309 research outputs found
Entropic Steering Criteria: Applications to Bipartite and Tripartite Systems
The effect of quantum steering describes a possible action at a distance via
local measurements. Whereas many attempts on characterizing steerability have
been pursued, answering the question as to whether a given state is steerable
or not remains a difficult task. Here, we investigate the applicability of a
recently proposed method for building steering criteria from generalized
entropic uncertainty relations. This method works for any entropy which satisfy
the properties of (i) (pseudo-) additivity for independent distributions; (ii)
state independent entropic uncertainty relation (EUR); and (iii) joint
convexity of a corresponding relative entropy. Our study extends the former
analysis to Tsallis and R\'enyi entropies on bipartite and tripartite systems.
As examples, we investigate the steerability of the three-qubit GHZ and W
states.Comment: 27 pages, 8 figures. Published version. Title change
Estimating Mixture Entropy with Pairwise Distances
Mixture distributions arise in many parametric and non-parametric settings --
for example, in Gaussian mixture models and in non-parametric estimation. It is
often necessary to compute the entropy of a mixture, but, in most cases, this
quantity has no closed-form expression, making some form of approximation
necessary. We propose a family of estimators based on a pairwise distance
function between mixture components, and show that this estimator class has
many attractive properties. For many distributions of interest, the proposed
estimators are efficient to compute, differentiable in the mixture parameters,
and become exact when the mixture components are clustered. We prove this
family includes lower and upper bounds on the mixture entropy. The Chernoff
-divergence gives a lower bound when chosen as the distance function,
with the Bhattacharyya distance providing the tightest lower bound for
components that are symmetric and members of a location family. The
Kullback-Leibler divergence gives an upper bound when used as the distance
function. We provide closed-form expressions of these bounds for mixtures of
Gaussians, and discuss their applications to the estimation of mutual
information. We then demonstrate that our bounds are significantly tighter than
well-known existing bounds using numeric simulations. This estimator class is
very useful in optimization problems involving maximization/minimization of
entropy and mutual information, such as MaxEnt and rate distortion problems.Comment: Corrects several errata in published version, in particular in
Section V (bounds on mutual information
Unified entropic measures of quantum correlations induced by local measurements
We introduce quantum correlations measures based on the minimal change in
unified entropies induced by local rank-one projective measurements, divided by
a factor that depends on the generalized purity of the system in the case of
non-additive entropies. In this way, we overcome the issue of the artificial
increasing of the value of quantum correlations measures based on non-additive
entropies when an uncorrelated ancilla is appended to the system without
changing the computability of our entropic correlations measures with respect
to the previous ones. Moreover, we recover as limiting cases the quantum
correlations measures based on von Neumann and R\'enyi entropies (i.e.,
additive entropies), for which the adjustment factor becomes trivial. In
addition, we distinguish between total and semiquantum correlations and obtain
some relations between them. Finally, we obtain analytical expressions of the
entropic correlations measures for typical quantum bipartite systems.Comment: 10 pages, 1 figur
State-Dependent Approach to Entropic Measurement-Disturbance Relations
Heisenberg's intuition was that there should be a tradeoff between measuring
a particle's position with greater precision and disturbing its momentum.
Recent formulations of this idea have focused on the question of how well two
complementary observables can be jointly measured. Here, we provide an
alternative approach based on how enhancing the predictability of one
observable necessarily disturbs a complementary one. Our
measurement-disturbance relation refers to a clear operational scenario and is
expressed by entropic quantities with clear statistical meaning. We show that
our relation is perfectly tight for all measurement strengths in an existing
experimental setup involving qubit measurements.Comment: 9 pages, 2 figures. v4: published versio
Entropy of quantum channel in the theory of quantum information
Quantum channels, also called quantum operations, are linear, trace
preserving and completely positive transformations in the space of quantum
states. Such operations describe discrete time evolution of an open quantum
system interacting with an environment. The thesis contains an analysis of
properties of quantum channels and different entropies used to quantify the
decoherence introduced into the system by a given operation. Part I of the
thesis provides a general introduction to the subject. In Part II, the action
of a quantum channel is treated as a process of preparation of a quantum
ensemble. The Holevo information associated with this ensemble is shown to be
bounded by the entropy exchanged during the preparation process between the
initial state and the environment. A relation between the Holevo information
and the entropy of an auxiliary matrix consisting of square root fidelities
between the elements of the ensemble is proved in some special cases. Weaker
bounds on the Holevo information are also established. The entropy of a
channel, also called the map entropy, is defined as the entropy of the state
corresponding to the channel by the Jamiolkowski isomorphism. In Part III of
the thesis, the additivity of the entropy of a channel is proved. The minimal
output entropy, which is difficult to compute, is estimated by an entropy of a
channel which is much easier to obtain. A class of quantum channels is
specified, for which additivity of channel capacity is conjectured. The last
part of the thesis contains characterization of Davies channels, which
correspond to an interaction of a state with a thermal reservoir in the week
coupling limit, under the condition of quantum detailed balance and
independence of rotational and dissipative evolutions. The Davies channels are
characterized for one-qubit and one-qutrit systems
On the quantum Renyi relative entropies and related capacity formulas
We show that the quantum -relative entropies with parameter
can be represented as generalized cutoff rates in the sense
of [I. Csiszar, IEEE Trans. Inf. Theory 41, 26-34, (1995)], which provides a
direct operational interpretation to the quantum -relative entropies.
We also show that various generalizations of the Holevo capacity, defined in
terms of the -relative entropies, coincide for the parameter range
, and show an upper bound on the one-shot epsilon-capacity of
a classical-quantum channel in terms of these capacities.Comment: v4: Cutoff rates are treated for correlated hypotheses, some proofs
are given in greater detai
A family of generalized quantum entropies: definition and properties
We present a quantum version of the generalized (h, φ)-entropies, introduced by Salicrú et al. for the study of classical probability distributions.We establish their basic properties and show that already known quantum entropies such as von Neumann, and quantum versions of Rényi, Tsallis, and unified entropies, constitute particular classes of the present general quantum Salicrú form. We exhibit that majorization plays a key role in explaining most of their common features. We give a characterization of the quantum (h, φ)-entropies under the action of quantum operations and study their properties for composite systems. We apply these generalized entropies to the problem of detection of quantum entanglement and introduce a discussion on possible generalized conditional entropies as well.Facultad de Ciencias ExactasInstituto de Física La Plat
Generalised entropy accumulation
Consider a sequential process in which each step outputs a system and
updates a side information register . We prove that if this process
satisfies a natural "non-signalling" condition between past outputs and future
side information, the min-entropy of the outputs conditioned
on the side information at the end of the process can be bounded from below
by a sum of von Neumann entropies associated with the individual steps. This is
a generalisation of the entropy accumulation theorem (EAT), which deals with a
more restrictive model of side information: there, past side information cannot
be updated in subsequent rounds, and newly generated side information has to
satisfy a Markov condition. Due to its more general model of side-information,
our generalised EAT can be applied more easily and to a broader range of
cryptographic protocols. As examples, we give the first multi-round security
proof for blind randomness expansion and a simplified analysis of the E91 QKD
protocol. The proof of our generalised EAT relies on a new variant of Uhlmann's
theorem and new chain rules for the Renyi divergence and entropy, which might
be of independent interest.Comment: 42 pages; v2 expands introduction but does not change any results; in
FOCS 202
- …