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

Russian Mathematics

On Majorization of Quantum Channels by Probability Distributions

Grigori Gennadievich Amosov

Published Jan 1, 2025• 1 citation
Publisher

Abstract

Abstract Based on the concept of majorization for the probability distribution, a definition of the majorization of a quantum channel by a probability distribution is introduced. It is shown that the proposed approach makes it possible to solve the problem of taking the extremes of convex functions from the output eigenvalues of mixed unitary channels in the case when summation in the definition of the channel is carried out according to the Heisenberg–Weyl group.

DOI
10.3103/s1066369x25700070

Related papers

On majorization of quantum channels by probability distributions

Grigori Gennadievich Amosov

Based on the concept of majorization for the probability distribution, a definition of the majorization of a quantum channel by a probability distribution is introduced. It is shown that the proposed approach makes it possible to solve the problem of taking the extremes of convex functions from the output eigenvalues of mixed unitary channels in the case when summation in the definition of the channel is carried out according to the Heisenberg-Weyl group.

Russian Mathematics•Feb 5, 2025•0 citations

On Condition of Majorization for Mixed Unitary Channels

Grigori Gennadievich Amosov, L. A. Ryskin

Abstract We investigate the majorization procedure for mixed unitary quantum channels. The conditions under which we get the exact estimation are found. The example based upon projective unitary representations of Abelian groups is considered.

Lobachevskii Journal of Mathematics•Jun 1, 2025•0 citations

On characteristics of mixed unitary channels being additive or multiplicative with respect to taking tensor products

Grigori Gennadievich Amosov

We study mixed unitary channels generated by finite subgroups of the group of all unitary operators in a Hilbert space. Based on the majorization theory we introduce techniques allowing to calculate different characteristics of output states of channels. A class of channels has been allocated for which the use of entangled states doesn't give any advantage under taking supremum and infimum for output characteristics of channels. In particular, $l_p$-norms are multiplicative and the minimal entropy is additive with respect to taking tensor products of channels. As an important application of the obtained results the classical capacity of channel is calculated in the evident form. We compare our techniques with the informational characteristics of Boson quantum channels.

arXiv:2401.05104•Jan 10, 2024•2 citations

On the convergence of output sets of quantum channels

Benoı̂t Collins, Motohisa Fukuda, Ion Nechita

We study the asymptotic behavior of the output states of sequences of quantum channels. Under a natural assumption, we show that the output set converges to a compact convex set, clarifying and substantially generalizing results in [BCN13]. Random mixed unitary channels satisfy the assumption; we give a formula for the asymptotic maximum output infinity norm and we show that the minimum output entropy and the Holevo capacity have a simple relation for the complementary channels. We also give non-trivial examples of sequences $Φ_n$ such that along with any other quantum channel $Ξ$, we have convergence of the output set of $Φ_n$ and $Φ_n\otimes Ξ$ simultaneously; the case when $Ξ$ is entanglement breaking is investigated in details.

arXiv:1311.7571•Journal of Operator Theory•May 1, 2015•8 citations

Comparison of Quantum Channels by Superchannels

Gilad Gour

We extend the definition of the conditional min-entropy from bipartite quantum states to bipartite quantum channels. We show that many of the properties of the conditional min-entropy carry over to the extended version, including an operational interpretation as a guessing probability when one of the subsystems is classical. We then show that the extended conditional min-entropy can be used to fully characterize when two bipartite quantum channels are related to each other via a superchannel (also known as supermap or a comb) that is acting on one of the subsystems. This relation is a pre-order that extends the definition of “quantum majorization” from bipartite states to bipartite channels, and can also be characterized with semidefinite programming. As a special case, our characterization provides necessary and sufficient conditions for when a set of quantum channels is related to another set of channels via a single superchannel. We discuss the applications of our results to channel discrimination, and to resource theories of quantum processes. Along the way we study channel divergences, entropy functions of quantum channels, and noise models of superchannels, including random unitary superchannels and doubly-stochastic superchannels. For the latter we give a physical meaning as being completely-uniformity preserving.

arXiv:1808.02607•IEEE Transactions on Information Theory•Apr 1, 2019•125 citations

Derivation of quantum probability from measurement

Fedor Herbut

To begin with, it is pointed out that the form of the quantum probabil- ity formula originates in the very initial state of the object system as seen when the state is expanded with the eigen-projectors of the measured ob- servable. Making use of the probability reproducibility condition, which is a key concept in unitary measurement theory, one obtains the relevant coher- ent distribution of the complete-measurement results in the final unitary- measurement state in agreement with the mentioned probability formula. Treating the transition from the final unitary, or premeasurement, state, where all possible results are present, to one complete-measurement result sketchily in the usual way, the well-known probability formula is derived. In conclusion it is pointed out that the entire argument is only formal unless one makes it physical assuming that the quantum probability law is valid in the extreme case of probability-one (certain) events (projectors).

arXiv:1512.04028•Dec 13, 2015•0 citations
Abstract Based on the concept of majorization for the probability distribution, a definition of the majorization of a quantum channel by a probability distribution is introduced. It is shown that the proposed approach makes it possible to solve the problem of taking the extremes of convex functions from the output eigenvalues of mixed unitary channels in the case when summation in the definition of the channel is carried out according to the Heisenberg–Weyl group.
We study mixed unitary channels generated by finite subgroups of the group of all unitary operators in a Hilbert space. Based on the majorization theory we introduce techniques allowing to calculate different characteristics of output states of channels. A class of channels has been allocated for which the use of entangled states doesn't give any advantage under taking supremum and infimum for output characteristics of channels. In particular, lpl_plp​-norms are multiplicative and the minimal entropy is additive with respect to taking tensor products of channels. As an important application of the obtained results the classical capacity of channel is calculated in the evident form. We compare our techniques with the informational characteristics of Boson quantum channels.
We study the asymptotic behavior of the output states of sequences of quantum channels. Under a natural assumption, we show that the output set converges to a compact convex set, clarifying and substantially generalizing results in [BCN13]. Random mixed unitary channels satisfy the assumption; we give a formula for the asymptotic maximum output infinity norm and we show that the minimum output entropy and the Holevo capacity have a simple relation for the complementary channels. We also give non-trivial examples of sequences ΦnΦ_nΦn​ such that along with any other quantum channel ΞΞΞ, we have convergence of the output set of ΦnΦ_nΦn​ and Φn⊗ΞΦ_n\otimes ΞΦn​⊗Ξ simultaneously; the case when ΞΞΞ is entanglement breaking is investigated in details.
We extend the definition of the conditional min-entropy from bipartite quantum states to bipartite quantum channels. We show that many of the properties of the conditional min-entropy carry over to the extended version, including an operational interpretation as a guessing probability when one of the subsystems is classical. We then show that the extended conditional min-entropy can be used to fully characterize when two bipartite quantum channels are related to each other via a superchannel (also known as supermap or a comb) that is acting on one of the subsystems. This relation is a pre-order that extends the definition of “quantum majorization” from bipartite states to bipartite channels, and can also be characterized with semidefinite programming. As a special case, our characterization provides necessary and sufficient conditions for when a set of quantum channels is related to another set of channels via a single superchannel. We discuss the applications of our results to channel discrimination, and to resource theories of quantum processes. Along the way we study channel divergences, entropy functions of quantum channels, and noise models of superchannels, including random unitary superchannels and doubly-stochastic superchannels. For the latter we give a physical meaning as being completely-uniformity preserving.
To begin with, it is pointed out that the form of the quantum probabil- ity formula originates in the very initial state of the object system as seen when the state is expanded with the eigen-projectors of the measured ob- servable. Making use of the probability reproducibility condition, which is a key concept in unitary measurement theory, one obtains the relevant coher- ent distribution of the complete-measurement results in the final unitary- measurement state in agreement with the mentioned probability formula. Treating the transition from the final unitary, or premeasurement, state, where all possible results are present, to one complete-measurement result sketchily in the usual way, the well-known probability formula is derived. In conclusion it is pointed out that the entire argument is only formal unless one makes it physical assuming that the quantum probability law is valid in the extreme case of probability-one (certain) events (projectors).