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arXiv:2403.03667 · Annales Henri Poincaré

Random Covariant Quantum Channels

Ion Nechita, Sang-Jun Park

Published Mar 26, 2025• 4 citations
arXivHTMLPDF

Abstract

Abstract The group symmetries inherent in quantum channels often make them tractable and applicable to various problems in quantum information theory. In this paper, we introduce natural probability distributions for covariant quantum channels. Specifically, this is achieved through the application of “twirling operations” on random quantum channels derived from the Stinespring representation that uses Haar-distributed random isometries. We explore various types of group symmetries, including unitary and orthogonal covariance, hyperoctahedral covariance, and diagonal orthogonal covariance (DOC), and analyze their properties related to quantum entanglement based on the model parameters. In particular, we discuss the threshold phenomenon for positive partial transpose and entanglement breaking properties, comparing thresholds among different classes of random covariant channels. Finally, we contribute to the $$\hbox {PPT}^2$$ PPT 2 conjecture by showing that the composition between two random DOC channels is generically entanglement breaking.

DOI
10.1007/s00023-025-01558-y

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Abstract The group symmetries inherent in quantum channels often make them tractable and applicable to various problems in quantum information theory. In this paper, we introduce natural probability distributions for covariant quantum channels. Specifically, this is achieved through the application of “twirling operations” on random quantum channels derived from the Stinespring representation that uses Haar-distributed random isometries. We explore various types of group symmetries, including unitary and orthogonal covariance, hyperoctahedral covariance, and diagonal orthogonal covariance (DOC), and analyze their properties related to quantum entanglement based on the model parameters. In particular, we discuss the threshold phenomenon for positive partial transpose and entanglement breaking properties, comparing thresholds among different classes of random covariant channels. Finally, we contribute to the PPT2\hbox {PPT}^2PPT2 PPT 2 conjecture by showing that the composition between two random DOC channels is generically entanglement breaking.
Cartan-covariant quantum channels are introduced and studied using the Choi-Jamiołkowski isomorphism. The channels are Cartan-covariant as the covariance groups considered form symmetric pairs with the special unitary group SU(nnn), and their associated Cartan involution provides a route to exactly compute the eigenspectrum of their Choi states and the partial transpose for any n∈Nn\in \mathbb{N}n∈N. These channels include previously studied SO(n)(n)(n)-covariant channels, and further include Sp(n2)(\frac{n}{2})(2n​)-covariant channels (when nnn is even) and S(U(ppp) ×\times× U(qqq))-covariant channels (where n ⁣= ⁣p ⁣+ ⁣qn\!=\!p\!+\!qn=p+q). We show that all Cartan-covariant channels satisfy the PPT2^{2}2-conjecture and further demonstrate the nontrivial nature of this result for the class of Sp(n2)(\frac{n}{2})(2n​)-covariant and S(U(p) ⁣× ⁣U(q)){\rm S}({\rm U}(p) \!\times\!{\rm U}(q))S(U(p)×U(q))-covariant channels
Abstract We investigate unitary operators acting on a tensor product space, with the property that the quantum channels they generate, via the Stinespring dilation theorem, are of a particular type, independently of the state of the ancilla system in the Stinespring relation. The types of quantum channels we consider are those of interest in quantum information theory: unitary conjugations, constant channels, unital channels, mixed unitary channels, positive partial transpose channels, and entanglement breaking channels. For some of the classes of bipartite unitary operators corresponding to the above types of channels, we provide explicit characterizations, necessary and/or sufficient conditions for membership, and we compute the dimension of the corresponding algebraic variety. Inclusions between these classes are considered, and we show that for small dimensions, many of these sets are identical.
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We consider the sequences of random quantum channels defined by using the Stinespring formula with Haar-distributed random orthogonal matrices. For any fixed sequence of input states, we study the asymptotic eigenvalue distribution of the outputs through the tensor powers of random channels. We show that the input states achieving minimum output entropy are tensor products of maximally entangled states (Bell states) when the tensor power is even. This phenomenon is completely different from the one for random quantum channels constructed from Haar-distributed random unitary matrices, which leads us to formulate some conjectures about the regularized minimum output entropy.
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.
Abstract 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.