On privacy amplification, lossy compression, and their duality to channel coding

We examine the task of privacy amplification from information-theoretic and coding-theoretic points of view. In the former, we give a one-shot characterization of the optimal rate of privacy amplification against classical adversaries in terms of the optimal type-II error in asymmetric hypothesis testing. This formulation can be easily computed to give finite-blocklength bounds and turns out to be equivalent to smooth min-entropy bounds by Renner and Wolf [Asiacrypt 2005] and Watanabe and Hayashi [ISIT 2013], as well as a bound in terms of the $E_\gamma$ divergence by Yang, Schaefer, and Poor [arXiv:1706.03866 [cs.IT]]. In the latter, we show that protocols for privacy amplification based on linear codes can be easily repurposed for channel simulation. Combined with known relations between channel simulation and lossy source coding, this implies that privacy amplification can be understood as a basic primitive for both channel simulation and lossy compression. Applied to symmetric channels or lossy compression settings, our construction leads to proto- cols of optimal rate in the asymptotic i.i.d. limit. Finally, appealing to the notion of channel duality recently detailed by us in [IEEE Trans. Info. Theory 64, 577 (2018)], we show that linear error-correcting codes for symmetric channels with quantum output can be transformed into linear lossy source coding schemes for classical variables arising from the dual channel. This explains a "curious duality" in these problems for the (self-dual) erasure channel observed by Martinian and Yedidia [Allerton 2003; arXiv:cs/0408008] and partly anticipates recent results on optimal lossy compression by polar and low-density generator matrix codes.Comment: v3: updated to include equivalence of the converse bound with smooth entropy formulations. v2: updated to include comparison with the one-shot bounds of arXiv:1706.03866. v1: 11 pages, 4 figure

Efficient Approximation of Quantum Channel Capacities

We propose an iterative method for approximating the capacity of classical-quantum channels with a discrete input alphabet and a finite dimensional output, possibly under additional constraints on the input distribution. Based on duality of convex programming, we derive explicit upper and lower bounds for the capacity. To provide an $\varepsilon$-close estimate to the capacity, the presented algorithm requires $O(\tfrac{(N \vee M) M^3 \log(N)^{1/2}}{\varepsilon})$, where $N$ denotes the input alphabet size and $M$ the output dimension. We then generalize the method for the task of approximating the capacity of classical-quantum channels with a bounded continuous input alphabet and a finite dimensional output. For channels with a finite dimensional quantum mechanical input and output, the idea of a universal encoder allows us to approximate the Holevo capacity using the same method. In particular, we show that the problem of approximating the Holevo capacity can be reduced to a multidimensional integration problem. For families of quantum channels fulfilling a certain assumption we show that the complexity to derive an $\varepsilon$-close solution to the Holevo capacity is subexponential or even polynomial in the problem size. We provide several examples to illustrate the performance of the approximation scheme in practice.Comment: 36 pages, 1 figur

Approaching the Kosterlitz-Thouless transition for the classical XY model with tensor networks

We apply variational tensor-network methods for simulating the Kosterlitz-Thouless phase transition in the classical two-dimensional XY model. In particular, using uniform matrix product states (MPS) with non-Abelian O(2) symmetry, we compute the universal drop in the spin stiffness at the critical point. In the critical low-temperature regime, we focus on the MPS entanglement spectrum to characterize the Luttinger-liquid phase. In the high-temperature phase, we confirm the exponential divergence of the correlation length and estimate the critical temperature with high precision. Our MPS approach can be used to study generic two-dimensional phase transitions with continuous symmetries