The Uncertainty Relation for Smooth Entropies

Uncertainty relations give upper bounds on the accuracy by which the outcomes of two incompatible measurements can be predicted. While established uncertainty relations apply to cases where the predictions are based on purely classical data (e.g., a description of the system's state before measurement), an extended relation which remains valid in the presence of quantum information has been proposed recently [Berta et al., Nat. Phys. 6, 659 (2010)]. Here, we generalize this uncertainty relation to one formulated in terms of smooth entropies. Since these entropies measure operational quantities such as extractable secret key length, our uncertainty relation is of immediate practical use. To illustrate this, we show that it directly implies security of a family of quantum key distribution protocols including BB84. Our proof remains valid even if the measurement devices used in the experiment deviate arbitrarily from the theoretical model.Comment: Weakened claim concerning semi device-independence in the application to QKD. A full security proof for this setup without any restrictions on the measurement devices can be found in arXiv:1210.435

The Fidelity of Recovery is Multiplicative

Fawzi and Renner [Commun. Math. Phys. 340(2):575, 2015] recently established a lower bound on the conditional quantum mutual information (CQMI) of tripartite quantum states $ABC$ in terms of the fidelity of recovery (FoR), i.e. the maximal fidelity of the state $ABC$ with a state reconstructed from its marginal $BC$ by acting only on the $C$ system. The FoR measures quantum correlations by the local recoverability of global states and has many properties similar to the CQMI. Here we generalize the FoR and show that the resulting measure is multiplicative by utilizing semi-definite programming duality. This allows us to simplify an operational proof by Brandao et al. [Phys. Rev. Lett. 115(5):050501, 2015] of the above-mentioned lower bound that is based on quantum state redistribution. In particular, in contrast to the previous approaches, our proof does not rely on de Finetti reductions.Comment: v2: 9 pages, published versio

Investigating Properties of a Family of Quantum Renyi Divergences

Audenaert and Datta recently introduced a two-parameter family of relative R\'{e}nyi entropies, known as the $\alpha$-$z$-relative R\'{e}nyi entropies. The definition of the $\alpha$-$z$-relative R\'{e}nyi entropy unifies all previously proposed definitions of the quantum R\'{e}nyi divergence of order $\alpha$ under a common framework. Here we will prove that the $\alpha$-$z$-relative R\'{e}nyi entropies are a proper generalization of the quantum relative entropy by computing the limit of the $\alpha$-$z$ divergence as $\alpha$ approaches one and $z$ is an arbitrary function of $\alpha$. We also show that certain operationally relevant families of R\'enyi divergences are differentiable at $\alpha = 1$. Finally, our analysis reveals that the derivative at $\alpha = 1$ evaluates to half the relative entropy variance, a quantity that has attained operational significance in second-order quantum hypothesis testing.Comment: 15 pages, v2: journal versio

Second-Order Asymptotics for the Classical Capacity of Image-Additive Quantum Channels

We study non-asymptotic fundamental limits for transmitting classical information over memoryless quantum channels, i.e. we investigate the amount of classical information that can be transmitted when a quantum channel is used a finite number of times and a fixed, non-vanishing average error is permissible. We consider the classical capacity of quantum channels that are image-additive, including all classical to quantum channels, as well as the product state capacity of arbitrary quantum channels. In both cases we show that the non-asymptotic fundamental limit admits a second-order approximation that illustrates the speed at which the rate of optimal codes converges to the Holevo capacity as the blocklength tends to infinity. The behavior is governed by a new channel parameter, called channel dispersion, for which we provide a geometrical interpretation.Comment: v2: main results significantly generalized and improved; v3: extended to image-additive channels, change of title, journal versio

Entropic uncertainty from effective anti-commutators

We investigate entropic uncertainty relations for two or more binary measurements, for example spin-$\frac{1}{2}$ or polarisation measurements. We argue that the effective anti-commutators of these measurements, i.e. the anti-commutators evaluated on the state prior to measuring, are an expedient measure of measurement incompatibility. Based on the knowledge of pairwise effective anti-commutators we derive a class of entropic uncertainty relations in terms of conditional R\'{e}nyi entropies. Our uncertainty relations are formulated in terms of effective measures of incompatibility, which can be certified device-independently. Consequently, we discuss potential applications of our findings to device-independent quantum cryptography. Moreover, to investigate the tightness of our analysis we consider the simplest (and very well-studied) scenario of two measurements on a qubit. We find that our results outperform the celebrated bound due to Maassen and Uffink [Phys. Rev. Lett. 60, 1103 (1988)] and provide a new analytical expression for the minimum uncertainty which also outperforms some recent bounds based on majorisation.Comment: 5 pages, 3 figures (excluding Supplemental Material), revte

Second-Order Coding Rates for Channels with State

We study the performance limits of state-dependent discrete memoryless channels with a discrete state available at both the encoder and the decoder. We establish the epsilon-capacity as well as necessary and sufficient conditions for the strong converse property for such channels when the sequence of channel states is not necessarily stationary, memoryless or ergodic. We then seek a finer characterization of these capacities in terms of second-order coding rates. The general results are supplemented by several examples including i.i.d. and Markov states and mixed channels

A decoupling approach to classical data transmission over quantum channels

Most coding theorems in quantum Shannon theory can be proven using the decoupling technique: to send data through a channel, one guarantees that the environment gets no information about it; Uhlmann's theorem then ensures that the receiver must be able to decode. While a wide range of problems can be solved this way, one of the most basic coding problems remains impervious to a direct application of this method: sending classical information through a quantum channel. We will show that this problem can, in fact, be solved using decoupling ideas, specifically by proving a "dequantizing" theorem, which ensures that the environment is only classically correlated with the sent data. Our techniques naturally yield a generalization of the Holevo-Schumacher-Westmoreland Theorem to the one-shot scenario, where a quantum channel can be applied only once

A Fully Quantum Asymptotic Equipartition Property

The classical asymptotic equipartition property is the statement that, in the limit of a large number of identical repetitions of a random experiment, the output sequence is virtually certain to come from the typical set, each member of which is almost equally likely. In this paper, we prove a fully quantum generalization of this property, where both the output of the experiment and side information are quantum. We give an explicit bound on the convergence, which is independent of the dimensionality of the side information. This naturally leads to a family of Renyi-like quantum conditional entropies, for which the von Neumann entropy emerges as a special case.Comment: Main claim is updated with improved bound