By Masahito Hayashi
This e-book is the 1st one addressing quantum info from the point of view of crew symmetry. Quantum platforms have a gaggle symmetrical constitution. This constitution permits to address systematically quantum info processing. besides the fact that, there isn't any different textbook concentrating on crew symmetry for quantum details even though there exist many textbooks for workforce illustration. After the mathematical education of quantum details, this booklet discusses quantum entanglement and its quantification by utilizing staff symmetry. staff symmetry tremendously simplifies the calculation of a number of entanglement measures even supposing their calculations are typically very tough to deal with. This e-book treats optimum details tactics together with quantum nation estimation, quantum kingdom cloning, estimation of workforce motion and quantum channel and so on. often it's very tough to derive the optimum quantum info strategies with out asymptotic environment of those themes. despite the fact that, staff symmetry permits to derive those optimum suggestions with no assuming the asymptotic atmosphere. subsequent, this e-book addresses the quantum mistakes correcting code with the symmetric constitution of Weyl-Heisenberg teams. This constitution results in comprehend the quantum mistakes correcting code systematically. ultimately, this e-book specializes in the quantum common info protocols by utilizing the gang SU(d). This subject may be considered as a quantum model of the Csiszar-Korner's common coding thought with the kind strategy. the mandatory mathematical wisdom approximately crew illustration is summarized within the significant other ebook, workforce illustration for Quantum thought.
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Extra resources for A Group Theoretic Approach to Quantum Information
D −1 } as follows. 4) where ωZ := ei2π/d . This definition is equivalent to the following definition. XZ = |x + 1 x|, ZZ = ωZx |x x|. 5) x Then, we define W as ˜ Z (s, t) := Xs Zt = W ωZxt |x + s x|, ∀(s, t) ∈ Z2d . 7) ˜ Z is a projective unitary representation of Z2 . holds. So, W d (d+1)/2 , and when d is even, we set τZ := eiπ/d . 8) s where st takes values in integers between 0 and d − 1. 9) where is 0 or 1 dependently of s, t, s , t . WZ (s, t) is called discrete Heisenberg representation of Z2d .
71). 73) is a negative number for s < 1. Thus, sup s<1 s R − ψ(s) s R − ψ(s) = lim = − log p1 − R. 72). Summarizing the above lemmas, we obtain the following theorem. 76) D(q p) = min D(ρ ρ) H (ρ )≥R s R − ψ(s) s = max (R − H1−s (ρ)). 8 Rényi Mutual Information In Sect. 3, we discussed the mutual information Iρ (A : B) on the composite system H A ⊗ H B when ρ is a density matrix on H A ⊗ H B . Recall tho two expressions for the mutual information Iρ (A : B) as Iρ (A : B) = D(ρ ρ A ⊗ ρ B ) = min D(ρ ρ A ⊗ σ B ).
Proof First, we notice that the averaged state ρ is invariant with respect to f. So, since any density matrix ρ satisfies D(f(g)ρf(g)† σ) = D(ρ σ), we have D(f(g)ρf(g)† σ)μG (dg) D(ρ σ) = G D(f(g)ρf(g)† ρ)μG (dg) + D(ρ σ) = G =D(ρ ρ) + D(ρ σ) = H (ρ) − H (ρ) + D(ρ σ). 5 Show that Iρ (A : B) = D(ρ ρ A ⊗ ρ B ). 43) where h(x) := −x log x − (1 − x) log(1 − x). 7 Show the following type of Pythagorean theorem. D(ρ σ A ⊗ σ B ) = D(ρ σ A ⊗ ρ B ) + D(ρ B σ B ). 8 We consider the pinching Λ E with respect to the spectral decomposition E of σ.
A Group Theoretic Approach to Quantum Information by Masahito Hayashi