Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2002.11452 · Open Systems & Information Dynamics

A Class of Quasi-Eternal Non-Markovian Pauli Channels and Their Measure

Shrikant Utagi, Vinod N. Rao, R. Srikanth, Subhashish Banerjee

Published Dec 1, 2020• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1142/s1230161220500195

Related papers

On the eternal non-Markovianity of non-unital quantum channels

Shrikant Utagi, Subhashish Banerjee, R. Srikanth

The eternally non-Markovian (ENM) Pauli channel is an example of a unital channel characterized by a negative decay rate for all time [Formula: see text]. Here, we consider the problem of constructing an analogous non-unital channel, and show in particular that a [Formula: see text]-dimensional generalized amplitude damping (GAD) channel cannot be ENM when the non-Markovianity originates solely from the non-unital part of the channel. We study specific ramifications of this result for qubit GAD. Specifically, we construct a quasi-ENM qubit GAD channel, characterized by a time [Formula: see text], such that the channel is non-Markovian (NM) only and for all time [Formula: see text]. We further point out that our negative result for the qudit GAD channel, namely, the impossibility of the eternal NM property, does not hold for a general qubit or higher-dimensional non-unital channel.

arXiv:2203.10625•International Journal of Quantum Information•Oct 6, 2023•3 citations

Singularities, mixing, and non-Markovianity of Pauli dynamical maps

Shrikant Utagi, Vinod N. Rao, R. Srikanth +1 more

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.

arXiv:2011.04053•Physical Review A•Apr 19, 2021•11 citations

Non-Markovian dephasing and depolarizing channels

Shrikant Utagi, R. Srikanth, Subhashish Banerjee

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.

arXiv:1805.11411•Physical Review A•Sep 26, 2018•55 citations

Markovian semigroup from mixing noninvertible dynamical maps

Katarzyna Siudzińska

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.

arXiv:2012.00385•Physical Review A•Feb 10, 2021•14 citations

Noninvertibility and non-Markovianity of quantum dynamical maps

Vinayak Jagadish, R. Srikanth, Francesco Petruccione

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.

arXiv:2306.12773•Physical Review A•Oct 5, 2023•6 citations

The quantum non-Markovianity for a special class of generalized Weyl channel

Wen Xu, Mao-Sheng Li, Bo Li +2 more

Abstract A quantum channel is usually represented as a sum of Kraus operators. The recent study [Phys. Rev. A 98, 032328 (2018)] has shown that applying a perturbation to the Kraus operators in qubit Pauli channels, the dynamical maps exhibit interesting properties, such as non-Markovianity, singularity. This has sparked our interest in studying the properties of other quantum channels. In this work, we study a special class of generalized Weyl channel where the Kraus operators are proportional to the Weyl diagonal matrices and the rest are vanishing. We use the Choi matrix of intermediate map to study quantum non-Markovianity. The crossover point of the eigenvalues of Choi matrix is a singularity of the decoherence rates in the canonical form of the master equation. Moreover, we identify the non-Markovianity based on the methods of CP divisibility and distinguishability. We also quantify the non-Markovianity in terms of the Hall-Cresser-Li-Andersson (HCLA) measure and the Breuer-Laine-Piilo (BLP) measure, respectively. In particular, we choose mutually unbiased bases as a pair of orthogonal initial states to quantify the non-Markovianity based on the BLP measure.

arXiv:2503.10338•Physica Scripta•Apr 7, 2025•2 citations
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.
The eternally non-Markovian (ENM) Pauli channel is an example of a unital channel characterized by a negative decay rate for all time [Formula: see text]. Here, we consider the problem of constructing an analogous non-unital channel, and show in particular that a [Formula: see text]-dimensional generalized amplitude damping (GAD) channel cannot be ENM when the non-Markovianity originates solely from the non-unital part of the channel. We study specific ramifications of this result for qubit GAD. Specifically, we construct a quasi-ENM qubit GAD channel, characterized by a time [Formula: see text], such that the channel is non-Markovian (NM) only and for all time [Formula: see text]. We further point out that our negative result for the qudit GAD channel, namely, the impossibility of the eternal NM property, does not hold for a general qubit or higher-dimensional non-unital channel.
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 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.
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.
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.
Abstract A quantum channel is usually represented as a sum of Kraus operators. The recent study [Phys. Rev. A 98, 032328 (2018)] has shown that applying a perturbation to the Kraus operators in qubit Pauli channels, the dynamical maps exhibit interesting properties, such as non-Markovianity, singularity. This has sparked our interest in studying the properties of other quantum channels. In this work, we study a special class of generalized Weyl channel where the Kraus operators are proportional to the Weyl diagonal matrices and the rest are vanishing. We use the Choi matrix of intermediate map to study quantum non-Markovianity. The crossover point of the eigenvalues of Choi matrix is a singularity of the decoherence rates in the canonical form of the master equation. Moreover, we identify the non-Markovianity based on the methods of CP divisibility and distinguishability. We also quantify the non-Markovianity in terms of the Hall-Cresser-Li-Andersson (HCLA) measure and the Breuer-Laine-Piilo (BLP) measure, respectively. In particular, we choose mutually unbiased bases as a pair of orthogonal initial states to quantify the non-Markovianity based on the BLP measure.