Michał Horodecki

Abstract Epsilon-nets and approximate unitary t-designs are natural notions that capture properties of unitary operations relevant for numerous applications in quantum information and quantum computing. The former constitute subsets of unitary channels that are epsilon-close to any unitary channel in the diamond norm. The latter are ensembles of unitaries that (approximately) recover Haar averages of polynomials in entries of unitary channels up to order t. In this work we systematically study quantitative connections between these two notions. Specifically, we prove that, for a fixed dimension d of the Hilbert space, unitaries constituting delta-approximate t-expanders form epsilon-nets for t~(d^(5/2))/epsilon and delta~[(epsilon^(3/2))/d]^(d^2). We also show that epsilon-nets can be used to construct delta-approximate unitary t-designs for delt~epsilon*t, where the notion of approximation is based on the diamond norm. Finally, we prove that the degree of an exact unitary t-design necessary to obtain an epsilon-net must grow at least fast as 1/epsilon (for fixed dimension) and not slower than d^2 (for fixed epsilon). This shows near optimality of our result connecting t-designs and epsilon-nets. We further apply our findings in conjunction with the recent results of Varju 2013 in the context of quantum computing. First, we show that that approximate t-designs can be generated by shallow random circuits formed from a set of universal two-qudit gates in the parallel and sequential local architectures considered in Brandao-Harrow-Horodecki 2016. Importantly, our gate sets need not to be symmetric (i.e. contains gates together with their inverses) or consist of gates with algebraic entries. Second, we consider a problem of compilation of quantum gates and prove a non-constructive version of the Solovay-Kitaev theorem for general universal gate sets. Our main technical contribution is a new construction of efficient polynomial approximations to the Dirac delta in the space of quantum channels, which can be of independent interest.

Abstract We introduce and discuss a novel multi-port based teleportation schemes performing transmission of a number of unknown quantum states or one composite system in one go. We fully characterize the probabilistic and deterministic case by presenting expressions for the average probability of success and entanglement fidelity in both non-optimal and optimal variant. We also deliver explicit forms of the measurements and the resource state exploited by parties to perform the process. To obtain our results, i.e. explicit expressions for the performance of the new schemes, we deliver novel mathematical tools concerning representation theory of the algebra of partially transposed permutation operators, where the transposition acts on more than one subsystem. Additionally, the optimal values of the entanglement fidelity and probability success emerge from formulated and solved primal and dual semidefinite problems, which due to existing symmetries and delivered mathematical tools could be solved analytically. Next, we have applied the obtained formulas for the performance of multi-port based teleportation schemes to get a qualitative improvement of asymptotic "teleportation capacities" of multi-port based teleportation schemes over the pre-existing port-based teleportation schemes.

A direct analysis of the protocol of randomness amplification using Bell inequality violation is performed in terms of the convex combination of no-signaling boxes required to simulate quantum violation of the inequality. The probability distributions of bits generated by a Santha-Vazirani source are shown to be mixtures of permutations of Bernoulli distributions with parameter defined by the source. An intuitive proof is provided for the range of partial randomness from which perfect randomness can be extracted using quantum correlations violating the chain inequalities. Exact values are derived in the asymptotic limit of a large number of measurement settings.