By Masahito Hayashi
This ebook is the 1st one addressing quantum details from the point of view of crew symmetry. Quantum platforms have a bunch symmetrical constitution. This constitution permits to address systematically quantum details processing. even if, there isn't any different textbook targeting crew symmetry for quantum info even if there exist many textbooks for crew illustration. After the mathematical coaching of quantum details, this e-book discusses quantum entanglement and its quantification through the use of team symmetry. crew symmetry vastly simplifies the calculation of a number of entanglement measures even if their calculations are typically very tough to deal with. This e-book treats optimum info strategies together with quantum kingdom estimation, quantum kingdom cloning, estimation of staff motion and quantum channel and so forth. often it's very tough to derive the optimum quantum details strategies with no asymptotic atmosphere of those themes. even though, team symmetry permits to derive those optimum suggestions with no assuming the asymptotic atmosphere. subsequent, this booklet addresses the quantum blunders correcting code with the symmetric constitution of Weyl-Heisenberg teams. This constitution ends up in comprehend the quantum errors correcting code systematically. ultimately, this ebook 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 conception with the kind approach. the necessary mathematical wisdom approximately workforce illustration is summarized within the better half e-book, workforce illustration for Quantum conception.
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Additional resources for A Group Theoretic Approach to Quantum Information
As another type of measures, we can measure the difference between two density matrices ρ1 and ρ2 by modifying the information quantity in the previous subsection. Firstly, based on von Neumann entropy, we define relative entropy D(ρ1 ρ2 ) := s Tr ρ1 (log ρ1 − log ρ2 ). Also, we define ψ(s|ρ1 ρ2 ) := log Tr ρ1−s 1 ρ2 . Using this function, we define relative Renyi entropy of order 1 − s: D1−s (ρ1 ρ2 ) := − ψ(s|ρs1 ρ2 ) . Taking the limit s → 0, we can recover the relative entropy D(ρ1 ρ2 ) = lims→0 D1−s (ρ1 ρ2 ).
When a POVM can be realized by one-way LOCC, two-way LOCC, or separable, it is called a one-way LOCC POVM, a two-way LOCC POVM, or a separable POVM, respectively. 1 Locality Conditions 41 separable operation two-way LOCC one-way LOCC Fig. 1 Relation among local operations where M m,l is a positive semi definite matrix on the system Hm . Now, we recall the j majorization relation between two density matrices ρ1 and ρ2 defined in Sect. 2. 1 (Nielsen ) Given two vector states |Φ and |Ψ on the composite system H A ⊗ H B , we define the two states ρ1 := Tr B |Φ Φ| and ρ2 := Tr B |Ψ Ψ |.
58) hold for R ∈ [0, log k]. 54). 54) also guarR )) antees that S R = R−ψ(s(S . 56). 57). 59) 34 2 Quantum Channel, and Information Quantity, and Their Mathematical Structure and − H (ρ ) + H (ρs ) = Tr ρ log ρ − log ρ1−s Tr ρ1−s =D(ρ ρs ) + (1 − s) Tr ρ − + Tr ρ − ρ1−s Tr ρ1−s ρ1−s Tr ρ1−s log ρ1−s Tr ρ1−s log ρ. 54) implies H (ρs(SR ) ) = R. 60) imply D(ρ ρs(SR ) ) = − Tr ρ − ρs(SR ) log ρ 1 − s(S R ) D(ρ ρ) − D(ρs(SR ) ρ) − D(ρ ρs(SR ) ) . = s(S R ) That is, we have D(ρ ρ) − D(ρs(SR ) ρ) = 1 D(ρ ρs(SR ) ) ≥ 0.
A Group Theoretic Approach to Quantum Information by Masahito Hayashi