64 research outputs found
Duality Between Smooth Min- and Max-Entropies
In classical and quantum information theory, operational quantities such as
the amount of randomness that can be extracted from a given source or the
amount of space needed to store given data are normally characterized by one of
two entropy measures, called smooth min-entropy and smooth max-entropy,
respectively. While both entropies are equal to the von Neumann entropy in
certain special cases (e.g., asymptotically, for many independent repetitions
of the given data), their values can differ arbitrarily in the general case.
In this work, a recently discovered duality relation between (non-smooth)
min- and max-entropies is extended to the smooth case. More precisely, it is
shown that the smooth min-entropy of a system A conditioned on a system B
equals the negative of the smooth max-entropy of A conditioned on a purifying
system C. This result immediately implies that certain operational quantities
(such as the amount of compression and the amount of randomness that can be
extracted from given data) are related. Such relations may, for example, have
applications in cryptographic security proofs
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
Toward correlation self-testing of quantum theory in the adaptive Clauser-Horne-Shimony-Holt game
Correlation self-testing of a theory addresses the question of whether we can
identify the set of correlations realisable in a theory from its performance in
a particular information processing task. Applied to quantum theory it aims to
identify an information processing task whose optimal performance is achieved
only by theories realising the same correlations as quantum theory in any
causal structure. In [Phys. Rev. Lett. 125 060406 (2020)] we introduced a
candidate task for this, the adaptive CHSH game. Here, we analyse the maximum
probability of winning this game in different generalised probabilistic
theories. We show that theories with a joint state space given by the minimal
or the maximal tensor product are inferior to quantum theory, before
considering other tensor products in theories whose elementary systems have
various two-dimensional state spaces. For these, we find no theories that
outperform quantum theory in the adaptive CHSH game and prove that it is
impossible to recover the quantum performance in various cases. This is the
first step towards a general solution that, if successful, will have
wide-ranging consequences, in particular, enabling an experiment that could
rule out all theories in which the set of realisable correlations does not
coincide with the quantum set.Comment: 12+2 pages, 2 figures; v2: typos correcte
Analysing causal structures in generalised probabilistic theories
Causal structures give us a way to understand the origin of observed
correlations. These were developed for classical scenarios, but quantum
mechanical experiments necessitate their generalisation. Here we study causal
structures in a broad range of theories, which include both quantum and
classical theory as special cases. We propose a method for analysing
differences between such theories based on the so-called measurement entropy.
We apply this method to several causal structures, deriving new relations that
separate classical, quantum and more general theories within these causal
structures. The constraints we derive for the most general theories are in a
sense minimal requirements of any causal explanation in these scenarios. In
addition, we make several technical contributions that give insight for the
entropic analysis of quantum causal structures. In particular, we prove that
for any causal structure and for any generalised probabilistic theory, the set
of achievable entropy vectors form a convex cone.Comment: 11+13 pages, 5 figures, v2: new examples and additional discussion
added, v3 (published version): presentation improve
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