AN OPERATIONAL ALGEBRAIC APPROACH TO QUANTUM CHANNEL CAPACITY

2008 ◽  
Vol 06 (05) ◽  
pp. 981-996 ◽  
Author(s):  
V. P. BELAVKIN ◽  
X. DAI
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Quantum Channel ◽  
Algebraic Approach ◽  
Quantum Information Theory ◽  
Quantum Channels ◽  
Operational Approach ◽  
Quantum Relative Entropy ◽  
Quantum Mutual Information ◽  
Arbitrary Quantum

An elementary introduction into algebraic approach to unified quantum information theory and operational approach to quantum entanglement as generalized encoding is given. After introducing compound quantum state and two types of informational divergences, namely, Araki–Umegaki (a-type) and of Belavkin–Staszewski (b-type) quantum relative entropic information, this paper treats two types of quantum mutual information via entanglement and defines two types of corresponding quantum channel capacities as the supremum via the generalized encodings. It proves the additivity property of quantum channel capacities via entanglement, which extends the earlier results of Belavkin to products of arbitrary quantum channels for quantum relative entropy of any type.

scholarly journals On the Mathematical Boundaries of Communication with Zero-Capacity Quantum Channels

10.29007/7h1q ◽  
2018 ◽  
Author(s):  
Laszlo Gyongyosi ◽  
Sandor Imre
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
21St Century ◽  
Quantum Information Theory ◽  
Quantum Mechanical ◽  
Quantum Channels ◽  
Classical Systems ◽  
Quantum Relative Entropy ◽  
Mathematical Properties ◽  
Zero Capacity

In the first decade of the 21st century, many revolutionary properties of quantum channels were discovered. These phenomena are purely quantum mechanical and completely unimaginable in classical systems. Recently, the most important discovery in Quantum Information Theory was the possibility of transmitting quantum information over zero-capacity quantum channels. In this work we prove that the possibility of superactivation of quantum channel capacities is determined by the mathematical properties of the quantum relative entropy function.

THE QUANTUM CAPACITY BOUNDS OF A LOSSY GAUSSIAN QUANTUM CHANNEL

2011 ◽  
Vol 09 (04) ◽  
pp. 1081-1090
Author(s):  
XIAO-YU CHEN ◽  
LI-ZHEN JIANG
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Lower Bound ◽  
Quantum Channel ◽  
Quantum Information Theory ◽  
Upper And Lower Bounds ◽  
Gaussian State ◽  
Quantum Capacity ◽  
Capacity Bounds ◽  
N Use

Quantum capacity of the lossy Gaussian quantum channel remains an open problem in quantum information theory, although the upper and lower bounds are well-known. We show that for the n-use of the channel, the input of entangled commutative Gaussian state does not improve the lower bound of the capacity. When the total energy is limited, an unfair distribution of the energy among the n-use will improve the lower bound.

QUANTUM ENTROPIES AND ENTANGLEMENT

2005 ◽  
Vol 03 (01) ◽  
pp. 99-104 ◽  
Author(s):  
J. BATLE ◽  
M. CASAS ◽  
A. R. PLASTINO ◽  
A. PLASTINO
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Relative Entropy ◽  
Conditional Entropy ◽  
Quantum Information Theory ◽  
Total Entropy ◽  
Separability Criterion ◽  
Quantum Relative Entropy ◽  
Entanglement Of Formation

The nature of quantum entropies, and its use in Quantum Information Theory in the form of (i) total entropy, (ii) relative entropy and (iii) conditional entropy is revisited. In this ordering, we first show the correlations existing between the total q-entropy and entanglement, quantified in the form of entanglement of formation. Then, we revisit the use of the quantum relative entropy as a measure of entanglement, and we finally discuss some features of the quantum conditional q-entropies, which are used in turn as a separability criterion.

scholarly journals SINGULAR VALUE DECOMPOSITION AND MATRIX REORDERINGS IN QUANTUM INFORMATION THEORY

2011 ◽  
Vol 22 (09) ◽  
pp. 897-918 ◽  
Author(s):  
JAROSŁAW ADAM MISZCZAK
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Singular Value Decomposition ◽  
Computing System ◽  
Quantum Information Theory ◽  
Singular Value ◽  
Quantum States ◽  
Quantum Channels ◽  
Basic Functions ◽  
Value Decomposition

We review Schmidt and Kraus decompositions in the form of singular value decomposition using operations of reshaping, vectorization and reshuffling. We use the introduced notation to analyze the correspondence between quantum states and operations with the help of Jamiołkowski isomorphism. The presented matrix reorderings allow us to obtain simple formulae for the composition of quantum channels and partial operations used in quantum information theory. To provide examples of the discussed operations, we utilize a package for the Mathematica computing system implementing basic functions used in the calculations related to quantum information theory.

scholarly journals Some applications of the Hermit-Hadamard inequality for log-convex functions in quantum divergence

2021 ◽  
Author(s):  
Fatemeh Hassanzad ◽  
Hossien Mehri-Dehnavi ◽  
Hamzeh Agahi
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Relative Entropy ◽  
Convex Functions ◽  
Quantum Information Theory ◽  
Density Matrices ◽  
Hadamard Inequality ◽  
Quantum Relative Entropy ◽  
Nonlinear Anal ◽  
Jensen Shannon Divergence

One of the beautiful and very simple inequalities for a convex function is the Hermit-Hadamard inequality [S. Mehmood, et. al. Math. Methods Appl. Sci., 44 (2021) 3746], [S. Dragomir, et. al., Math. Methods Appl. Sci., in press]. The concept of log-convexity is a stronger property of convexity. Recently, the refined Hermit-Hadamard’s inequalities for log-convex functions were introduced by researchers [C. P. Niculescu, Nonlinear Anal. Theor., 75 (2012) 662]. In this paper, by the Hermit-Hadamard inequality, we introduce two parametric Tsallis quantum relative entropy, two parametric Tsallis-Lin quantum relative entropy and two parametric quantum Jensen-Shannon divergence in quantum information theory. Then some properties of quantum Tsallis-Jensen-Shannon divergence for two density matrices are investigated by this inequality. \newline \textbf{Keywords:} \textit{ Hermit-Hadamard’s inequality; log-convexity; Density matrices; Quantum relative entropy; Tsallis quantum relative entropy; quantum Jensen-Shannon divergence divergence.

scholarly journals Recoverability in quantum information theory

2015 ◽  
Vol 471 (2182) ◽  
pp. 20150338 ◽  
Author(s):  
Mark M. Wilde
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Relative Entropy ◽  
Entropy Inequality ◽  
Quantum Information Theory ◽  
The Other ◽  
Complex Interpolation ◽  
The Core ◽  
Quantum Relative Entropy ◽  
Entropy Inequalities

The fact that the quantum relative entropy is non-increasing with respect to quantum physical evolutions lies at the core of many optimality theorems in quantum information theory and has applications in other areas of physics. In this work, we establish improvements of this entropy inequality in the form of physically meaningful remainder terms. One of the main results can be summarized informally as follows: if the decrease in quantum relative entropy between two quantum states after a quantum physical evolution is relatively small, then it is possible to perform a recovery operation, such that one can perfectly recover one state while approximately recovering the other. This can be interpreted as quantifying how well one can reverse a quantum physical evolution. Our proof method is elementary, relying on the method of complex interpolation, basic linear algebra and the recently introduced Rényi generalization of a relative entropy difference. The theorem has a number of applications in quantum information theory, which have to do with providing physically meaningful improvements to many known entropy inequalities.

scholarly journals Tensorization of the strong data processing inequality for quantum chi-square divergences

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 199
Author(s):  
Yu Cao ◽  
Jianfeng Lu
Keyword(s):  
Data Processing ◽  
Relative Entropy ◽  
Quantum Channel ◽  
Quantum States ◽  
Quantum Channels ◽  
Chi Square ◽  
Quantum Regime ◽  
Quantum Relative Entropy ◽  
Data Processing Inequality ◽  
Arbitrary Quantum

It is well-known that any quantum channel E satisfies the data processing inequality (DPI), with respect to various divergences, e.g., quantum χκ2 divergences and quantum relative entropy. More specifically, the data processing inequality states that the divergence between two arbitrary quantum states ρ and σ does not increase under the action of any quantum channel E. For a fixed channel E and a state σ, the divergence between output states E(ρ) and E(σ) might be strictly smaller than the divergence between input states ρ and σ, which is characterized by the strong data processing inequality (SDPI). Among various input states ρ, the largest value of the rate of contraction is known as the SDPI constant. An important and widely studied property for classical channels is that SDPI constants tensorize. In this paper, we extend the tensorization property to the quantum regime: we establish the tensorization of SDPIs for the quantum χκ1/22 divergence for arbitrary quantum channels and also for a family of χκ2 divergences (with κ≥κ1/2) for arbitrary quantum-classical channels.

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