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.

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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.