A short impossibility proof of Quantum Bit Commitment

TL;DR

This paper presents a simplified, new proof demonstrating the impossibility of unconditionally secure quantum bit commitment protocols, using quantum combs and a worst-case concealment assumption.

Contribution

It introduces a Schrodinger picture proof utilizing quantum combs, simplifying the classification of protocols and strategies compared to previous Heisenberg picture approaches.

Findings

Proves the impossibility of secure quantum bit commitment under worst-case concealment.

Uses quantum combs to model and analyze protocols and strategies.

Restates the tradeoff in terms of continuity of dilations of probabilistic quantum combs.

Abstract

Bit commitment protocols, whose security is based on the laws of quantum mechanics alone, are generally held to be impossible on the basis of a concealment-bindingness tradeoff. A strengthened and explicit impossibility proof has been given in: G. M. D'Ariano, D. Kretschmann, D. Schlingemann, and R. F. Werner, Phys. Rev. A 76, 032328 (2007), in the Heisenberg picture and in a C*-algebraic framework, considering all conceivable protocols in which both classical and quantum information are exchanged. In the present paper we provide a new impossibility proof in the Schrodinger picture, greatly simplifying the classification of protocols and strategies using the mathematical formulation in terms of quantum combs, with each single-party strategy represented by a conditional comb. We prove that assuming a stronger notion of concealment--worst-case over the classical information histories--allows Alice's cheat to pass also the worst-case Bob's test. The present approach allows us to restate the concealment-bindingness tradeoff in terms of the continuity of dilations of probabilistic quantum combs with respect to the comb-discriminability distance.