aqora

Some More Aspects of Quantum Information Theory

Harish Parthasarathy

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 "/>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 "/>

CRC Press eBooksNov 1, 20220 citations

Quantum Information Theory, A Selection of Matrix Inequalities

Harish Parthasarathy

Statement of the result: Let 0 ≤ s ≤ 1 and let K be a TPCP map acting on the convex set of states in a finite dimensional Hilbert space. Then, for any two states +ρ, σ, we have https://www.w3.org/1998/Math/MathML"> T r ( K ( σ ) s K ( ρ ) 1 − s ) ≥ T r ( σ s ρ 1 − s ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353454/e21648bc-3c5c-46b7-b52d-2c5637ecef01/content/math1_1_B.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>Statement of the result: Let 0 ≤ s ≤ 1 and let K be a TPCP map acting on the convex set of states in a finite dimensional Hilbert space. Then, for any two states +ρ, σ, we have https://www.w3.org/1998/Math/MathML"> T r ( K ( σ ) s K ( ρ ) 1 − s ) ≥ T r ( σ s ρ 1 − s ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353454/e21648bc-3c5c-46b7-b52d-2c5637ecef01/content/math1_1_B.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>

CRC Press eBooksOct 17, 20220 citations

Some Results in Quantum Information Theory

Debashree Chakraborty

Jan 1, 20150 citations

More Problems in Classical and Quantum Information Theory

Harish Parthasarathy

[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 "/>[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 "/>

CRC Press eBooksOct 17, 20220 citations

Quantum Information Theory: Mathematical Foundation

Masahito Hayashi

Nov 17, 201689 citations

Basic Notions of Quantum Information

John Watrous

Cambridge University Press eBooksMay 17, 20182 citations

Some More Concepts and Results in Quantum Information Theory

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Some More Concepts and Results in Quantum Information Theory

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