CWI Cryptology Group

Abstract: We study the uncertainty of the measurement outcome when measuring an arbitrary n-qubit quantum state in a basis that is chosen at random from some fixed family of bases. We discuss several canonical cases and obtain (tight) lower bounds on the uncertainty of the measurement outcome, where the uncertainty is measured in terms of the min-entropy. Finally, we briefly point out how these quantum uncertainty relations can be used for designing quantum cryptographic scheme.

Abstract: We will introduce new techniques coming from pure Functional Analysis in the study of correlation Bell inequalities. As a result we will be able to show that there are tripartite Bell inequalities allowing an arbitrarily large quantum violation, solving and old question of Tsirelson. We will also discuss some of the implications of our result, for instance in the problem of measuring the Hilbert space dimension of a quantum system.

Abstract: I will show that the Bell inequality can be recast in a form appropriate to Bell-type experiments with any number of outcomes in the two wings of the experiment. It is then interesting to study the maximal violation possible by quantum mechanics, as the number of outcomes increases (or if you like, as the dimension of the Hilbert spaces increase). I will present numerical evidence for the optimal quantum measurements and the optimal quantum state for larger and larger dimension. It appears that quantum mechanics can asymptotically do as well as is allowed by the mere requirement of non-signalling, which is: perfectly. The main aim of my talk is to interest the readers in the many open problems which are suggested by this work.
Reference: arXiv:quant-ph/0612020, joint work with Stefan Zohren, appeared 2008 in PRL.

Abstract: We show that the conditional min-entropy Hmin(A|B) of a bipartite state rho_AB is directly related to the maximum achievable overlap with a maximally entangled state if only local actions on the B-part of rho_AB are allowed. In the special case where A is classical, this overlap corresponds to the probability of guessing A given B. In a similar vein, we connect the conditional max-entropy Hmax(A|B) to the maximum fidelity of rho_AB with a product state that is completely mixed on A. In the case where A is classical, this corresponds to the security of A when used as a secret key in the presence of an adversary holding B. Because min- and max-entropies are known to characterize information-processing tasks such as randomness extraction and state merging, our results establish a direct connection between these tasks and basic operational problems. For example, they imply that the (logarithm of the) probability of guessing A given B is a lower bound on the number of uniform secret bits that can be extracted from A relative to an adversary holding B. (Joint work with Renato Renner and Robert Koenig)