scholarly journals Relating Entropies of Quantum Channels

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 1028
Author(s):  
Dariusz Kurzyk ◽  
Łukasz Pawela ◽  
Zbigniew Puchała
Keyword(s):  
Relative Entropy ◽  
Upper Bound ◽  
Quantum Channel ◽  
Von Neumann Entropy ◽  
Quantum Channels ◽  
Von Neumann ◽  
Channel One ◽  
Depolarizing Channel

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.

scholarly journals Trade-off relations for operation entropy of complementary quantum channels

2019 ◽  
Vol 17 (05) ◽  
pp. 1950046
Author(s):  
Jakub Czartowski ◽  
Daniel Braun ◽  
Karol Życzkowski
Keyword(s):  
Quantum Channel ◽  
Lower Boundary ◽  
Von Neumann Entropy ◽  
Input State ◽  
Quantum Channels ◽  
Quantum Operation ◽  
Trade Off ◽  
Von Neumann ◽  
State Of The Environment ◽  
Complementary Channel

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.

scholarly journals Symmetry-like Relation of Relative Entropy Measure of Quantum Coherence

2020 ◽  
Author(s):  
Cheng-yang Zhang ◽  
Zhi-hua Guo ◽  
H.X. Cao
Keyword(s):  
Relative Entropy ◽  
Quantum Coherence ◽  
Quantum Physics ◽  
Information Science ◽  
Upper Bounds ◽  
Von Neumann Entropy ◽  
Entropy Measure ◽  
Lower And Upper Bounds ◽  
Von Neumann ◽  
Natural Logarithm

Quantum coherence is an important physical resource in quantum information science, and also as one of the most fundamental and striking features in quantum physics. In this paper, we obtain a symmetry-like relation of relative entropy measure $C_r(\rho)$ of coherence for $n$-partite quantum states $\rho$, which gives lower and upper bounds for $C_r(\rho)$. Meanwhile, we discuss the conjecture about the validity of the inequality $C_r(\rho)\leq C_{\ell_1}(\rho)$ for any state $\rho$. We observe that every mixture $\eta$ of a state $\rho$ satisfying $C_r(\rho)\leq C_{\ell_1}(\rho)$ and any incoherent state $\sigma$ also satisfies the conjecture. We also note that if the von Neumann entropy is defined by the natural logarithm $\ln$ instead of $\log_2$, then the reduced relative entropy measure of coherence $\bar{C}_r(\rho)=-\rho_{\rm{diag}}\ln \rho_{\rm{diag}}+\rho\ln \rho$ satisfies the inequality ${\bar{C}}_r(\rho)\leq C_{\ell_1}(\rho)$ for any mixed state $\rho$.

LIMITATION ON THE ACCESSIBLE INFORMATION FOR QUANTUM CHANNELS WITH INEFFICIENT MEASUREMENTS

2005 ◽  
Vol 03 (supp01) ◽  
pp. 87-95
Author(s):  
KURT JACOBS
Keyword(s):  
Quantum System ◽  
Von Neumann Entropy ◽  
Quantum Channels ◽  
Initial State ◽  
Von Neumann ◽  
Classical Information ◽  
Quantum Operations ◽  
Amount Of Information ◽  
Accessible Information

To transmit classical information using a quantum system, the sender prepares the system in one of a set of possible states and sends it to the receiver. The receiver then makes a measurement on the system to obtain information about the senders choice of state. The amount of information which is accessible to the receiver depends upon the encoding and the measurement. Here we derive a bound on this information which generalizes the bound derived by Schumacher, Westmoreland and Wootters [Schumacher, Westmoreland and Wootters, Phys. Rev. Lett. 76, 3452 (1996)] to include inefficient measurements, and thus all quantum operations. This also allows us to obtain a generalization of a bound derived by Hall [Hall, Phys. Rev. A 55, 100 (1997)], and to show that the average reduction in the von Neumann entropy which accompanies a measurement is concave in the initial state, for all quantum operations.

scholarly journals MONOTONICITY OF QUANTUM RELATIVE ENTROPY REVISITED

2003 ◽  
Vol 15 (01) ◽  
pp. 79-91 ◽  
Author(s):  
DÉNES PETZ
Keyword(s):  
Relative Entropy ◽  
Coarse Graining ◽  
Von Neumann Entropy ◽  
Trace Inequality ◽  
Von Neumann ◽  
Quantum Relative Entropy ◽  
Crucial Property

Monotonicity under coarse-graining is a crucial property of the quantum relative entropy. The aim of this paper is to investigate the condition of equality in the monotonicity theorem and in its consequences as the strong sub-additivity of von Neumann entropy, the Golden–Thompson trace inequality and the monotonicity of the Holevo quantitity. The relation to quantum Markov states is briefly indicated.

Mixing doubly stochastic quantum channels with the completely depolarizing channel

2009 ◽  
Vol 9 (5&6) ◽  
pp. 406-413
Author(s):  
J. Watrous
Keyword(s):  
Quantum Channel ◽  
Convex Combination ◽  
Quantum Channels ◽  
Doubly Stochastic ◽  
Depolarizing Channel

It is proved that every doubly stochastic quantum channel that is properly averaged with the completely depolarizing channel can be written as a convex combination of unitary channels. It follows that within the space of doubly stochastic quantum channels, there is a ball with positive radius around the completely depolarizing channel within which all channels are convex combinations of unitary channels.

scholarly journals Depolarizing behavior of quantum channels in higher dimensions

2011 ◽  
Vol 11 (5&6) ◽  
pp. 466-484
Author(s):  
Easwar Magesan
Keyword(s):  
Quantum Channel ◽  
Higher Dimensions ◽  
Uniqueness Result ◽  
Quantum Gate ◽  
Distance Measures ◽  
Independent Interest ◽  
Quantum Channels ◽  
Quantum Operations ◽  
Pure States ◽  
Depolarizing Channel

The paper analyzes the behavior of quantum channels, particularly in large dimensions, by proving various properties of the quantum gate fidelity. Many of these properties are of independent interest in the theory of distance measures on quantum operations. A non-uniqueness result for the gate fidelity is proven, a consequence of which is the existence of non-depolarizing channels that produce a constant gate fidelity on pure states. Asymptotically, the gate fidelity associated with any quantum channel is shown to converge to that of a depolarizing channel. Methods for estimating the minimum of the gate fidelity are also presented.

Some bounds on the minimum number of queries required for quantum channel perfect discrimination

2012 ◽  
Vol 12 (1&2) ◽  
pp. 138-148
Author(s):  
Cheng Lu ◽  
Jianxin Chen ◽  
Runyao Duan
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Quantum Channel ◽  
Necessary Condition ◽  
Quantum Channels ◽  
Sequential Scheme ◽  
Minimum Number ◽  
Perfect Discrimination ◽  
Sufficient And Necessary Condition

We prove a lower bound on the $q$-maximal fidelities between two quantum channels $\E_0$ and $\E_1$ and an upper bound on the $q$-maximal fidelities between a quantum channel $\E$ and an identity $\I$. Then we apply these two bounds to provide a simple sufficient and necessary condition for sequential perfect distinguishability between $\E$ and $\I$ and provide both a lower bound and an upper bound on the minimum number of queries required to sequentially perfectly discriminating $\E$ and $\I$. Interestingly, in the $2$-dimensional case, both bounds coincide. Based on the optimal perfect discrimination protocol presented in \cite{DFY09}, we can further generalize the lower bound and upper bound to the minimum number of queries to perfectly discriminating $\E$ and $I$ over all possible discrimination schemes. Finally the two lower bounds are shown remain working for perfectly discriminating general two quantum channels $\E_0$ and $\E_1$ in sequential scheme and over all possible discrimination schemes respectively.

scholarly journals ENTROPIC CHARACTERIZATION OF QUANTUM OPERATIONS

2011 ◽  
Vol 09 (04) ◽  
pp. 1031-1045 ◽  
Author(s):  
WOJCIECH ROGA ◽  
KAROL ŻYCZKOWSKI ◽  
MARK FANNES
Keyword(s):  
Von Neumann Entropy ◽  
Input State ◽  
Dimensional System ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Output Entropy ◽  
The Individual ◽  
Depolarizing Channel ◽  
Minimal Output Entropy

We investigate decoherence induced by a quantum channel in terms of minimal output entropy and map entropy. The latter is the von Neumann entropy of the Jamiołkowski state of the channel. Both quantities admit q-Renyi versions. We prove additivity of the map entropy for all q. For the case q = 2, we show that the depolarizing channel has the smallest map entropy among all channels with a given minimal output Renyi entropy of order two. This allows us to characterize pairs of channels such that the output entropy of their tensor product acting on a maximally entangled input state is larger than the sum of the minimal output entropies of the individual channels. We conjecture that for any channel Φ1 acting on a finite dimensional system, there exists a class of channels Φ2 sufficiently close to a unitary map such that additivity of minimal output entropy for Ψ1 ⊗ Ψ2 holds.

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