stationary States of Quantum Channels

Cited by 24 publications

References 26 publications

Thermodynamically-efficient local computation and the inefficiency of quantum memory compression

Thermodynamically-efficient local computation and the inefficiency of quantum memory compression

Invariant measure for quantum trajectories

Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states

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