A Group Theoretic Approach to Quantum Information by Masahito Hayashi

By Masahito Hayashi

This publication is the 1st one addressing quantum info from the point of view of staff symmetry. Quantum structures have a bunch symmetrical constitution. This constitution permits to deal with systematically quantum details processing. even if, there's no different textbook targeting team symmetry for quantum info even if there exist many textbooks for workforce illustration. After the mathematical training of quantum details, this e-book discusses quantum entanglement and its quantification by utilizing crew symmetry. staff symmetry tremendously simplifies the calculation of numerous entanglement measures even though their calculations are typically very tricky to address. This e-book treats optimum details tactics together with quantum nation estimation, quantum nation cloning, estimation of crew motion and quantum channel and so on. frequently it's very tough to derive the optimum quantum details methods with no asymptotic atmosphere of those issues. even though, crew symmetry permits to derive those optimum recommendations with out assuming the asymptotic environment. subsequent, this booklet addresses the quantum mistakes correcting code with the symmetric constitution of Weyl-Heisenberg teams. This constitution results in comprehend the quantum errors correcting code systematically. eventually, this booklet specializes in the quantum common details protocols by utilizing the gang SU(d). This subject should be considered as a quantum model of the Csiszar-Korner's common coding concept with the sort approach. the necessary mathematical wisdom approximately workforce illustration is summarized within the spouse e-book, crew illustration for Quantum Theory. 

Show description

Read or Download A Group Theoretic Approach to Quantum Information PDF

Best abstract books

Representation Theory and Complex Analysis: Lectures given at the C.I.M.E. Summer School held in Venice, Italy June 10–17, 2004

Six best specialists lecture on a large spectrum of contemporary effects with reference to the name, offering either an effective reference and deep insights on present study job. Michael Cowling offers a survey of assorted interactions among illustration conception and harmonic research on semisimple teams and symmetric areas.

Foundations of Analysis: A Straightforward Introduction: Book 2, Topological Ideas

This ebook is an creation to the guidelines from common topology which are utilized in ordinary research. it really is written at a degree that's meant to make the majority of the fabric obtainable to scholars within the latter a part of their first 12 months of analysis at a school or collage even if scholars will mostly meet lots of the paintings of their moment or later years.

Lie Groups and Algebraic Groups (Springer Series in Soviet Mathematics)

This can be a particularly impressive booklet on Lie teams and algebraic teams. produced from hectographed notes in Russian from Moscow college, which for plenty of Soviet mathematicians were whatever similar to a "bible", the booklet has been considerably prolonged and arranged to improve the fabric during the posing of difficulties and to demonstrate it via a wealth of examples.

Towards the Mathematics of Quantum Field Theory, 1st Edition

This formidable and unique publication units out to introduce to mathematicians (even together with graduate scholars ) the mathematical equipment of theoretical and experimental quantum box idea, with an emphasis on coordinate-free displays of the mathematical items in use. This in flip promotes the interplay among mathematicians and physicists via providing a standard and versatile language for the nice of either groups, notwithstanding mathematicians are the first goal.

Additional resources for A Group Theoretic Approach to Quantum Information

Sample text

32) In particular, since there exist unitaries U and U such that (I − E i )ρ 1 = Tr(I − E i )ρU and E i ρ(I − E i ) 1 = Tr E i ρ(I − E i )U , Schwartz inequality for the inner product Tr X † ρY between two matrices X and Y implies that ρ − E i ρE i 1 = (I − E i )ρ + E i ρ(I − E i ) 1 = (I − E i )ρ 1 + E i ρ(I − E i ) 1 = Tr(I − E i )ρU + Tr E i ρ(I − E i )U ≤ Tr(I − E i )ρ(I − E i ) Tr U † ρU + Tr E i ρE i Tr U † (I − E i )ρ(I − E i )U = Tr(I − E i )ρ + Tr E i ρ Tr(I − E i )ρ ≤ 2 Tr(I − E i )ρ.

Pk ) and q = ( p1 , . . , pk ), we can define relative entropy D( p q) := D(ρ( p) ρ(q)) by using the diagonal density matrices ρ( p) and ρ(q). As a comparison between the states before and after the application of pinching Λ X (ρ) = i E i ρE i with respect to the Hermitian matrix X = i xi E i , we obtain the following relation. D(ρ Λ X (ρ)) = H (Λ X (ρ)) − H (ρ). 32) In particular, since there exist unitaries U and U such that (I − E i )ρ 1 = Tr(I − E i )ρU and E i ρ(I − E i ) 1 = Tr E i ρ(I − E i )U , Schwartz inequality for the inner product Tr X † ρY between two matrices X and Y implies that ρ − E i ρE i 1 = (I − E i )ρ + E i ρ(I − E i ) 1 = (I − E i )ρ 1 + E i ρ(I − E i ) 1 = Tr(I − E i )ρU + Tr E i ρ(I − E i )U ≤ Tr(I − E i )ρ(I − E i ) Tr U † ρU + Tr E i ρE i Tr U † (I − E i )ρ(I − E i )U = Tr(I − E i )ρ + Tr E i ρ Tr(I − E i )ρ ≤ 2 Tr(I − E i )ρ.

54), we have 1 = − ds(S SR + dR d SR (1 − s(S R )) d R s (S R )ψ (s(S R )). 52) into this equation, we have 1 = (1 − s(S R )) d SR . 68), we obtain ds(S R ) 1 = < 0. 69) yield d s(S R ) 1 (S R − R) = −1= dR 1 − s(S R ) 1 − s(S R ) d2 1 (S R − R) = ψ (s(S R )) > 0. d R2 (1 − s(S R ))3 Hence, the function R → S R − R is a convex function. , R = H (ρ). Since S H (ρ) − H (ρ) = 0, this function realizes the minimum value 0 when R = H (ρ). 65). 66). We also have the following lemma with respect to the function ψ.

Download PDF sample

Rated 4.01 of 5 – based on 38 votes