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 $\rho$.

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

Entropy ◽  
2020 ◽  
Vol 22 (3) ◽  
pp. 297 ◽  
Author(s):  
Chengyang Zhang ◽  
Zhihua Guo ◽  
Huaixin Cao
Keyword(s):  
Relative Entropy ◽  
Quantum Coherence ◽  
Quantum Physics ◽  
Information Science ◽  
Von Neumann Entropy ◽  
Entropy Measure ◽  
Lower And Upper Bounds ◽  
Von Neumann ◽  
Natural Logarithm ◽  
The One

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. To quantify coherence, two proper measures were introduced in the literature, the one is the relative entropy of coherence C r ( ρ ) = S ( ρ diag ) − S ( ρ ) and the other is the ℓ 1 -norm of coherence C ℓ 1 ( ρ ) = ∑ i ≠ j | ρ i j | . In this paper, we obtain a symmetry-like relation of relative entropy measure C r ( ρ A 1 A 2 ⋯ A n ) of coherence for an n-partite quantum states ρ A 1 A 2 ⋯ A n , which gives lower and upper bounds for C r ( ρ ) . As application of our inequalities, we conclude that when each reduced states ρ A i is pure, ρ A 1 ⋯ A n is incoherent if and only if the reduced states ρ A i and tr A i ρ A 1 ⋯ A n ( i = 1 , 2 , … , n ) are all incoherent. Meanwhile, we discuss the conjecture that C r ( ρ ) ≤ C ℓ 1 ( ρ ) for any state ρ , which was proved to be valid for any mixed qubit state and any pure state, and open for a general state. We observe that every mixture η of a state ρ satisfying the conjecture with any incoherent state σ also satisfies the conjecture. We also observe that when the von Neumann entropy is defined by the natural logarithm ln instead of log 2 , the reduced relative entropy measure of coherence C ¯ r ( ρ ) = − ρ diag ln ρ diag + ρ ln ρ satisfies the inequality C ¯ r ( ρ ) ≤ C ℓ 1 ( ρ ) for any state ρ .

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.

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 Quantum reversibility is relative, or does a quantum measurement reset initial conditions?

2018 ◽  
Vol 376 (2123) ◽  
pp. 20170315 ◽  
Author(s):  
Wojciech H. Zurek
Keyword(s):  
Quantum Dynamics ◽  
Quantum Physics ◽  
Information Gain ◽  
Initial Conditions ◽  
Foundations Of Quantum Mechanics ◽  
Von Neumann Entropy ◽  
Quantum Evolution ◽  
Von Neumann ◽  
The Universe

I compare the role of the information in classical and quantum dynamics by examining the relation between information flows in measurements and the ability of observers to reverse evolutions. I show that in the Newtonian dynamics reversibility is unaffected by the observer’s retention of the information about the measurement outcome. By contrast—even though quantum dynamics is unitary, hence, reversible—reversing quantum evolution that led to a measurement becomes, in principle, impossible for an observer who keeps the record of its outcome. Thus, quantum irreversibility can result from the information gain rather than just its loss—rather than just an increase of the (von Neumann) entropy. Recording of the outcome of the measurement resets, in effect, initial conditions within the observer’s (branch of) the Universe. Nevertheless, I also show that the observer’s friend—an agent who knows what measurement was successfully carried out and can confirm that the observer knows the outcome but resists his curiosity and does not find out the result—can, in principle, undo the measurement. This relativity of quantum reversibility sheds new light on the origin of the arrow of time and elucidates the role of information in classical and quantum physics. Quantum discord appears as a natural measure of the extent to which dissemination of information about the outcome affects the ability to reverse the measurement. This article is part of a discussion meeting issue ‘Foundations of quantum mechanics and their impact on contemporary society’.

scholarly journals Upper bounds on mixing rates

2013 ◽  
Vol 13 (11&12) ◽  
pp. 986-994
Author(s):  
Elliott H. Lieb ◽  
Anna Vershynina
Keyword(s):  
Density Operator ◽  
Random Variable ◽  
Upper Bounds ◽  
Von Neumann Entropy ◽  
Mixing Rate ◽  
Unitary Evolution ◽  
Von Neumann ◽  
Mixing Rates ◽  
Non Local ◽  
Expected Density

We prove upper bounds on the rate, called "mixing rate", at which the von Neumann entropy of the expected density operator of a given ensemble of states changes under non-local unitary evolution. For an ensemble consisting of two states, with probabilities of p and 1-p, we prove that the mixing rate is bounded above by 4\sqrt{p(1-p)} for any Hamiltonian of norm 1. For a general ensemble of states with probabilities distributed according to a random variable X and individually evolving according to any set of bounded Hamiltonians, we conjecture that the mixing rate is bounded above by a Shannon entropy of a random variable $X$. For this general case we prove an upper bound that is independent of the dimension of the Hilbert space on which states in the ensemble act.

An entropy inequality

2009 ◽  
Vol 9 (7&8) ◽  
pp. 622-627
Author(s):  
M. Hellmund ◽  
A. Uhlmann
Keyword(s):  
Entropy Inequality ◽  
Quantum Information Theory ◽  
Upper Bounds ◽  
Symmetric Polynomial ◽  
Von Neumann Entropy ◽  
Elementary Symmetric Polynomial ◽  
Von Neumann ◽  
Entanglement Of Formation ◽  
Output Entropy ◽  
Quantities Of Interest

Let $S(\rho)=-\Tr (\rho\log\rho)$ be the von Neumann entropy of an $N$-dimensional quantum state $\rho$ and $e_2(\rho)$ the second elementary symmetric polynomial of the eigenvalues of $\rho$. We prove the inequality S(\rho) \;\le \; c(N) \; \sqrt{e_2(\rho)} where $c(N)=\log(N) \, \sqrt{\frac{2N}{N-1}}$. This generalizes an inequality given by Fuchs and Graaf~\cite{fuchsgraaf} for the case of one qubit, i.e., $N=2$. Equality is achieved if and only if $\rho$ is either a pure or the maximally mixed state. This inequality delivers new bounds for quantities of interest in quantum information theory, such as upper bounds for the minimum output entropy and the entanglement of formation as well as a lower bound for the Holevo channel capacity.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

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