Families of completely positive maps associated with monotone metrics

Abstract

An operator convex function on (0, ∞) which satisfies the symmetry condition k(x −1 ) = xk(x) can be used to define a type of non-commutative multiplication by a positive definite matrix (or its inverse) using the primitive concepts of left and right multiplication and the functional calculus. The operators for the inverse can be used to define quadratic forms associated with Riemannian metrics which contract under the action of completely positive trace-preserving maps.We study the question of when these operators define maps which are also completely positive (CP). Although A → D −1/2 AD −1/2 is the only case for which both the map and its inverse are CP, there are several well-known one parameter families for which either the map or its inverse is CP. We present a complete analysis of the behavior of these families, as well as the behavior of lines connecting an extreme point with the smallest one and some results for geometric bridges between these points.Our primary tool is an order relation based on the concept of positive definite functions. Although some results can be obtained from known properties, we also prove new results based on the positivity of the Fourier transforms of certain functions. Concrete computations of certain Fourier transforms not only yield new examples of positive *

Contraction coefficients for noisy quantum channels

“…The functions κ WYD t given in (26) showed up through the representation of the Wigner-Yanase-Dyson skew information in terms of monotone metrics, as described in [22,Section 2.4,Example 4.8]. Furthermore, as studied in [29], the trace functional of WYD concavity/convexity [39,2] is recovered by the quasi-entropy for g (t) given in (25) as follows:…”

“…Proof. For (b) ⇒ (a), see [22,Theorem 2.4]. As for (c), Kraus' theorem (see, e.g., [21,Corollary 2.7.8]) implies that g is operator convex if and only if…”

“…which may be considered as the center of K from some aspects. For instance, it is known [22,Theorem 3.5] that x −1/2 is the only function κ ∈ K such that both Ω κ ρ and (Ω κ ρ ) −1 are CP for all ρ ∈ D d . The function in G sym corresponding to x −1/2 is (x − 1) 2 x −1/2 and the corresponding divergence is…”

Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance

“…There is a well-developed theory of monotone metrics beginning with work of Chentsov and Morozova [51] for classical Markov processes and its non-commutative extension initiated by Petz [61], and further developed in [39,33,59,34,69]. Other results from this theory will be useful in further developments.…”

Some families of operator norm inequalities

“…It is clear α w β. If we choose as i 1 = 1, i 2 = 4, j 1 = 2 and j 2 = 5, then we have |||M (8,7,3), (10,6,4) (L H , R K )X||| ≤ |||M (9,2), (8,5) (L H , R K )X|||.…”