A graphical calculus for integration over random diagonal unitary matrices

Nechita, Ion; Singh, Satvik

doi:10.1016/j.laa.2020.12.014

Cited by 11 publications

(27 citation statements)

References 27 publications

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“…We shall consider three situations: the first two correspond to the equations above, with the unitary U restricted to the class of diagonal unitary matrices (diagonal matrices with arbitrary complex phases), while the third one corresponds to U being restricted to diagonal orthogonal matrices (diagonal matrices with arbitrary signs). These classes of states, called respectively LDUI, CLDUI, and LDOI, have been introduced in [14,36,52]. We provide a detailed analysis of these matrices, from various points of views: linear algebra, convexity, positivity, separability, etc.…”

Section: Introductionmentioning

confidence: 99%

“…It turns out that the classes of states we investigate are still rich enough for the separability problem to be intractable [67,63,36], but the situation is simpler, since the search space for separable decompositions is smaller. We describe the separability properties of these invariant states in terms of two cones of pairs and triples of matrices, called the pairwise completely positive [36] and the triplewise completely positive [52] cone, respectively. Both these notions can be understood as extensions of the classical case of completely positive matrices [4], which is a key notion in combinatorics and optimization [1].…”

Section: Introductionmentioning

confidence: 99%

“…Although the presentation is self-contained, we refer the reader for the background material and the proofs of some results to our previous work [52] as well as to the paper of Johnston and MacLean [36], where the notions of CLDUI and PCP matrices were introduced. The paper has two main parts, one focusing on bipartite matrices, the other one on linear maps.…”

Section: Introductionmentioning

confidence: 99%

“…The sections are structured as follows. In Section 2 we review the main background material on invariant states from [52], recalling several key results from this paper. Section 3 contains a list of salient examples, already present in the literature, or new.…”

Section: Introductionmentioning

confidence: 99%

“…At several instances in this paper, we employ the diagrammatic language of boxes and strings to represent tensor equations, in order to ease proofs and make the presentation more visually intuitive. For readers who are unfamiliar with this language, Section 3 of our previous work [52] should suffice for a quick introduction.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Diagonal unitary and orthogonal symmetries in quantum theory

Singh

Nechita

2021

Quantum

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We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the literature, which includes notable entries like the Diagonal Symmetric states and the Choi-type maps, we show that this class of matrices (and maps) encompasses a wide variety of scenarios, thereby unifying their study. We examine their linear algebraic structure and investigate different notions of positivity through their convex conic manifestations. In particular, we generalize the well-known cone of completely positive matrices to that of triplewise completely positive matrices and connect it to the separability of the relevant invariant states (or the entanglement breaking property of the corresponding quantum channels). For linear maps, we provide explicit characterizations of the stated covariance in terms of their Kraus, Stinespring, and Choi representations, and systematically analyze the usual properties of positivity, decomposability, complete positivity, and the like. We also describe the invariant subspaces of these maps and use their structure to provide necessary and sufficient conditions for separability of the associated invariant bipartite states.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Diagonal unitary and orthogonal symmetries in quantum theory

Singh

Nechita

2021

Quantum

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The PPT$$^2$$ Conjecture Holds for All Choi-Type Maps

Singh

Nechita

2022

Ann. Henri Poincaré

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We prove that the PPT$$^2$$ 2 conjecture holds for linear maps between matrix algebras which are covariant under the action of the diagonal unitary group. Many salient examples, like the Choi-type maps, depolarizing maps, dephasing maps, amplitude damping maps, and mixtures thereof, lie in this class. Our proof relies on a generalization of the matrix-theoretic notion of factor width for pairwise completely positive matrices, and a complete characterization in the case of factor width two.

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Ergodic theory of diagonal orthogonal covariant quantum channels

Singh,

Datta,

Nechita

2024

Lett Math Phys

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We analyse 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.

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A graphical calculus for integration over random diagonal unitary matrices

Cited by 11 publications

References 27 publications

Diagonal unitary and orthogonal symmetries in quantum theory

Diagonal unitary and orthogonal symmetries in quantum theory

The PPT$$^2$$ Conjecture Holds for All Choi-Type Maps

Ergodic theory of diagonal orthogonal covariant quantum channels

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