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arXiv:1903.12448 · Physics Letters A

Pauli semigroups and unistochastic quantum channels

Zbigniew Puchała, Łukasz Rudnicki, Karol Życzkowski

Published May 3, 2019• 18 citations
arXivHTMLPDF
DOI
10.1016/j.physleta.2019.04.057

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We analyze the set ANQ of mixed unitary channels represented in the Weyl basis and accessible by a Lindblad semigroup acting on an N-level quantum system. General necessary and sufficient conditions for a mixed Weyl quantum channel of an arbitrary dimension to be accessible by a semigroup are established. The set ANQ is shown to be log-convex and star-shaped with respect to the completely depolarizing channel. A decoherence supermap acting in the space of Lindblad operators transforms them into the space of Kolmogorov generators of classical semigroups. We show that for mixed Weyl channels, the super-decoherence commutes with the dynamics so that decohering a quantum accessible channel, we obtain a bistochastic matrix from the set ANC of classical maps accessible by a semigroup. Focusing on three-level systems, we investigate the geometry of the sets of quantum accessible maps, its classical counterpart, and the support of their spectra. We demonstrate that the set A3Q is not included in the set U3Q of quantum unistochastic channels, although an analogous relation holds for N = 2. The set of transition matrices obtained by super-decoherence of unistochastic channels of order N ≥ 3 is shown to be larger than the set of unistochastic matrices of this order and yields a motivation to introduce the larger sets of k-unistochastic matrices.
We explore the symmetry properties of Stinespring dilations of single-qubit Pauli channels, addressing both the generic case and the specific examples of phase damping and depolarizing channels. For each scenario, we derive the representation of the Pauli group acting on the Hilbert space of the environment. We then focus on dilations that are continuous in time and driven by a time-independent Hamiltonian, and on collision models that generate a Pauli dynamical semigroup in the limit of fast collisions. First, we complement some recent general results on these types of dilations (M. Cattaneo, Phys. Rev. A 111, 022209 (2025)) with some additions and clarifications, including the case of covariant channels with strongly conserved quantities. Next, we show that the covariance property of Pauli channels impose strong constraints on both the dilation Hamiltonian and the initial state of the environment, and demonstrate how these constraints can be exploited to explicitly construct the time-dependent dilations in all considered cases. Our results are relevant for the quantum simulation of Pauli channels via unitary dilations and of Pauli semigroups via collision models, both in the laboratory and on quantum computers.