DE FINETTI THEOREMS FOR EASY QUANTUM GROUPS

Vitonofrio Crismale, Yun Gang Lu

Fernando G. S. L. Brandão, Aram W. Harrow

Giulio Chiribella

R. L. Hudson

Louis Desruisseaux, Simon Ducharme, Gurleen Padda +1 more

For many information processing tasks, de Finetti-style theorems can often simplify the analysis in worst-case input scenarios for which the task exhibits some permutation-invariance symmetry, as they can allow for a reduction from an analysis on worst-case inputs to that of i.i.d. inputs. If further information is available on the inputs, it might be advantageous to reflect this information in the de Finetti reduction. In our work, we focus on a form of such constraint, based on the type of the input. This allows us to obtain a conceptually simple proof of a new de Finetti reduction for classical probability distributions, derived from elementary properties from the method of types. We apply our constrained de Finetti reduction to the compression of quantum interactive communication protocols with classical inputs, and prove that the prior-free quantum information cost equals the worst-case input amortized quantum communication cost.

Rotem Arnon-Friedman, Renato Renner

de Finetti-type theorems enable a substantially simplified analysis of information-processing tasks in various areas, such as quantum cryptography and quantum tomography. The idea is that instead of carrying out the analysis for any possible state it is sufficient to consider one particular de Finetti state, that is, a convex combination of i.i.d. states. It is thus interesting to see whether such de Finetti-type theorems are unique for quantum theory or whether they hold for more general theories. Here we prove a de Finetti-type theorem in the framework of conditional probability distributions. In this framework, a system is described by a conditional probability distribution P_A|X where X denotes the measurement and A the outcome. We show that any permutation invariant system P_A|X can be reduced to a de Finetti system. We then demonstrate how the theorem can be applied to security proofs of device independent cryptographic protocols.