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Register a new userSimple proof of the impossibility of bit-commitment in generalised probabilistic theories using cone programming
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Bit-commitment is a fundamental cryptographic task, in which Alice commits a bit to Bob such that she cannot later change the value of the bit, while, simultaneously, the bit is hidden from Bob. It is known that ideal bit-commitment is impossible within quantum theory. In this work, we show that it is also impossible in generalised probabilistic theories (under a small set of assumptions) by presenting a quantitative trade-off between Alices and Bobs cheating probabilities. Our proof relies crucially on a formulation of cheating strategies as cone programs, a natural generalisation of semidefinite programs. In fact, using the generality of this technique, we prove that this result holds for the more general task of integer-commitment.
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Unconditionally secure non-relativistic bit commitment is known to be impossible in both the classical and the quantum worlds. But when committing to a string of n bits at once, how far can we stretch the quantum limits? In this paper, we introduce a framework for quantum schemes where Alice commits a string of n bits to Bob in such a way that she can only cheat on a bits and Bob can learn at most b bits of information before the reveal phase. Our results are two-fold: we show by an explicit construction that in the traditional approach, where the reveal and guess probabilities form the security criteria, no good schemes can exist: a+b is at least n. If, however, we use a more liberal criterion of security, the accessible information, we construct schemes where a=4log n+O(1) and b=4, which is impossible classically. We furthermore present a cheat-sensitive quantum bit string commitment protocol for which we give an explicit tradeoff between Bobs ability to gain information about the committed string, and the probability of him being detected cheating.
Generalized probabilistic theories (GPT) provide a general framework that includes classical and quantum theories. It is described by a cone $C$ and its dual $C^*$. We show that whether some one-way communication complexity problems can be solved within a GPT is equivalent to the recently introduced cone factorisation of the corresponding communication matrix $M$. We also prove an analogue of Holevos theorem: when the cone $C$ is contained in $mathbb{R}^{n}$, the classical capacity of the channel realised by sending GPT states and measuring them is bounded by $log n$. Polytopes and optimising functions over polytopes arise in many areas of discrete mathematics. A conic extension of a polytope is the intersection of a cone $C$ with an affine subspace whose projection onto the original space yields the desired polytope. Extensions of polytopes can sometimes be much simpler geometric objects than the polytope itself. The existence of a conic extension of a polytope is equivalent to that of a cone factorisation of the slack matrix of the polytope, on the same cone. We show that all $0/1$ polytopes whose vertices can be recognized by a polynomial size circuit, which includes as a special case the travelling salesman polytope and many other polytopes from combinatorial optimisation, have small conic extension complexity when the cone is the completely positive cone. Using recent exponential lower bounds on the linear extension complexity of polytopes, this provides an exponential gap between the communication complexity of GPT based on the completely positive cone and classical communication complexity, and a conjectured exponential gap with quantum communication complexity. Our work thus relates the communication complexity of generalisations of quantum theory to questions of mainstream interest in the area of combinatorial optimisation.
Unconditionally secure quantum bit commitment (QBC) was widely believed to be impossible for more than two decades. But recently, basing on an anomalous behavior found in quantum steering, we proposed a QBC protocol which can be unconditionally secure in principle. The protocol requires the use of infinite-dimensional systems, thus it may seem less feasible at first glance. Here we show that such infinite-dimensional systems can be implemented with quantum optical methods, and propose an experimental scheme using Mach-Zehnder interferometer.
Quantum bit commitment has been known to be impossible by the independent proofs of Mayers, and Lo and Chau, under the assumption that the whole quantum states right before the unveiling phase are static to users. We here provide an unconditionally secure non-static quantum bit commitment protocol with a trusted third party, which is not directly involved in any communications between users and can be limited not to get any information of commitment without being detected by users. We also prove that our quantum bit commitment protocol is not secure without the help of the trusted third party. The proof is basically different from the Mayers-Lo-Chaus no-go theorem, because we do not assume the staticity of the finally shared quantum states between users.
What singles out quantum mechanics as the fundamental theory of Nature? Here we study local measurements in generalised probabilistic theories (GPTs) and investigate how observational limitations affect the production of correlations. We find that if only a subset of typical local measurements can be made then all the bipartite correlations produced in a GPT can be simulated to a high degree of accuracy by quantum mechanics. Our result makes use of a generalisation of Dvoretzkys theorem for GPTs. The tripartite correlations can go beyond those exhibited by quantum mechanics, however.
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