Diagonal unitary and orthogonal symmetries in quantum theory: II. Evolution operators

Singh, Satvik; Nechita, Ion

doi:10.1088/1751-8121/ac7017

Cited by 4 publications

(5 citation statements)

References 37 publications

(27 reference statements)

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“…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.…”

Section: Spatially Symmetric Statesmentioning

confidence: 99%

“…The matrix elements of Z are identical with those of the Fourier matrix (48) (up to an overall constant), but now the N ×N matrix elements are arranged in a diagonal matrix of size N 2 ×N 2 . In a certain sense the matrix Z is the dual of the Fourier matrix (48). The transposition symmetry of the matrix F translates into a space reflection symmetry of Z.…”

Section: Graph Statesmentioning

confidence: 99%

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

Mestyán,

Pozsgay,

Wanless

2024

SciPost Phys.

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We consider dual unitary operators and their multi-leg generalizations that have appeared at various places in the literature. These objects can be related to multi-party quantum states with special entanglement patterns: the sites are arranged in a spatially symmetric pattern and the states have maximal entanglement for all bipartitions that follow from the reflection symmetries of the given geometry. We consider those cases where the state itself is invariant with respect to the geometrical symmetry group. The simplest examples are those dual unitary operators which are also self dual and reflection invariant, but we also consider the generalizations in the hexagonal, cubic, and octahedral geometries. We provide a number of constructions and concrete examples for these objects for various local dimensions. All of our examples can be used to build quantum cellular automata in 1+1 or 2+1 dimensions, with multiple equivalent choices for the “direction of time”.

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Section: Spatially Symmetric Statesmentioning

confidence: 99%

Section: Graph Statesmentioning

confidence: 99%

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

Mestyán,

Pozsgay,

Wanless

2024

SciPost Phys.

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“…However, a complete description of the family of dual unitary matrices in U(d ⊗ d) is currently unavailable, although several different constructions of such operators have been proposed [ARL21, CL21, BP22] 3 . In this regard, two authors of this paper came up with an interesting family of dual unitary matrices having certain local diagonal symmetries [SN22] as described in Table 1, which we again list below in Table 2.…”

Section: Acronym Symmetrymentioning

confidence: 99%

“…( 12), (13), and (14), respectively. With the appropriate definitions in hand, we can state a few main results from [SN22].…”

Section: Acronym Symmetrymentioning

confidence: 99%

“…then the resulting subgroup of local diagonal orthogonal invariant (LDOI) unitary gates provides a rich and explicitly parameterized family of unitary gates which can be used to construct new families of highly symmetric dual unitary gates [SN22]. Moreover, the long term behaviour of two-point spatio-temporal correlations between local observables in a circuit built from such LDOI dual unitary gates is completely determined by the ergodic properties of a specific DOC quantum channel (Lemma 4.9).…”

Section: Introductionmentioning

confidence: 99%

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Ergodic theory of diagonal orthogonal covariant quantum channels

Singh¹,

Datta²,

Nechita³

2022

Preprint

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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.

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Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics

Claeys

Lamacraft

2022

Quantum

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Recent works have investigated the emergence of a new kind of random matrix behaviour in unitary dynamics following a quantum quench. Starting from a time-evolved state, an ensemble of pure states supported on a small subsystem can be generated by performing projective measurements on the remainder of the system, leading to a projected ensemble. In chaotic quantum systems it was conjectured that such projected ensembles become indistinguishable from the uniform Haar-random ensemble and lead to a quantum state design. Exact results were recently presented by Ho and Choi [Phys. Rev. Lett. 128, 060601 (2022)] for the kicked Ising model at the self-dual point. We provide an alternative construction that can be extended to general chaotic dual-unitary circuits with solvable initial states and measurements, highlighting the role of the underlying dual-unitarity and further showing how dual-unitary circuit models exhibit both exact solvability and random matrix behaviour. Building on results from biunitary connections, we show how complex Hadamard matrices and unitary error bases both lead to solvable measurement schemes.

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Diagonal unitary and orthogonal symmetries in quantum theory: II. Evolution operators

Cited by 4 publications

References 37 publications

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

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

Ergodic theory of diagonal orthogonal covariant quantum channels

Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics

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