129 research outputs found
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 in terms of the fidelity of recovery (FoR),
i.e. the maximal fidelity of the state with a state reconstructed from
its marginal by acting only on the 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 --relative R\'{e}nyi entropies.
The definition of the --relative R\'{e}nyi entropy unifies all
previously proposed definitions of the quantum R\'{e}nyi divergence of order
under a common framework. Here we will prove that the
--relative R\'{e}nyi entropies are a proper generalization of the
quantum relative entropy by computing the limit of the - divergence
as approaches one and is an arbitrary function of . We
also show that certain operationally relevant families of R\'enyi divergences
are differentiable at . Finally, our analysis reveals that the
derivative at 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 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
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- 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
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
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