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.

scholarly journals Notes on entropic characteristics of quantum channels

Open Physics ◽  
2013 ◽  
Vol 11 (1) ◽  
Author(s):  
Alexey Rastegin
Keyword(s):  
Information Theory ◽  
Quantum Information Theory ◽  
Upper Bounds ◽  
Quantum Channels ◽  
Average Output ◽  
Von Neumann ◽  
Channel Characteristics ◽  
Transmission Of Information ◽  
Output Entropy ◽  
Generalized Entropies

AbstractOne 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.

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 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.

scholarly journals Quantifying Disorder in Networks

2011 ◽  
pp. 66-76 ◽  
Author(s):  
Filippo Passerini ◽  
Simone Severini
Keyword(s):  
Information Theory ◽  
Spectral Properties ◽  
Quantum Information Theory ◽  
Von Neumann Entropy ◽  
Connected Components ◽  
Quantum Mechanical ◽  
Von Neumann ◽  
Regularity Properties ◽  
Wide Interface ◽  
Edge Addition

The authors introduce a novel entropic notion with the purpose of quantifying disorder/uncertainty in networks. This is based on the Laplacian and it is exactly the von Neumann entropy of certain quantum mechanical states. It is remarkable that the von Neumann entropy depends on spectral properties and it can be computed efficiently. The analytical results described here and the numerical computations lead us to conclude that the von Neumann entropy increases under edge addition, increases with the regularity properties of the network and with the number of its connected components. The notion opens the perspective of a wide interface between quantum information theory and the study of complex networks at the statistical level.

scholarly journals Order-Stability in Complex Biological, Social, and AI-Systems from Quantum Information Theory

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 355
Author(s):  
Andrei Khrennikov ◽  
Noboru Watanabe
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Quantum Information Theory ◽  
Von Neumann Entropy ◽  
Order Structure ◽  
Compound System ◽  
Von Neumann ◽  
Local Disorder ◽  
Low Entropy ◽  
The Brain

This paper is our attempt, on the basis of physical theory, to bring more clarification on the question “What is life?” formulated in the well-known book of Schrödinger in 1944. According to Schrödinger, the main distinguishing feature of a biosystem’s functioning is the ability to preserve its order structure or, in mathematical terms, to prevent increasing of entropy. However, Schrödinger’s analysis shows that the classical theory is not able to adequately describe the order-stability in a biosystem. Schrödinger also appealed to the ambiguous notion of negative entropy. We apply quantum theory. As is well-known, behaviour of the quantum von Neumann entropy crucially differs from behaviour of classical entropy. We consider a complex biosystem S composed of many subsystems, say proteins, cells, or neural networks in the brain, that is, S=(Si). We study the following problem: whether the compound system S can maintain “global order” in the situation of an increase of local disorder and if S can preserve the low entropy while other Si increase their entropies (may be essentially). We show that the entropy of a system as a whole can be constant, while the entropies of its parts rising. For classical systems, this is impossible, because the entropy of S cannot be less than the entropy of its subsystem Si. And if a subsystems’s entropy increases, then a system’s entropy should also increase, by at least the same amount. However, within the quantum information theory, the answer is positive. The significant role is played by the entanglement of a subsystems’ states. In the absence of entanglement, the increasing of local disorder implies an increasing disorder in the compound system S (as in the classical regime). In this note, we proceed within a quantum-like approach to mathematical modeling of information processing by biosystems—respecting the quantum laws need not be based on genuine quantum physical processes in biosystems. Recently, such modeling found numerous applications in molecular biology, genetics, evolution theory, cognition, psychology and decision making. The quantum-like model of order stability can be applied not only in biology, but also in social science and artificial intelligence.

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.

Quantifying Complexity in Networks

2009 ◽  
Vol 1 (4) ◽  
pp. 58-67 ◽  
Author(s):  
Filippo Passerini ◽  
Simone Severini
Keyword(s):  
Spectral Properties ◽  
Quantum Information Theory ◽  
Von Neumann Entropy ◽  
Connected Components ◽  
Quantum Mechanical ◽  
Von Neumann ◽  
Regularity Properties ◽  
Statistical Level ◽  
Wide Interface ◽  
Edge Addition

The authors introduce a novel entropic notion with the purpose of quantifying disorder/uncertainty in networks. This is based on the Laplacian and it is exactly the von Neumann entropy of certain quantum mechanical states. It is remarkable that the von Neumann entropy depends on spectral properties and it can be computed efficiently. The analytical results described here and the numerical computations lead us to conclude that the von Neumann entropy increases under edge addition, increases with the regularity properties of the network and with the number of its connected components. The notion opens the perspective of a wide interface between quantum information theory and the study of complex networks at the statistical level.

scholarly journals Islands in linear dilaton black holes

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Georgios K. Karananas ◽  
Alex Kehagias ◽  
John Taskas
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Entanglement Entropy ◽  
Von Neumann Entropy ◽  
Two Dimensional ◽  
Extremal Case ◽  
Von Neumann ◽  
Dimensional Black Hole ◽  
Linear Dilaton ◽  
Linear Dilaton Black Hole

Abstract We derive a novel four-dimensional black hole with planar horizon that asymptotes to the linear dilaton background. The usual growth of its entanglement entropy before Page’s time is established. After that, emergent islands modify to a large extent the entropy, which becomes finite and is saturated by its Bekenstein-Hawking value in accordance with the finiteness of the von Neumann entropy of eternal black holes. We demonstrate that viewed from the string frame, our solution is the two-dimensional Witten black hole with two additional free bosons. We generalize our findings by considering a general class of linear dilaton black hole solutions at a generic point along the σ-model renormalization group (RG) equations. For those, we observe that the entanglement entropy is “running” i.e. it is changing along the RG flow with respect to the two-dimensional worldsheet length scale. At any fixed moment before Page’s time the aforementioned entropy increases towards the infrared (IR) domain, whereas the presence of islands leads the running entropy to decrease towards the IR at later times. Finally, we present a four-dimensional charged black hole that asymptotes to the linear dilaton background as well. We compute the associated entanglement entropy for the extremal case and we find that an island is needed in order for it to follow the Page curve.

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