By Masahito Hayashi

ISBN-10: 3319452398

ISBN-13: 9783319452395

ISBN-10: 331945241X

ISBN-13: 9783319452418

This booklet is the 1st one addressing quantum details from the perspective of staff symmetry. Quantum structures have a gaggle symmetrical constitution. This constitution permits to deal with systematically quantum info processing. even though, there's no different textbook concentrating on workforce symmetry for quantum info even supposing there exist many textbooks for workforce illustration. After the mathematical guidance of quantum info, this ebook discusses quantum entanglement and its quantification by utilizing workforce symmetry. crew symmetry enormously simplifies the calculation of a number of entanglement measures even if their calculations tend to be very tricky to address. This ebook treats optimum details methods together with quantum country estimation, quantum kingdom cloning, estimation of staff motion and quantum channel and so on. often it's very tough to derive the optimum quantum info strategies with out asymptotic surroundings of those themes. besides the fact that, staff symmetry permits to derive those optimum ideas with no assuming the asymptotic surroundings. subsequent, this e-book addresses the quantum errors correcting code with the symmetric constitution of Weyl-Heisenberg teams. This constitution results in comprehend the quantum mistakes correcting code systematically. ultimately, this ebook makes a speciality of the quantum common details protocols by utilizing the crowd SU(d). This subject could be considered as a quantum model of the Csiszar-Korner's common coding thought with the kind technique. the mandatory mathematical wisdom approximately team illustration is summarized within the significant other ebook, staff illustration for Quantum thought.

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83) x=1 for t ∈ (0, 1). Then, we find that φW, p (t) is a concave function. This function will be used for a universal code for classical-quantum channel in Sect. 7. 11 Since (Tr B (Tr A ρ1+s ρ−s A ) have es D1+s (ρ ρ A ⊗σ B ) −s 1+s −s 1+s −s = Tr ρ1+s ρ−s ρA ) σB A σ B = Tr B (Tr A ρ 1+s 1+s (Tr A ρ1+s ρ−s A ) 1 =e s D1+s (ρ ρ A ⊗σ ∗B (1+s)) = es D1+s (ρ Tr B ( 1+s Tr B (Tr A ρ1+s ρ−s A ) ρ A ⊗σ ∗B (1+s)) s D1+s (σ ∗B (1+s) σ B ) e 1 )1+s σ −s B . 22). Chapter 3 Entanglement and Its Quantification Abstract Quantum mechanics is completely far from everyday intuition not only because the measured outcome can be predicted only probabilistically but also because of a quantum-specific correlation called entanglement.

8]. First of all, we review discrete Heisenberg representation of the algebra Zd , which is a d-dimensional irreducible projective unitary representation as follows. We define two matrices XZ and ZZ on the system with the computational basis {|0 , . . , |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 .

6 (matrix Hölder inequality) When 1/ p + 1/q = 1 and ∞ > p, q > 1, two matrix A and B satisfy AB 1 ≤ A p B q. 21) Using this inequality, we can show the reverse matrix Hölder inequality as follows. 7 (matrix reverse Hölder inequality) When 1/ p + 1/q = 1 and 1 > p > 0 > q, two positive semi definite matrices A and B satisfy AB 1 ≥ A p B q. 22) Proof We show the desired argument in the case when A and B are invertible. Otherwise, we can show the argument by taking the limit. Choose two matrices a := log A and b := log B and a real number s := − q1 .

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A Group Theoretic Approach to Quantum Information by Masahito Hayashi

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