42,209 research outputs found
Two Results about Quantum Messages
We show two results about the relationship between quantum and classical
messages. Our first contribution is to show how to replace a quantum message in
a one-way communication protocol by a deterministic message, establishing that
for all partial Boolean functions we
have . This bound was previously
known for total functions, while for partial functions this improves on results
by Aaronson, in which either a log-factor on the right hand is present, or the
left hand side is , and in which also no entanglement is
allowed.
In our second contribution we investigate the power of quantum proofs over
classical proofs. We give the first example of a scenario, where quantum proofs
lead to exponential savings in computing a Boolean function. The previously
only known separation between the power of quantum and classical proofs is in a
setting where the input is also quantum.
We exhibit a partial Boolean function , such that there is a one-way
quantum communication protocol receiving a quantum proof (i.e., a protocol of
type QMA) that has cost for , whereas every one-way quantum
protocol for receiving a classical proof (protocol of type QCMA) requires
communication
Quantum Information Complexity and Amortized Communication
We define a new notion of information cost for quantum protocols, and a
corresponding notion of quantum information complexity for bipartite quantum
channels, and then investigate the properties of such quantities. These are the
fully quantum generalizations of the analogous quantities for bipartite
classical functions that have found many applications recently, in particular
for proving communication complexity lower bounds. Our definition is strongly
tied to the quantum state redistribution task.
Previous attempts have been made to define such a quantity for quantum
protocols, with particular applications in mind; our notion differs from these
in many respects. First, it directly provides a lower bound on the quantum
communication cost, independent of the number of rounds of the underlying
protocol. Secondly, we provide an operational interpretation for quantum
information complexity: we show that it is exactly equal to the amortized
quantum communication complexity of a bipartite channel on a given state. This
generalizes a result of Braverman and Rao to quantum protocols, and even
strengthens the classical result in a bounded round scenario. Also, this
provides an analogue of the Schumacher source compression theorem for
interactive quantum protocols, and answers a question raised by Braverman.
We also discuss some potential applications to quantum communication
complexity lower bounds by specializing our definition for classical functions
and inputs. Building on work of Jain, Radhakrishnan and Sen, we provide new
evidence suggesting that the bounded round quantum communication complexity of
the disjointness function is \Omega (n/M + M), for M-message protocols. This
would match the best known upper bound.Comment: v1, 38 pages, 1 figur
The quantum one-time pad in the presence of an eavesdropper
A classical one-time pad allows two parties to send private messages over a
public classical channel -- an eavesdropper who intercepts the communication
learns nothing about the message. A quantum one-time pad is a shared quantum
state which allows two parties to send private messages or private quantum
states over a public quantum channel. If the eavesdropper intercepts the
quantum communication she learns nothing about the message. In the classical
case, a one-time pad can be created using shared and partially private
correlations. Here we consider the quantum case in the presence of an
eavesdropper, and find the single letter formula for the rate at which the two
parties can send messages using a quantum one-time pad
Quantum Arthur-Merlin Games
This paper studies quantum Arthur-Merlin games, which are Arthur-Merlin games
in which Arthur and Merlin can perform quantum computations and Merlin can send
Arthur quantum information. As in the classical case, messages from Arthur to
Merlin are restricted to be strings of uniformly generated random bits. It is
proved that for one-message quantum Arthur-Merlin games, which correspond to
the complexity class QMA, completeness and soundness errors can be reduced
exponentially without increasing the length of Merlin's message. Previous
constructions for reducing error required a polynomial increase in the length
of Merlin's message. Applications of this fact include a proof that logarithmic
length quantum certificates yield no increase in power over BQP and a simple
proof that QMA is contained in PP. Other facts that are proved include the
equivalence of three (or more) message quantum Arthur-Merlin games with
ordinary quantum interactive proof systems and some basic properties concerning
two-message quantum Arthur-Merlin games.Comment: 22 page
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