On a measure of distance for quantum strategies

Abstract

The present paper studies an operator norm that captures the distinguishability of quantum strategies in the same sense that the trace norm captures the distinguishability of quantum states or the diamond norm captures the distinguishability of quantum channels. Characterizations of its unit ball and dual norm are established via strong duality of a semidefinite optimization problem. A full, formal proof of strong duality is presented for the semidefinite optimization problem in question. This norm and its properties are employed to generalize a state discrimination result of Ref. [GW05]. The generalized result states that for any two convex sets S,T of strategies there exists a fixed interactive measurement scheme that successfully distinguishes any choice of s in S from any choice of t in T with bias proportional to the minimal distance between the sets S and T as measured by this norm. A similar discrimination result for channels then follows as a special case.Comment: 25 pages: 17 main body, 6 appendix, 2 references. Final version, minor change

Optimal networks for quantum metrology: semidefinite programs and product rules

“…For prior probabilities π 0 and π 1 , the probability of success and the norm are linked by the relation [34] p succ = 1 2 1 + ||π 0 P 0 − π 1 P 1 || op , which generalizes the well-known expression of Helstrom [16] for the optimal discrimination between two quantum states. In the binary case the dual expression for the maximum success probability given by theorem 1 coincides with the dual expression presented by Gutoski in [35].…”

“…The proof of the theorem, given in the appendix, follows the same lines used by Gutoski [35] to prove strong duality for the minimum error discrimination of two quantum processes, which the special instance of our problem corresponding to X := {0, 1} and g(x, x) = δx ,x . Here we illustrate the result of theorem 1 in a few special examples.…”

“…The discrimination of two multi-time processes P 0 and P 1 corresponds to the special case where X = {0, 1}. In this case, the maximum probability of successful discrimination defines an operational norm in the real vector space generated by quantum processes [34,35]. For prior probabilities π 0 and π 1 , the probability of success and the norm are linked by the relation [34] p succ = 1 2 1 + ||π 0 P 0 − π 1 P 1 || op , which generalizes the well-known expression of Helstrom [16] for the optimal discrimination between two quantum states.…”

Composable security in relativistic quantum cryptography

“…Similar formalisms (e.g. [39][40][41]) have been previously developed but they are only suitable for modelling systems where the order of messages is predefined, they fail to be closed under composition when considering simple cryptographic protocols that involve dynamical ordering of messages during runtime [22]. In particular, the formalism allows us to model quantum causal systems in Minkowski space and construct new causal systems by composing them.…”

Classical cryptographic protocols in a quantum world

“…Therefore, we can have a well-de¯ned statistically interactively indistinguishability using the jj Á jj ¦r norm for quantum strategies. See Gutoski 58 and Chiribella et al 59 for details about characterizing distinguishability of quantum strategies using the jj Á jj ¦r norm.…”