Necessary and sufficient optimality conditions for classical simulations of quantum communication processes

Alberto Montina and Stefan Wolf
Phys. Rev. A 90, 012309 – Published 8 July 2014

Abstract

We consider the process consisting of preparation, transmission through a quantum channel, and subsequent measurement of quantum states. The communication complexity of the channel is the minimal amount of classical communication required for classically simulating it. Recently, we reduced the computation of this quantity to a convex minimization problem with linear constraints. Every solution of the constraints provides an upper bound on the communication complexity. In this paper, we derive the dual maximization problem of the original one. The feasible points of the dual constraints, which are inequalities, give lower bounds on the communication complexity, as illustrated with an example. The optimal values of the two problems turn out to be equal (zero duality gap). By this property, we provide necessary and sufficient conditions for optimality in terms of a set of equalities and inequalities. We use these conditions and two reasonable but unproven hypotheses to derive the lower bound n×2n1 for a noiseless quantum channel with capacity equal to n qubits. This lower bound can have interesting consequences in the context of the recent debate on the reality of the quantum state.

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  • Received 2 March 2014

DOI:https://doi.org/10.1103/PhysRevA.90.012309

©2014 American Physical Society

Authors & Affiliations

Alberto Montina and Stefan Wolf

  • Facoltà di Informatica, Università della Svizzera Italiana, Via G. Buffi 13, 6900 Lugano, Switzerland

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Vol. 90, Iss. 1 — July 2014

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