9,703 research outputs found
Lower bounds on the communication complexity of two-party (quantum) processes
The process of state preparation, its transmission and subsequent measurement
can be classically simulated through the communication of some amount of
classical information. Recently, we proved that the minimal communication cost
is the minimum of a convex functional over a space of suitable probability
distributions. It is now proved that this optimization problem is the dual of a
geometric programming maximization problem, which displays some appealing
properties. First, the number of variables grows linearly with the input size.
Second, the objective function is linear in the input parameters and the
variables. Finally, the constraints do not depend on the input parameters.
These properties imply that, once a feasible point is found, the computation of
a lower bound on the communication cost in any two-party process is linearly
complex. The studied scenario goes beyond quantum processes and includes the
communication complexity scenario introduced by Yao. We illustrate the method
by analytically deriving some non-trivial lower bounds. Finally, we conjecture
the lower bound for a noiseless quantum channel with capacity
qubits. This bound can have an interesting consequence in the context of the
recent quantum-foundational debate on the reality of the quantum state.Comment: Conference version. A more extensive version with more details will
be available soo
Tensor Norms and the Classical Communication Complexity of Nonlocal Quantum Measurement
We initiate the study of quantifying nonlocalness of a bipartite measurement
by the minimum amount of classical communication required to simulate the
measurement. We derive general upper bounds, which are expressed in terms of
certain tensor norms of the measurement operator. As applications, we show that
(a) If the amount of communication is constant, quantum and classical
communication protocols with unlimited amount of shared entanglement or shared
randomness compute the same set of functions; (b) A local hidden variable model
needs only a constant amount of communication to create, within an arbitrarily
small statistical distance, a distribution resulted from local measurements of
an entangled quantum state, as long as the number of measurement outcomes is
constant.Comment: A preliminary version of this paper appears as part of an article in
Proceedings of the the 37th ACM Symposium on Theory of Computing (STOC 2005),
460--467, 200
Optimal measurements for nonlocal correlations
A problem in quantum information theory is to find the experimental setup
that maximizes the nonlocality of correlations with respect to some suitable
measure such as the violation of Bell inequalities. The latter has however some
drawbacks. First and foremost it is unfeasible to determine the whole set of
Bell inequalities already for a few measurements and thus unfeasible to find
the experimental setup maximizing their violation. Second, the Bell violation
suffers from an ambiguity stemming from the choice of the normalization of the
Bell coefficients. An alternative measure of nonlocality with a direct
information-theoretic interpretation is the minimal amount of classical
communication required for simulating nonlocal correlations. In the case of
many instances simulated in parallel, the minimal communication cost per
instance is called nonlocal capacity, and its computation can be reduced to a
convex-optimization problem. This quantity can be computed for a higher number
of measurements and turns out to be useful for finding the optimal experimental
setup. Focusing on the bipartite case, in this paper, we present a simple
method for maximizing the nonlocal capacity over a given configuration space
and, in particular, over a set of possible measurements, yielding the
corresponding optimal setup. Furthermore, we show that there is a functional
relationship between Bell violation and nonlocal capacity. The method is
illustrated with numerical tests and compared with the maximization of the
violation of CGLMP-type Bell inequalities on the basis of entangled two-qubit
as well as two-qutrit states. Remarkably, the anomaly of nonlocality displayed
by qutrits turns out to be even stronger if the nonlocal capacity is employed
as a measure of nonlocality.Comment: Some typos and errors have been corrected, especially in the section
concerning the relation between Bell violation and communication complexit
Quantum states cannot be transmitted efficiently classically
We show that any classical two-way communication protocol with shared
randomness that can approximately simulate the result of applying an arbitrary
measurement (held by one party) to a quantum state of qubits (held by
another), up to constant accuracy, must transmit at least bits.
This lower bound is optimal and matches the complexity of a simple protocol
based on discretisation using an -net. The proof is based on a lower
bound on the classical communication complexity of a distributed variant of the
Fourier sampling problem. We obtain two optimal quantum-classical separations
as easy corollaries. First, a sampling problem which can be solved with one
quantum query to the input, but which requires classical queries
for an input of size . Second, a nonlocal task which can be solved using
Bell pairs, but for which any approximate classical solution must communicate
bits.Comment: 24 pages; v3: accepted version incorporating many minor corrections
and clarification
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