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:1302.4525 · Journal of Physics A Mathematical and Theoretical

Unified-entropy trade-off relations for a single quantum channel

Alexey E. Rastegin

Published Jun 27, 2013• 9 citations
arXivHTMLPDF

Abstract

Many important properties of quantum channels are quantified by means of entropic functionals. Characteristics of such a kind are closely related to different representations of a quantum channel. In the Jamiołkowski–Choi representation, the given quantum channel is described by the so-called dynamical matrix. Entropies of the rescaled dynamical matrix known as map entropies describe a degree of introduced decoherence. Within the so-called natural representation, the quantum channel is formally posed by another matrix, obtained as the reshuffling of the dynamical matrix. The corresponding entropies characterize an amount, with which the receiver a priori knows the channel output. As was previously shown, the map and receiver entropies are mutually complementary characteristics. Indeed, there exists a non-trivial lower bound on their sum. First, we extend the concept of receiver entropy to the family of unified entropies. Developing the previous results, we further derive non-trivial lower bounds on the sum of the map and receiver ( q , s )-entropies. The derivation is essentially based on some inequalities with the Schatten norms and anti-norms.

DOI
10.1088/1751-8113/46/28/285301

Related papers

Unified-entropy uncertainty relations for a single quantum channel

Alexey E. Rastegin

Many important properties of quantum channels are quantified by means of entropic functionals. Characteristics of such a kind are closely related to different representations of a quantum channel. In the Jamio{\l}kowski-Choi representation, the given quantum channel is described by the so-called dynamical matrix. Entropies of the rescaled dynamical matrix known as map entropies describe a degree of introduced decoherence. Within the so-called natural representation, the quantum channel is formally posed by another matrix obtained as reshuffling of the dynamical matrix. The corresponding entropies characterize an amount, with which the receiver a priori knows the channel output. As was previously shown, the map and receiver entropies are mutually complementary characteristics. Indeed, there exists a non-trivial lower bound on their sum. First, we extend the concept of receiver entropy to the family of unified entropies. Developing the previous results, we further derive non-trivial lower bounds on the sum of the map and receiver $(q,s)$-entropies. The derivation is essentially based on some inequalities with the Schatten norms and anti-norms.

•Feb 19, 2013•0 citations

Entropic trade-off relations for quantum operations

Wojciech Roga, Zbigniew Puchała, Łukasz Rudnicki +1 more

Spectral properties of an arbitrary matrix can be characterized by the entropy of its rescaled singular values. Any quantum operation can be described by the associated dynamical matrix or by the corresponding superoperator. The entropy of the dynamical matrix describes the degree of decoherence introduced by the map, while the entropy of the superoperator characterizes the a priori knowledge of the receiver of the outcome of a quantum channel $\ensuremath{\Phi}$. We prove that for any map acting on an $N$-dimensional quantum system the sum of both entropies is not smaller than $\mathrm{ln}N$. For any bistochastic map this lower bound reads $2\mathrm{ln}N$. We investigate also the corresponding R\'enyi entropies, providing an upper bound for their sum, and analyze the entanglement of the bi-partite quantum state associated with the channel.

arXiv:1206.2536•Physical Review A•Mar 7, 2013•23 citations

On unified-entropy characterization of quantum channels

Alexey E. Rastegin

We consider properties of quantum channels with the use of unified entropies. Extremal unravelings of quantum channel with respect to these entropies are examined. The concept of map entropy is extended in terms of the unified entropies. The map ( q , s )-entropy is naturally defined as the unified ( q , s )-entropy of a rescaled dynamical matrix of given quantum channel. Inequalities of Fannes type are obtained for introduced entropies in terms of both the trace and Frobenius norms of difference between corresponding dynamical matrices. Additivity properties of introduced map entropies are discussed. The known inequality of Lindblad with the entropy exchange is generalized to many of the unified entropies. For the tensor product of a pair of quantum channels, we derive a two-sided estimate on the output entropy of a maximally entangled input state.

arXiv:1109.2197•Journal of Physics A Mathematical and Theoretical•Dec 23, 2011•14 citations

Trade-off relations for operation entropy of complementary quantum channels

Jakub Czartowski, Daniel Braun, Karol Życzkowski

The entropy of a quantum operation, defined as the von Neumann entropy of the corresponding Choi–Jamiołkowski state, characterizes the coupling of the principal system with the environment. For any quantum channel acting on a state of a given size, one defines the complementary channel, which sends the input state into the state of the environment after the operation. Making use of subadditivity of entropy, we show that for any dimension the sum of both entropies is bounded from below. This result characterizes the trade-off between the information on the initial quantum state accessible to the principal system and the information leaking to the environment. For one qubit maps we describe the interpolating family of depolarizing maps, for which the sum of both entropies gives the lower boundary of the region allowed in the space spanned by both entropies.

arXiv:1908.03492•International Journal of Quantum Information•Aug 1, 2019•2 citations

Notes on entropic characteristics of quantum channels

Alexey E. Rastegin

Abstract One of most important issues in quantum information theory concerns transmission of information through noisy quantum channels. We discuss a few channel characteristics expressed by means of generalized entropies. Such characteristics can often be treated in line with more usual treatment based on the von Neumann entropies. For any channel, we show that the q-average output entropy of degree q ≥ 1 is bounded from above by the q-entropy of the input density matrix. The concavity properties of the (q, s)-entropy exchange are considered. Fano type quantum bounds on the (q, s)-entropy exchange are derived. We also give upper bounds on the map (q, s)-entropies in terms of the output entropy, corresponding to the completely mixed input.

arXiv:1206.3056•Open Physics•Oct 24, 2012•4 citations

Relating Entropies of Quantum Channels

Dariusz Kurzyk, Łukasz Pawela, Zbigniew Puchała

In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi-Jamiołkowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture and provide numerical intuitions that the bound can also be saturated for random channels as their dimension tends to infinity.

arXiv:2106.09629•Entropy•Aug 10, 2021•3 citations
Many important properties of quantum channels are quantified by means of entropic functionals. Characteristics of such a kind are closely related to different representations of a quantum channel. In the Jamiołkowski–Choi representation, the given quantum channel is described by the so-called dynamical matrix. Entropies of the rescaled dynamical matrix known as map entropies describe a degree of introduced decoherence. Within the so-called natural representation, the quantum channel is formally posed by another matrix, obtained as the reshuffling of the dynamical matrix. The corresponding entropies characterize an amount, with which the receiver a priori knows the channel output. As was previously shown, the map and receiver entropies are mutually complementary characteristics. Indeed, there exists a non-trivial lower bound on their sum. First, we extend the concept of receiver entropy to the family of unified entropies. Developing the previous results, we further derive non-trivial lower bounds on the sum of the map and receiver ( q , s )-entropies. The derivation is essentially based on some inequalities with the Schatten norms and anti-norms.
Many important properties of quantum channels are quantified by means of entropic functionals. Characteristics of such a kind are closely related to different representations of a quantum channel. In the Jamio{\l}kowski-Choi representation, the given quantum channel is described by the so-called dynamical matrix. Entropies of the rescaled dynamical matrix known as map entropies describe a degree of introduced decoherence. Within the so-called natural representation, the quantum channel is formally posed by another matrix obtained as reshuffling of the dynamical matrix. The corresponding entropies characterize an amount, with which the receiver a priori knows the channel output. As was previously shown, the map and receiver entropies are mutually complementary characteristics. Indeed, there exists a non-trivial lower bound on their sum. First, we extend the concept of receiver entropy to the family of unified entropies. Developing the previous results, we further derive non-trivial lower bounds on the sum of the map and receiver (q,s)(q,s)(q,s)-entropies. The derivation is essentially based on some inequalities with the Schatten norms and anti-norms.
Spectral properties of an arbitrary matrix can be characterized by the entropy of its rescaled singular values. Any quantum operation can be described by the associated dynamical matrix or by the corresponding superoperator. The entropy of the dynamical matrix describes the degree of decoherence introduced by the map, while the entropy of the superoperator characterizes the a priori knowledge of the receiver of the outcome of a quantum channel $\ensuremath{\Phi}$. We prove that for any map acting on an NNN-dimensional quantum system the sum of both entropies is not smaller than lnN\mathrm{ln}NlnN. For any bistochastic map this lower bound reads 2lnN2\mathrm{ln}N2lnN. We investigate also the corresponding R\'enyi entropies, providing an upper bound for their sum, and analyze the entanglement of the bi-partite quantum state associated with the channel.
We consider properties of quantum channels with the use of unified entropies. Extremal unravelings of quantum channel with respect to these entropies are examined. The concept of map entropy is extended in terms of the unified entropies. The map ( q , s )-entropy is naturally defined as the unified ( q , s )-entropy of a rescaled dynamical matrix of given quantum channel. Inequalities of Fannes type are obtained for introduced entropies in terms of both the trace and Frobenius norms of difference between corresponding dynamical matrices. Additivity properties of introduced map entropies are discussed. The known inequality of Lindblad with the entropy exchange is generalized to many of the unified entropies. For the tensor product of a pair of quantum channels, we derive a two-sided estimate on the output entropy of a maximally entangled input state.
The entropy of a quantum operation, defined as the von Neumann entropy of the corresponding Choi–Jamiołkowski state, characterizes the coupling of the principal system with the environment. For any quantum channel acting on a state of a given size, one defines the complementary channel, which sends the input state into the state of the environment after the operation. Making use of subadditivity of entropy, we show that for any dimension the sum of both entropies is bounded from below. This result characterizes the trade-off between the information on the initial quantum state accessible to the principal system and the information leaking to the environment. For one qubit maps we describe the interpolating family of depolarizing maps, for which the sum of both entropies gives the lower boundary of the region allowed in the space spanned by both entropies.
Abstract One of most important issues in quantum information theory concerns transmission of information through noisy quantum channels. We discuss a few channel characteristics expressed by means of generalized entropies. Such characteristics can often be treated in line with more usual treatment based on the von Neumann entropies. For any channel, we show that the q-average output entropy of degree q ≥ 1 is bounded from above by the q-entropy of the input density matrix. The concavity properties of the (q, s)-entropy exchange are considered. Fano type quantum bounds on the (q, s)-entropy exchange are derived. We also give upper bounds on the map (q, s)-entropies in terms of the output entropy, corresponding to the completely mixed input.
In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi-Jamiołkowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture and provide numerical intuitions that the bound can also be saturated for random channels as their dimension tends to infinity.