52,278 research outputs found
Bell Inequalities with Auxiliary Communication
What is the communication cost of simulating the correlations produced by
quantum theory? We generalize Bell inequalities to the setting of local
realistic theories augmented by a fixed amount of classical communication.
Suppose two parties choose one of M two-outcome measurements and exchange 1 bit
of information. We present the complete set of inequalities for M = 2, and the
complete set of inequalities for the joint correlation observable for M = 3. We
find that correlations produced by quantum theory satisfy both of these sets of
inequalities. One bit of communication is therefore sufficient to simulate
quantum correlations in both of these scenarios.Comment: 5 page
Contextual advantage for state discrimination
Finding quantitative aspects of quantum phenomena which cannot be explained
by any classical model has foundational importance for understanding the
boundary between classical and quantum theory. It also has practical
significance for identifying information processing tasks for which those
phenomena provide a quantum advantage. Using the framework of generalized
noncontextuality as our notion of classicality, we find one such nonclassical
feature within the phenomenology of quantum minimum error state discrimination.
Namely, we identify quantitative limits on the success probability for minimum
error state discrimination in any experiment described by a noncontextual
ontological model. These constraints constitute noncontextuality inequalities
that are violated by quantum theory, and this violation implies a quantum
advantage for state discrimination relative to noncontextual models.
Furthermore, our noncontextuality inequalities are robust to noise and are
operationally formulated, so that any experimental violation of the
inequalities is a witness of contextuality, independently of the validity of
quantum theory. Along the way, we introduce new methods for analyzing
noncontextuality scenarios, and demonstrate a tight connection between our
minimum error state discrimination scenario and a Bell scenario.Comment: 18 pages, 9 figure
Quantum logarithmic Sobolev inequalities and rapid mixing
A family of logarithmic Sobolev inequalities on finite dimensional quantum
state spaces is introduced. The framework of non-commutative \bL_p-spaces is
reviewed and the relationship between quantum logarithmic Sobolev inequalities
and the hypercontractivity of quantum semigroups is discussed. This
relationship is central for the derivation of lower bounds for the logarithmic
Sobolev (LS) constants. Essential results for the family of inequalities are
proved, and we show an upper bound to the generalized LS constant in terms of
the spectral gap of the generator of the semigroup. These inequalities provide
a framework for the derivation of improved bounds on the convergence time of
quantum dynamical semigroups, when the LS constant and the spectral gap are of
the same order. Convergence bounds on finite dimensional state spaces are
particularly relevant for the field of quantum information theory. We provide a
number of examples, where improved bounds on the mixing time of several
semigroups are obtained; including the depolarizing semigroup and quantum
expanders.Comment: Updated manuscript, 30 pages, no figure
A Resource Framework for Quantum Shannon Theory
Quantum Shannon theory is loosely defined as a collection of coding theorems,
such as classical and quantum source compression, noisy channel coding
theorems, entanglement distillation, etc., which characterize asymptotic
properties of quantum and classical channels and states. In this paper we
advocate a unified approach to an important class of problems in quantum
Shannon theory, consisting of those that are bipartite, unidirectional and
memoryless.
We formalize two principles that have long been tacitly understood. First, we
describe how the Church of the larger Hilbert space allows us to move flexibly
between states, channels, ensembles and their purifications. Second, we
introduce finite and asymptotic (quantum) information processing resources as
the basic objects of quantum Shannon theory and recast the protocols used in
direct coding theorems as inequalities between resources. We develop the rules
of a resource calculus which allows us to manipulate and combine resource
inequalities. This framework simplifies many coding theorem proofs and provides
structural insights into the logical dependencies among coding theorems.
We review the above-mentioned basic coding results and show how a subset of
them can be unified into a family of related resource inequalities. Finally, we
use this family to find optimal trade-off curves for all protocols involving
one noisy quantum resource and two noiseless ones.Comment: 60 page
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