A short impossibility proof of quantum bit commitment

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 (Lo and Chau, 1997 [1], Mayers, 1997 [2]). A strengthened and explicit impossibility proof has been given in D'Ariano et al. (2007) [3] in the Heisenberg picture and in a C*-algebraic framework, considering all conceivable protocols in which both classical and quantum information is exchanged. In the present Letter we provide a new impossibility proof in the Schr{\"o}dinger picture, greatly simplifying the classification of protocols and strategies using the mathematical formulation in terms of quantum combs (Chiribella et al., 2008 [4]), with each single-party strategy represented by a conditioned comb. We prove that assuming a stronger notion of concealment---for each classical communication history, not in average---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 the metric given by the comb discriminability-distance