Quantum non-Markovianity of channels can be produced by mixing Markovian channels, as observed recently by various authors. We consider an analogous question of whether singularities of the channel can be produced by mixing nonsingular channels, i.e., ones that lack them. Here we answer the question in the negative in the context of qubit Pauli channels. On the other hand, mixing channels with a singularity can lead to the elimination of singularities in the resultant channel. We distinguish between two types of singular channels, which lead under mixing to broadly quite different properties of the singularity in the resultant channel. The connection to non-Markovianity (in the sense of completely positive indivisibility) is pointed out. These results impose nontrivial restrictions on the experimental realization of noninvertible quantum channels by a process of channel mixing.
Quantum non-Markovianity of channels can be produced by mixing Markovian channels, as observed recently by various authors. We consider an analogous question of whether singularities of the channel can be produced by mixing nonsingular channels, i.e., ones that lack them. Here we answer the question in the negative in the context of qubit Pauli channels. On the other hand, mixing channels with a singularity can lead to the elimination of singularities in the resultant channel. We distinguish between two types of singular channels, which lead under mixing to broadly quite different properties of the singularity in the resultant channel. The connection to non-Markovianity (in the sense of completely positive indivisibility) is pointed out. These results impose nontrivial restrictions on the experimental realization of noninvertible quantum channels by a process of channel mixing.
We identify two broad types of noninvertibilities in quantum dynamical maps, one necessarily associated with CP indivisibility and one not so. We study the production of (non-)Markovian, invertible maps by the process of mixing noninvertible Pauli maps and quantify the fraction of the same. The memory kernel perspective appears to be less transparent on the issue of invertibility than the approaches based on maps or master equations. Here we consider a related and potentially helpful issue: the identification of criteria of parameterized families of maps leading to the existence of a well-defined semigroup limit.
We introduce a method to construct non-Markovian variants of completely positive (CP) dynamical maps, particularly, qubit Pauli channels. We identify non-Markovianity with the breakdown in CP divisibility of the map, i.e., appearance of a not-completely positive intermediate map. In particular, we consider the case of non-Markovian dephasing in detail. The eigenvalues of the Choi matrix of the intermediate map crossover at a point which corresponds to a singularity in the canonical decoherence rate of the corresponding master equation and thus to a momentary noninvertibility of the map. Thereafter, the rate becomes negative, indicating non-Markovianity. We quantify the non-Markovianity by two methods, one based on CP divisibility [Hall et al., Phys. Rev. A 89, 042120 (2014)], which does not require optimization but requires normalization to handle the singularity, and another method, based on distinguishability [Breuer et al. Phys. Rev. Lett. 103, 210401 (2009)], which requires optimization but is insensitive to the singularity.
Pauli channels are widely used to describe errors in quantum computers, particularly when noise is shaped via Pauli twirling. A common assumption is that such channels admit a Markovian generator, namely a Pauli-Lindblad model with non-negative rates, but the validity of this assumption has not been systematically examined. Here, using CP-indivisibility as our criterion for non-Markovianity, we study multi-qubit Pauli channels from a single snapshot of the dynamics. We find that while the generator always has the same structure as the standard Pauli-Lindblad model, the rates may be negative or complex. We show that random Pauli channels are almost always non-Markovian, with the probability of encountering a negative rate converging doubly exponentially to unity with the number of qubits. For physically motivated noise models shaped by Pauli twirling, including single-qubit over-rotations and two-qubit amplitude damping errors, we find that negative rates are generic, even when the underlying physical noise is Markovian. We generalize probabilistic error amplification and cancellation to non-Markovian generators, and quantify the sampling overhead introduced by negative and complex rates. Experiments on superconducting qubits confirm that allowing negative rates in the learned noise model yields more accurate predictions than restricting to non-negative rates.
We study a class of qubit non-Markovian general Pauli dynamical maps with multiple singularities in the generator. We discuss a few easy examples involving trigonometric or other nonmonotonic time dependence of the map, and discuss in detail the structure of channels which don’t have any trigonometric functional dependence. We demystify the concept of a singularity here, showing that it corresponds to a point where the dynamics can be regular but the map is momentarily noninvertible, and this gives a basic guideline to construct such non-invertible non-Markovian channels. Most members of the channels in the considered family are quasi-eternally non-Markovian (QENM), which is a broader class of non-Markovian channels than the eternal non-Markovian channels. Specifically, the measure of quasi-eternal non-Markovian (QENM) channels in the considered class is shown to be [Formula: see text] in the isotropic case, and about 0.96 in the anisotropic case.
We analyze the convex combinations of noninvertible generalized Pauli dynamical maps. By manipulating the mixing parameters, one can produce a channel with shifted singularities, additional singularities, or even no singularities whatsoever. In particular, we show how to use noninvertible dynamical maps to produce the Markovian semigroup. Interestingly, the maps whose mixing results in a semigroup are generated by the time-local generators and time-homogeneous memory kernels that are not regular; that is, their formulas contain infinities. Finally, we show how the generators and memory kernels change after mixing the corresponding dynamical maps.
Finite-time Markovian channels, unlike their infinitesimal counterparts, do not form a convex set. As a particular instance of this observation, we consider the problem of mixing the three Pauli channels, conservatively assumed to be quantum dynamical semigroups, and fully characterize the resulting ``Pauli simplex.'' We show that neither the set of non-Markovian (completely positive indivisible) nor Markovian channels is convex in the Pauli simplex, and that the measure of non-Markovian channels is about 0.87. All channels in the Pauli simplex are P divisible. A potential application in the context of quantum resource theory is also discussed.