Completely positive linear mappings, non-Hamiltonian evolution, and quantum stochastic processes

Oseledets, V. I.

doi:10.1007/bf01101650

Cited by 3 publications

(14 citation statements)

References 131 publications

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“…As a matter of fact the original works on ergodicity for quantum stochastic processes were mostly discussed in this context, the ergodicity of the original channels been typically discussed as a derived property, see e.g. [5,[17][18][19].…”

Section: Ergodicity In Terms Of the Adjoint Mapmentioning

confidence: 99%

“…As already mentioned in the introduction, the majority of the results obtained in the field were explicitly derived for this specific maps (see e.g. [5,8]).…”

Section: Ergodicity and Mixing For Channels With Faithful Fixed Pointmentioning

confidence: 99%

“…This analysis is the subject of a quantum generalization of the Perron-Frobenius theory of classical Markov chains-see e.g. [5,6,30,42,43]. The following theorem summarizes some of these results and takes advantage of the complete positivity of quantum channels to give a convenient condition for the peripheral eigenvectors in terms of the Kraus operators: Proof.…”

Section: Remarkmentioning

confidence: 99%

“…For a review on the subject see e.g. [5][6][7]13]. A convenient definition of ergodic quantum channels can be given as follows: a observable on the fixed point [5,17,18] 7 .…”

Section: Introductionmentioning

confidence: 99%

“…[5][6][7]13]. A convenient definition of ergodic quantum channels can be given as follows: a observable on the fixed point [5,17,18] 7 . A related, but stronger property of an ergodic quantum channel, is the ability of transforming a generic input state into the fixed point ρ * after a sufficiently large number of repeated applications.…”

Section: Introductionmentioning

confidence: 99%

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Ergodic and mixing quantum channels in finite dimensions

et al. 2013

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The paper provides a systematic characterization of quantum ergodic and mixing channels in finite dimensions and a discussion of their structural properties. In particular, we discuss ergodicity in the general case where the fixed point of the channel is not a full-rank (faithful) density matrix. Notably, we show that ergodicity is stable under randomizations, namely that every random mixture of an ergodic channel with a generic channel is still ergodic. In addition, we prove several conditions under which ergodicity can be promoted to the stronger property of mixing. Finally, exploiting a suitable correspondence between quantum channels and generators of quantum dynamical semigroups, we extend our results to the realm of continuous-time quantum evolutions, providing a characterization of ergodic Lindblad generators and showing that they are dense in the set of all possible generators.quantum channel M is ergodic if and only if it admits a unique fixed point in the space of density matrices, that is, if there is only one density matrix ρ * that is unaltered by the action of M [1,[14][15][16]. The rationale behind such formulation is clear when we consider the discrete trajectories associated with the evolution of a generic input state ρ, evolving under iterated applications of the transformation M: in this case, the mean value of a generic observable A, averaged over the trajectories, converges asymptotically to the expectation value Tr[Aρ * ] of the

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Section: Ergodicity In Terms Of the Adjoint Mapmentioning

confidence: 99%

“…As already mentioned in the introduction, the majority of the results obtained in the field were explicitly derived for this specific maps (see e.g. [5,8]).…”

Section: Ergodicity and Mixing For Channels With Faithful Fixed Pointmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

“…For a review on the subject see e.g. [5][6][7]13]. A convenient definition of ergodic quantum channels can be given as follows: a observable on the fixed point [5,17,18] 7 .…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Ergodic and mixing quantum channels in finite dimensions

et al. 2013

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Universal quantum uncertainty relations between nonergodicity and loss of information

et al. 2018

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We establish uncertainty relations between information loss in general open quantum systems and the amount of non-ergodicity of the corresponding dynamics. The relations hold for arbitrary quantum systems interacting with an arbitrary quantum environment. The elements of the uncertainty relations are quantified via distance measures on the space of quantum density matrices. The relations hold for arbitrary distance measures satisfying a set of intuitively satisfactory axioms. The relations show that as the non-ergodicity of the dynamics increases, the lower bound on information loss decreases, which validates the belief that non-ergodicity plays an important role in preserving information of quantum states undergoing lossy evolution. We also consider a model of a central qubit interacting with a fermionic thermal bath and derive its reduced dynamics, to subsequently investigate the information loss and non-ergodicity in such dynamics. We comment on the "minimal" situations that saturate the uncertainty relations. arXiv:1707.08963v2 [quant-ph]

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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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Completely positive linear mappings, non-Hamiltonian evolution, and quantum stochastic processes

Cited by 3 publications

References 131 publications

Ergodic and mixing quantum channels in finite dimensions

Ergodic and mixing quantum channels in finite dimensions

Universal quantum uncertainty relations between nonergodicity and loss of information

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

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