109 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
Non-Asymptotic Analysis of Privacy Amplification via Renyi Entropy and Inf-Spectral Entropy
This paper investigates the privacy amplification problem, and compares the
existing two bounds: the exponential bound derived by one of the authors and
the min-entropy bound derived by Renner. It turns out that the exponential
bound is better than the min-entropy bound when a security parameter is rather
small for a block length, and that the min-entropy bound is better than the
exponential bound when a security parameter is rather large for a block length.
Furthermore, we present another bound that interpolates the exponential bound
and the min-entropy bound by a hybrid use of the Renyi entropy and the
inf-spectral entropy.Comment: 6 pages, 4 figure
On simultaneous min-entropy smoothing
In the context of network information theory, one often needs a multiparty
probability distribution to be typical in several ways simultaneously. When
considering quantum states instead of classical ones, it is in general
difficult to prove the existence of a state that is jointly typical. Such a
difficulty was recently emphasized and conjectures on the existence of such
states were formulated. In this paper, we consider a one-shot multiparty
typicality conjecture. The question can then be stated easily: is it possible
to smooth the largest eigenvalues of all the marginals of a multipartite state
{\rho} simultaneously while staying close to {\rho}? We prove the answer is yes
whenever the marginals of the state commute. In the general quantum case, we
prove that simultaneous smoothing is possible if the number of parties is two
or more generally if the marginals to optimize satisfy some non-overlap
property.Comment: 5 page
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
- …