Ergodic theory of diagonal orthogonal covariant quantum channels

Abstract

We analyze the ergodic properties of quantum channels that are covariant with respect to diagonal orthogonal transformations. We prove that the ergodic behaviour of a channel in this class is essentially governed by a classical stochastic matrix. This allows us to exploit tools from classical ergodic theory to study quantum ergodicity of such channels. As an application of our analysis, we study dual unitary brickwork circuits which have recently been proposed as minimal models of quantum chaos in many-body systems. Upon imposing a local diagonal orthogonal invariance symmetry on these circuits, the long-term behaviour of spatio-temporal correlations between local observables in such circuits is completely determined by the ergodic properties of a channel that is covariant under diagonal orthogonal transformations. We utilize this fact to show that such symmetric dual unitary circuits exhibit a rich variety of ergodic behaviours, thus emphasizing their importance. Contents 1. Introduction 1 1.1. Summary of main results 3 1.2. Outline of the paper 3 2. A survey of classical and quantum ergodic theory 3 2.1. Classical ergodic theory 3 2.2. Ergodic theory of quantum channels and stochastic matrices 5 2.3. Graph-theoretic description of ergodicity of stochastic matrices 8 3. Ergodicity of DOC channels 12 4. Lattice models 18 4.1. Non-ergodic behaviour 23 4.2. Ergodic and mixing behaviour 24 4.3. Ergodic and non-mixing behaviour 24 5. Conclusions 25 References 251 That is, besides exceptions due to symmetries and integrability.

From dual-unitary to biunitary: a 2-categorical model for exactly-solvable many-body quantum dynamics

“…We expect that similar calculations are possible for biunitary circuits, which would establish the connection between biunitary circuits and random matrix theory. Furthermore, while generic parametrizations of dual-unitary gates will generally lead to chaotic quantum dynamics, it is possible to fine-tune the underlying gates such that the full circuit is no longer chaotic and systematic parametrizations of dual-unitary gates remain an active topic of research [9,13,49,54,[57][58][59][60][61]. It is still an open question how parametrizations of these new biunitary connections can influence the level of ergodicity and the corresponding diagnostics of quantum chaos.…”

Exact dynamics in dual-unitary quantum circuits with projective measurements

“…We expect that similar calculations are possible for biunitary circuits using the presented graphical calculus, which would establish the connection between biunitary circuits and random matrix theory. Furthermore, while generic parametrizations of dual-unitary gates will generally lead to chaotic quantum dynamics, it is possible to fine-tune the underlying gates such that the full circuit is no longer chaotic and systematic parametrizations of dual-unitary gates remain an active topic of research [9,13,[48][49][50][51][52][53][54]. It is still an open question how parametrizations of these new biunitary connections can influence the level of ergodicity and the corresponding diagnostics of quantum chaos.…”

Multi-directional unitarity and maximal entanglement in spatially symmetric quantum states

“…In the case of dual unitary matrices certain solutions with other types of symmetry properties have already been considered in the literature. Perfect tensors with SU(2) invariance were studied in [46], whereas dual unitary matrices with diagonal SU(N ) or diagonal SO(N ) symmetries were analyzed in [47][48][49]. However, the spatial symmetry that we consider has not yet been investigated in the context of these maximally entangled states.…”