Theory of Ergodic Quantum Processes
- URL: http://arxiv.org/abs/2004.14397v3
- Date: Thu, 7 Jul 2022 15:01:33 GMT
- Title: Theory of Ergodic Quantum Processes
- Authors: Ramis Movassagh and Jeffrey Schenker
- Abstract summary: We consider general ergodic sequences of quantum channels with arbitrary correlations and non-negligible decoherence.
We compute the entanglement spectrum across any cut, by which the bipartite entanglement entropy can be computed exactly.
Other physical implications of our results are that most Floquet phases of matter are metastable and that noisy random circuits in the large depth limit will be trivial as far as their quantum entanglement is concerned.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: The generic behavior of quantum systems has long been of theoretical and
practical interest. Any quantum process is represented by a sequence of quantum
channels. We consider general ergodic sequences of stochastic channels with
arbitrary correlations and non-negligible decoherence. Ergodicity includes and
vastly generalizes random independence. We obtain a theorem which shows that
the composition of such a sequence of channels converges exponentially fast to
a replacement (rank-one) channel. Using this theorem, we derive the limiting
behavior of translation-invariant channels and stochastically independent
random channels. We then use our formalism to describe the thermodynamic limit
of ergodic matrix product states. We derive formulas for the expectation value
of a local observable and prove that the two-point correlations of local
observables decay exponentially. We then analytically compute the entanglement
spectrum across any cut, by which the bipartite entanglement entropy (i.e.,
R\'enyi or von Neumann) across an arbitrary cut can be computed exactly. Other
physical implications of our results are that most Floquet phases of matter are
metastable and that noisy random circuits in the large depth limit will be
trivial as far as their quantum entanglement is concerned. To obtain these
results, we bridge quantum information theory to dynamical systems and random
matrix theory.
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