By Bodo Pareigis

ISBN-10: 0125451504

ISBN-13: 9780125451505

**Read or Download Categories and Functors (Pure and Applied Mathematics, Vol. 39) PDF**

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**Additional resources for Categories and Functors (Pure and Applied Mathematics, Vol. 39) **

**Sample text**

39) Generally, we have seen that a bipartite state is separable only if it remains positive semi-definite under the partial transpose. The converse is not true in general. 3 Purification Consider any state ρ AB ∈ S (AB), and its marginals ρ A and ρ B . Then we say that ρ AB is an extension of ρ A and ρ B . Moreover, if ρ AB is pure, we call it a purification of ρ A and ρ B . Moreover, we can always construct a purification of a given state ρ A ∈ S (A). 40) x is a purification of ρ A . Here, A is an auxiliary system of the same dimension as A and {|ex A }x is any ONB of A .

3) These quantities are well-defined if ρ X and σ X have full support and otherwise we use the convention that 0 log 0 = 0 and 00 = 1, which ensures that the divergences ρ X . Finally, note that both are indeed continuous whenever ρ X = 0 and σ X quantities diverge to +∞ if the latter condition is not satisfied and α > 1. 2 Positive Definiteness and Data-Processing Unlike in the classical case, the above axioms do not uniquely determine a quantum generalization of these divergences. Hence, we first list some additional properties we would like a quantum generalization of the Rényi divergence to have.

Concavity and dataprocessing for all quantum Rényi divergences. It establishes that for functionals satisfying (I)–(VI), these properties are equivalent. 2 Let D be a functional satisfying (I)–(VI) and let g and Q be defined as in (VI). Then, the following two statements are equivalent. (1) Q is jointly convex (IXa) if g is monotonically increasing, or jointly concave (IXb) if g is monotonically decreasing. (2) D satisfies the data-processing inequality (VIII).

### Categories and Functors (Pure and Applied Mathematics, Vol. 39) by Bodo Pareigis

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