Andreas Winter

Arguably, the largest class of stochastic processes generated by means of a finite memory consists of those that are sequences of observations produced by sequential measurements in a suitable generalized probabilistic theory (GPT). These are constructed from a finite-dimensional memory evolving under a set of possible linear maps, and with probabilities of outcomes determined by linear functions of the memory state. Examples of such models are given by classical hidden Markov processes, where the memory state is a probability distribution, and at each step it evolves according to a non-negative matrix, and hidden quantum Markov processes, where the memory state is a finite dimensional quantum state, and at each step it evolves according to a completely positive map. Here we show that the set of processes admitting a finite-dimensional explanation do not need to be explainable in terms of either classical probability or quantum mechanics. To wit, we exhibit families of processes that have a finite-dimensional explanation, defined manifestly by the dynamics of explicitly given GPT, but that do not admit a quantum, and therefore not even classical, explanation in finite dimension. Furthermore, we present a family of quantum processes on qubits and qutrits that do not admit a classical finite-dimensional realization, which includes examples introduced earlier by Fox, Rubin, Dharmadikari and Nadkarni as functions of infinite dimensional Markov chains, and lower bound the size of the memory of a classical model realizing a noisy version of the qubit processes.

The last year has seen a series of dramatic progresses in the areas of quantum channel capacities, entanglement distillation and our understanding of how these problems relate to secret communication. In this talk I will first review the analogy between the theories of entanglement and key distillation. Then I go on to explain the elements of the recent progress, first describing Devetak's recent reduction of quantum state transmission to private communication and our subsequent joint work generalising this to entanglement/secret key distillation. These findings put into a new light the earlier observations (by Acin et al.and Bruss et al.) of qualitative equivalence between these two tasks, and also prepared the ground for the recent step forward by Horodecki et al., to formulate key distillation strictly within the LOCC paradigm and then to exhibit bound entangled states which nevertheless contain secret key.

We study unitary 2-designs and decoupling with random diagonal-unitaries. We first show that the alternate application of random diagonal-unitaries in the Pauli-Z and -X bases constitutes a unitary 2-design after a number of repetitions, implying that the process achieves decoupling. We then go on to show that even fewer repetitions are sufficient for achieving decoupling at the same rate as that with Haar random unitaries. These results imply that precise unitary 2-designs are not necessary for achieving the Haar decoupling rate, indicating the possibility to achieve the rate by random unitaries less uniform than 2-designs. We also provide a simple quantum circuit that implements a unitary 2-design and achieves decoupling, which is partitioned into a constant number of commuting parts.