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CRC Press eBooks

More Problems in Classical and Quantum Information Theory

Harish Parthasarathy

Published Oct 17, 2022• 0 citations
Publisher

Abstract

[1] If X is a Gaussian random vector with covariance R, then since the normal distribution maximizes the entropy of a probability distribution with second moment constraint, it follows that for any random variable Z having covariance R, we have https://www.w3.org/1998/Math/MathML"> H ( Z ) ≤ H ( X ) = ( n / 2 log ) ( 2 π e ) + ( 1 / 2 ) log | R | https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353454/e21648bc-3c5c-46b7-b52d-2c5637ecef01/content/math11_171_1_B.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>

DOI
10.1201/9781003353454-11

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[1] If X is a Gaussian random vector with covariance R, then since the normal distribution maximizes the entropy of a probability distribution with second moment constraint, it follows that for any random variable Z having covariance R, we have https://www.w3.org/1998/Math/MathML"> H ( Z ) ≤ H ( X ) = ( n / 2 log ) ( 2 π e ) + ( 1 / 2 ) log | R | https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353454/e21648bc-3c5c-46b7-b52d-2c5637ecef01/content/math11_171_1_B.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>
Let ρ, σ be two states and let spectral be https://www.w3.org/1998/Math/MathML"> ρ = ∑ a | e a > p ( a ) < e a | , σ = ∑ a | f a > q ( a ) < f a | https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353553/1603a02c-0268-4c9d-8b97-09135b05559e/content/math10_129_1_B.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> so that < e a | e b >= δ ( a, b ), < f a | f b >= δ ( a, b ). The relative entropy between these two states is https://www.w3.org/1998/Math/MathML"> D ( ρ | σ ) = T r ( ρ . ( log ( ρ ) − log ( σ ) ) ) = ∑ a p ( a ) log ( p ( a ) ) − ∑ a , b p ( a ) log ( q b ) | < e a | f b > | 2 = ∑ a , b ( p ( a ) | < e a | f b > | 2 ( log ( p ( a ) | < e a | f b > | 2 ) − log ( q b | < e a | f b > | 2 ) ) = ∑ a , b ( P ( a , b ) log ( P ( a , b ) ) − log ( Q ( a , b ) ) ) = D ( P | Q ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353553/1603a02c-0268-4c9d-8b97-09135b05559e/content/math10_129_2_B.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>
Relative entropy is an essential tool in quantum information theory. There are so many problems which are related to relative entropy. In this article, the optimal values which are defined by $\displaystyle\max_{U\in{U(\cX_{d})}} S(Uρ{U^{\ast}}\parallelσ)$ and $\displaystyle\min_{U\in{U(\cX_{d})}} S(Uρ{U^{\ast}}\parallelσ)$ for two positive definite operators $ρ,σ\in{\textmd{Pd}(\cX)}$ are obtained. And the set of S(UρU∗∥σ)S(Uρ{U^{\ast}}\parallelσ)S(UρU∗∥σ) for every unitary operator UUU is full of the interval $[\displaystyle\min_{U\in{U(\cX_{d})}} S(Uρ{U^{\ast}}\parallelσ),\displaystyle\max_{U\in{U(\cX_{d})}} S(Uρ{U^{\ast}}\parallelσ)]$
Since Q, P do not commute in quantum mechanics, we cannot talk of a joint probability density of these two at any time unlike the classical case. So how to interpret the Lindblad equation https://www.w3.org/1998/Math/MathML"> i ∂ t ρ = [ H , ρ ] − ( 1 / 2 ) ( L * L ρ + ρ L * L − 2 L ρ L * ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353454/e21648bc-3c5c-46b7-b52d-2c5637ecef01/content/math3_13_1_B.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>