By Masahito Hayashi
This publication is the 1st one addressing quantum details from the perspective of team symmetry. Quantum structures have a bunch symmetrical constitution. This constitution allows to address systematically quantum details processing. despite the fact that, there is not any different textbook targeting team symmetry for quantum details even though there exist many textbooks for crew illustration. After the mathematical coaching of quantum details, this ebook discusses quantum entanglement and its quantification by utilizing workforce symmetry. team symmetry enormously simplifies the calculation of a number of entanglement measures even though their calculations tend to be very tough to address. This ebook treats optimum info tactics together with quantum nation estimation, quantum country cloning, estimation of crew motion and quantum channel and so forth. frequently it's very tricky to derive the optimum quantum info methods with no asymptotic environment of those issues. even though, crew symmetry permits to derive those optimum options with no assuming the asymptotic atmosphere. subsequent, this ebook addresses the quantum mistakes correcting code with the symmetric constitution of Weyl-Heisenberg teams. This constitution ends up in comprehend the quantum blunders correcting code systematically. ultimately, this ebook makes a speciality of the quantum common info protocols through the use of the gang SU(d). This subject could be considered as a quantum model of the Csiszar-Korner's common coding thought with the sort approach. the mandatory mathematical wisdom approximately staff illustration is summarized within the spouse booklet, team illustration for Quantum concept.
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Extra info for A Group Theoretic Approach to Quantum Information
4 Information Quantities on Quantum System 23 Generally, when a function f (ρ1 , ρ2 ) is given as the square of a distance, f (ρ1 , ρ2 ) is a pseudo square distance. In the following, we discuss several examples for measures for the difference between two density matrices ρ1 and ρ2 . The first one is trace norm distance d1 (ρ1 , ρ2 ) := 21 ρ1 − ρ2 1 , which is given by the trace norm. The second one is √ √ the fidelity F(ρ1 , ρ2 ) := Tr | ρ1 ρ2 |. When ρ1 or ρ2 is pure, its square is Tr ρ1 ρ2 .
Proof It is sufficient to show that the vector space generated by the matrix g j the same as the vector space generated by the matrix r = 3. When g1 = Q 1 ⊕ P1 ⊕ P1 , I ζ is I . 3 Examples of Entangled States 49 ⎛ ⎞ ⎛ ⎞ 011 011 ζ = ⎝ 1 0 0 ⎠ , ζˆ = ⎝ 1 0 1 ⎠ , 100 110 with a simple calculation, we find that the vector space spanned by I ζ is converted I by the action g1 . Hence, we obtain the desired ζˆ statement with r = 3. Further, we can show the general case in the same way as the case of r = 3.
For example, when 1 > s > 0, the functions x → −x −s , x → x s , x → log x, x → −x 1+s , and x → −x log x are matrix concave. The functions x → x s , x → −x s , and x → log x are matrix monotone. 9]. 7]. 4 The following conditions are equivalent for a real-valued function f defined on [0, ∞). (1) f is matrix convex. (2) When a matrix C satisfies C † C ≤ I and the eigenvalues of a Hermitian matrix A belong to the domain of f , we have f (C † AC) ≤ C † f (A)C. † (3) Assume that a set of matrices C1 , .
A Group Theoretic Approach to Quantum Information by Masahito Hayashi