By Masahito Hayashi

This e-book is the 1st one addressing quantum info from the point of view of crew symmetry. Quantum structures have a gaggle symmetrical constitution. This constitution allows to address systematically quantum info processing. despite the fact that, there's no different textbook concentrating on workforce symmetry for quantum details even if there exist many textbooks for workforce illustration. After the mathematical training of quantum info, this booklet discusses quantum entanglement and its quantification through the use of workforce symmetry. workforce symmetry significantly simplifies the calculation of numerous entanglement measures even supposing their calculations are typically very tricky to address. This ebook treats optimum info procedures together with quantum kingdom estimation, quantum nation cloning, estimation of team motion and quantum channel and so on. often it's very tricky to derive the optimum quantum details procedures with out asymptotic surroundings of those issues. although, staff symmetry permits to derive those optimum recommendations with no assuming the asymptotic surroundings. subsequent, this publication addresses the quantum blunders correcting code with the symmetric constitution of Weyl-Heisenberg teams. This constitution ends up in comprehend the quantum mistakes correcting code systematically. ultimately, this publication specializes in the quantum common info protocols through the use of the gang SU(d). This subject might be considered as a quantum model of the Csiszar-Korner's common coding conception with the sort procedure. the necessary mathematical wisdom approximately workforce illustration is summarized within the significant other e-book, team illustration for Quantum idea.

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**Extra resources for A Group Theoretic Approach to Quantum Information**

**Example text**

15) := (s , t)X − (s, t )X . 16) where is 0 or 1 dependently of s, t, s , t . Hence, WrZ and WrF are projective unitary 2r representations of Z2r d and Fq , respectively. The projective unitary representations r r 2r WZ and WF are called discrete Heisenberg representation of Z2r d and Fq , respectively. 2 Stabilizer State As explained in Sect. 2, a vector state on the composite system H1 ⊗ H2 ( bipartite system) of two systems is described as |X by using a matrix X . In general, it is thought that the amount of entanglement is not changed by the applications of unitaries on individual systems, which is called a local unitary.

Hence, we have H (Λ ⊗ id R (|Φ Φ|)) = H (Λ E (ρ)), which implies Ic (ρ, Λ) = H (Λ(ρ)) − H (Λ E (ρ)). 47). 47). 11. 5 Qubit System Next, we consider a typical example of a quantum system, the two-dimensional system C2 , which can be regarded as a quantum analogue of “bit” and is called qubit. 49) we can characterize any density matrix on the qubit system as ρ x := 21 I + 3j=1 x j E j . Then, the eigenvectors of the Hermitian matrix ρ x are 1+2 x and 1−2 x . The Hermitian matrix ρ x is a density matrix if and only if x ≤ 1.

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 ). Further, larger relative Rényi entropy D1−s (ρ1 ρ2 ) implies larger difference between two density matrices ρ1 and ρ2 . When s = 1/2, eψ(s|ρ1 ρ2 ) is analogous to the fidelity F(ρ1 , ρ2 ). However, when ρ1 is not commutative with ρ2 , they are different.