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arXiv:2206.10742 · Journal of Physics A Mathematical and Theoretical

Phase-covariant mixtures of non-unital qubit maps

Katarzyna Siudzińska

Published Sep 8, 2022• 6 citations
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Abstract

Abstract We analyze convex combinations of non-unital qubit maps that are phase-covariant. In particular, we consider the behavior of maps that combine amplitude damping, inverse amplitude damping, and pure dephasing. We show that mixing non-unital channels can result in restoring the unitality, whereas mixing commutative maps can lead to non-commutativity. For the convex combinations of Markovian semigroups, we prove that classical uncertainties cannot break quantum Markovianity. Moreover, contrary to the Pauli channel case, the semigroup can be recovered only by mixing two other semigroups.

DOI
10.1088/1751-8121/ac909b

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Abstract We analyze convex combinations of non-unital qubit maps that are phase-covariant. In particular, we consider the behavior of maps that combine amplitude damping, inverse amplitude damping, and pure dephasing. We show that mixing non-unital channels can result in restoring the unitality, whereas mixing commutative maps can lead to non-commutativity. For the convex combinations of Markovian semigroups, we prove that classical uncertainties cannot break quantum Markovianity. Moreover, contrary to the Pauli channel case, the semigroup can be recovered only by mixing two other semigroups.
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In recent times, there has been a growing scholarly focus on investigating the intricacies of quantum channel mixing. It has been commonly believed, based on intuition in the literature, that every generalized Pauli channel with dimensionality ddd could be represented as a convex combination of (d+1)(d+1)(d+1) generalized Pauli dephasing channels (see [Phys. Rev. A 103, 022605 (2021)] as a reference). To our surprise, our findings indicate the inaccuracy of this intuitive perspective. This has stimulated our interest in exploring the properties of convex combinations of generalized Pauli channels, beyond the restriction to just (d+1)(d+1)(d+1) generalized Pauli dephasing channels. We demonstrate that many previously established properties still hold within this broader context. For instance, any mixture of invertible generalized Pauli channels retains its invertibility. It's worth noting that this property doesn't hold when considering the Weyl channels setting. Additionally, we demonstrate that every Pauli channel (for the case of d=2d=2d=2) can be represented as a mixture of (d+1)(d+1)(d+1) Pauli dephasing channels, but this generalization doesn't apply to higher dimensions. This highlights a fundamental distinction between qubit and general qudit cases. In contrast to prior understanding, we show that non-invertibility of mixed channels is not a prerequisite for the resulting mapping to constitute a Markovian semigroup.
Abstract We analyze quantum communication properties of phase-covariant channels depending on their degree of non-unitality. In particular, we derive analytical formulas for the minimal and maximal channel fidelity on pure states and maximal output purity. Next, we introduce a measure of non-unitality and show how to manipulate between unital and maximally non-unital maps by considering classical mixtures of quantum channels. Finally, we prove that maximal fidelity and maximal output purity increase with non-unitality and present several examples. Interestingly, non-unitality can also prolong quantum entanglement and lead to its rebirth.
Abstract We analyze the connections between the non-Markovianity degree of the most general phase-damping qubit maps and their legitimate mixtures. Using the results for image non-increasing dynamical maps, we formulate the necessary and sufficient conditions for the Pauli maps to satisfy specific divisibility criteria. Next, we examine how the non-Markovianity properties for (in general noninvertible) Pauli dynamical maps influence the properties of their convex combinations. Our results are illustrated with instructive examples. For P-divisible maps, we propose a legitimate time-local generator whose all decoherence rates are temporarily infinite.
Non-unital noise is a fundamental feature of open quantum systems, governing energy exchange and inducing state translations on the Bloch sphere. While such translations can be useful for tasks such as state preparation and channel capacity, unital dynamics where no translation occurs are often preferred in quantum information processing, particularly for error correction. Phase-covariant dynamics provide a general framework encompassing dissipation, excitation, and dephasing processes in qubit systems; however, commonly used models such as the generalized amplitude damping (GAD) channel offer only limited control over these features. In this work, we present a constructive framework for generating a broader class of phase-covariant dynamics by mixing amplitude-damping and anti-damping channels with unequal decay parameters and time-dependent mixing probabilities. This approach enables independent control over contraction and translation, allows continuous tuning between non-unital and unital regimes, and yields an effective dephasing contribution absent in GAD. To characterize these dynamics, we employ the general theorem for P-divisibility and show that it can be used constructively by evaluating it in suitably chosen Hilbert space bases, leading to simplified conditions for both unital and non-unital cases. The framework captures a wide range of behaviors, including CP-divisible, P-divisible but not CP-divisible, and non-P-divisible dynamics. We further show that appropriate mixing can reduce the deviation from the ideal noiseless evolution and that such improvement persists even after tuning the dynamics to the unital regime. Our results provide a flexible approach to engineering open system dynamics beyond standard thermal models, with potential applications in noise control and quantum information processing.