8 research outputs found
The communication complexity of non-signaling distributions
We study a model of communication complexity that encompasses many
well-studied problems, including classical and quantum communication
complexity, the complexity of simulating distributions arising from bipartite
measurements of shared quantum states, and XOR games. In this model, Alice gets
an input x, Bob gets an input y, and their goal is to each produce an output
a,b distributed according to some pre-specified joint distribution p(a,b|x,y).
We introduce a new technique based on affine combinations of lower-complexity
distributions. Specifically, we introduce two complexity measures, one which
gives lower bounds on classical communication, and one for quantum
communication. These measures can be expressed as convex optimization problems.
We show that the dual formulations have a striking interpretation, since they
coincide with maximum violations of Bell and Tsirelson inequalities. The dual
expressions are closely related to the winning probability of XOR games. These
lower bounds subsume many known communication complexity lower bound methods,
most notably the recent lower bounds of Linial and Shraibman for the special
case of Boolean functions.
We show that the gap between the quantum and classical lower bounds is at
most linear in the size of the support of the distribution, and does not depend
on the size of the inputs. This translates into a bound on the gap between
maximal Bell and Tsirelson inequality violations, which was previously known
only for the case of distributions with Boolean outcomes and uniform marginals.
Finally, we give an exponential upper bound on quantum and classical
communication complexity in the simultaneous messages model, for any
non-signaling distribution. One consequence is a simple proof that any quantum
distribution can be approximated with a constant number of bits of
communication.Comment: 23 pages. V2: major modifications, extensions and additions compared
to V1. V3 (21 pages): proofs have been updated and simplified, particularly
Theorem 10 and Theorem 22. V4 (23 pages): Section 3.1 has been rewritten (in
particular Lemma 10 and its proof), and various minor modifications have been
made. V5 (24 pages): various modifications in the presentatio
Simulating all non-signalling correlations via classical or quantum theory with negative probabilities
Many-party correlations between measurement outcomes in general probabilistic
theories are given by conditional probability distributions obeying the
non-signalling condition. We show that any such distribution can be obtained
from classical or quantum theory, by relaxing positivity constraints on either
the mixed state shared by the parties, or the local functions which generate
measurement outcomes. Our results apply to generic non-signalling correlations,
but in particular they yield two distinct quasi-classical models for quantum
correlations.Comment: 6 page
Better Non-Local Games from Hidden Matching
We construct a non-locality game that can be won with certainty by a quantum
strategy using log n shared EPR-pairs, while any classical strategy has winning
probability at most 1/2+O(log n/sqrt{n}). This improves upon a recent result of
Junge et al. in a number of ways.Comment: 11 pages, late
Bell Violations through Independent Bases Games
In a recent paper, Junge and Palazuelos presented two two-player games
exhibiting interesting properties. In their first game, entangled players can
perform notably better than classical players. The quantitative gap between the
two cases is remarkably large, especially as a function of the number of inputs
to the players. In their second game, entangled players can perform notably
better than players that are restricted to using a maximally entangled state
(of arbitrary dimension). This was the first game exhibiting such a behavior.
The analysis of both games is heavily based on non-trivial results from Banach
space theory and operator space theory. Here we present two games exhibiting a
similar behavior, but with proofs that are arguably simpler, using elementary
probabilistic techniques and standard quantum information arguments. Our games
also give better quantitative bounds.Comment: Minor update
Unbounded violations of bipartite Bell Inequalities via Operator Space theory
In this work we show that bipartite quantum states with local Hilbert space
dimension n can violate a Bell inequality by a factor of order (up
to a logarithmic factor) when observables with n possible outcomes are used. A
central tool in the analysis is a close relation between this problem and
operator space theory and, in particular, the very recent noncommutative
embedding theory. As a consequence of this result, we obtain better Hilbert
space dimension witnesses and quantum violations of Bell inequalities with
better resistance to noise
Classical and quantum partition bound and detector inefficiency
We study randomized and quantum efficiency lower bounds in communication
complexity. These arise from the study of zero-communication protocols in which
players are allowed to abort. Our scenario is inspired by the physics setup of
Bell experiments, where two players share a predefined entangled state but are
not allowed to communicate. Each is given a measurement as input, which they
perform on their share of the system. The outcomes of the measurements should
follow a distribution predicted by quantum mechanics; however, in practice, the
detectors may fail to produce an output in some of the runs. The efficiency of
the experiment is the probability that the experiment succeeds (neither of the
detectors fails).
When the players share a quantum state, this gives rise to a new bound on
quantum communication complexity (eff*) that subsumes the factorization norm.
When players share randomness instead of a quantum state, the efficiency bound
(eff), coincides with the partition bound of Jain and Klauck. This is one of
the strongest lower bounds known for randomized communication complexity, which
subsumes all the known combinatorial and algebraic methods including the
rectangle (corruption) bound, the factorization norm, and discrepancy.
The lower bound is formulated as a convex optimization problem. In practice,
the dual form is more feasible to use, and we show that it amounts to
constructing an explicit Bell inequality (for eff) or Tsirelson inequality (for
eff*). We give an example of a quantum distribution where the violation can be
exponentially bigger than the previously studied class of normalized Bell
inequalities.
For one-way communication, we show that the quantum one-way partition bound
is tight for classical communication with shared entanglement up to arbitrarily
small error.Comment: 21 pages, extended versio
Large violation of Bell inequalities with low entanglement
In this paper we obtain violations of general bipartite Bell inequalities of
order with inputs, outputs and
-dimensional Hilbert spaces. Moreover, we construct explicitly, up to a
random choice of signs, all the elements involved in such violations: the
coefficients of the Bell inequalities, POVMs measurements and quantum states.
Analyzing this construction we find that, even though entanglement is necessary
to obtain violation of Bell inequalities, the Entropy of entanglement of the
underlying state is essentially irrelevant in obtaining large violation. We
also indicate why the maximally entangled state is a rather poor candidate in
producing large violations with arbitrary coefficients. However, we also show
that for Bell inequalities with positive coefficients (in particular, games)
the maximally entangled state achieves the largest violation up to a
logarithmic factor.Comment: Reference [16] added. Some typos correcte
Violaciones de Bell en información cuántica: escenarios de no señalización, con comunicación y multipartitos
Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Análisis y Matemática Aplicada, leída el 27/01/2021In foundations of quantum information, the possible advantages of using quantum resources compared to classical resources are studied. Its relevance is due to the great current development of quantum technologies.One of the lines on which this study is based was started by Bell in 1964. He established that certain results obtained by measuring composite quantum systems separately are incompatible with a local and realistic description of nature, even in the assumption in which we allow the existence of hidden variables. This phenomenon is known as quantum non-locality. In addition to the theoretical interest that this has, it has been shown, since the 1990s, that this quantum phenomenon can be used to obtain advantages in cryptography, communication complexity and randomness amplification...En fundamentos de información cuántica se estudian las posibles ventajas de utilizar recursos cuánticos frente a recursos clásicos. Su relevancia se debe al gran desarrollo actual de las tecnologías cuánticas. Una de las líneas sobre las que se basa este estudio fue iniciada por Bell en el año 1964. Estableció que ciertos resultados obtenidos en mediciones separadas en sistemas cuánticos compuestos son incompatibles con una descripción de la naturaleza local y realista, incluso en el supuesto en el que permitamos la existencia de variables ocultas. Este fenómeno se conoce como no localidad cuántica. Además del interés teórico que esto tiene, se ha demostrado, desde la década de los 90, que este fenómeno cuántico se puede aprovechar para obtener ventajas en criptografía, complejidad de comunicación y amplificación de aleatoriedad...Fac. de Ciencias MatemáticasTRUEunpu