679 research outputs found
Quantum Entanglement and Communication Complexity
We consider a variation of the multi-party communication complexity scenario
where the parties are supplied with an extra resource: particles in an
entangled quantum state. We show that, although a prior quantum entanglement
cannot be used to simulate a communication channel, it can reduce the
communication complexity of functions in some cases. Specifically, we show
that, for a particular function among three parties (each of which possesses
part of the function's input), a prior quantum entanglement enables them to
learn the value of the function with only three bits of communication occurring
among the parties, whereas, without quantum entanglement, four bits of
communication are necessary. We also show that, for a particular two-party
probabilistic communication complexity problem, quantum entanglement results in
less communication than is required with only classical random correlations
(instead of quantum entanglement). These results are a noteworthy contrast to
the well-known fact that quantum entanglement cannot be used to actually
simulate communication among remote parties.Comment: 10 pages, latex, no figure
Substituting Quantum Entanglement for Communication
We show that quantum entanglement can be used as a substitute for
communication when the goal is to compute a function whose input data is
distributed among remote parties. Specifically, we show that, for a particular
function among three parties (each of which possesses part of the function's
input), a prior quantum entanglement enables one of them to learn the value of
the function with only two bits of communication occurring among the parties,
whereas, without quantum entanglement, three bits of communication are
necessary. This result contrasts the well-known fact that quantum entanglement
cannot be used to simulate communication among remote parties.Comment: 4 pages REVTeX, no figures. Minor correction
Bell inequalities stronger than the CHSH inequality for 3-level isotropic states
We show that some two-party Bell inequalities with two-valued observables are
stronger than the CHSH inequality for 3 \otimes 3 isotropic states in the sense
that they are violated by some isotropic states in the 3 \otimes 3 system that
do not violate the CHSH inequality. These Bell inequalities are obtained by
applying triangular elimination to the list of known facet inequalities of the
cut polytope on nine points. This gives a partial solution to an open problem
posed by Collins and Gisin. The results of numerical optimization suggest that
they are candidates for being stronger than the I_3322 Bell inequality for 3
\otimes 3 isotropic states. On the other hand, we found no Bell inequalities
stronger than the CHSH inequality for 2 \otimes 2 isotropic states. In
addition, we illustrate an inclusion relation among some Bell inequalities
derived by triangular elimination.Comment: 9 pages, 1 figure. v2: organization improved; less references to the
cut polytope to make the main results clear; references added; typos
corrected; typesetting style change
Classical simulation of quantum entanglement without local hidden variables
Recent work has extended Bell's theorem by quantifying the amount of communication required to simulate entangled quantum systems with classical information. The general scenario is that a bipartite measurement is given from a set of possibilities and the goal is to find a classical scheme that reproduces exactly the correlations that arise when an actual quantum system is measured. Previous results have shown that, using local hidden variables, a finite amount of communication suffices to simulate the correlations for a Bell state. We extend this in a number of ways. First, we show that, when the communication is merely required to be finite {\em on average}, Bell states can be simulated {\em without} any local hidden variables. More generally, we show that arbitrary positive operator valued measurements on systems of Bell states can be simulated with bits of communication on average (again, without local hidden variables). On the other hand, when the communication is required to be {\em absolutely bounded}, we show that a finite number of bits of local hidden variables is insufficent to simulate a Bell state. This latter result is based on an analysis of the non-deterministic communication complexity of the NOT-EQUAL function, which is constant in the quantum model and logarithmic in the classical model
Quantum Computation and Spin Physics
A brief review is given of the physical implementation of quantum computation
within spin systems or other two-state quantum systems. The importance of the
controlled-NOT or quantum XOR gate as the fundamental primitive operation of
quantum logic is emphasized. Recent developments in the use of quantum
entanglement to built error-robust quantum states, and the simplest protocol
for quantum error correction, are discussed.Comment: 21 pages, Latex, 3 eps figures, prepared for the Proceedings of the
Annual MMM Meeting, November, 1996, to be published in J. Appl. Phy
The quantum speed up as advanced knowledge of the solution
With reference to a search in a database of size N, Grover states: "What is
the reason that one would expect that a quantum mechanical scheme could
accomplish the search in O(square root of N) steps? It would be insightful to
have a simple two line argument for this without having to describe the details
of the search algorithm". The answer provided in this work is: "because any
quantum algorithm takes the time taken by a classical algorithm that knows in
advance 50% of the information that specifies the solution of the problem".
This empirical fact, unnoticed so far, holds for both quadratic and exponential
speed ups and is theoretically justified in three steps: (i) once the physical
representation is extended to the production of the problem on the part of the
oracle and to the final measurement of the computer register, quantum
computation is reduction on the solution of the problem under a relation
representing problem-solution interdependence, (ii) the speed up is explained
by a simple consideration of time symmetry, it is the gain of information about
the solution due to backdating, to before running the algorithm, a
time-symmetric part of the reduction on the solution; this advanced knowledge
of the solution reduces the size of the solution space to be explored by the
algorithm, (iii) if I is the information acquired by measuring the content of
the computer register at the end of the algorithm, the quantum algorithm takes
the time taken by a classical algorithm that knows in advance 50% of I, which
brings us to the initial statement.Comment: 23 pages, to be published in IJT
Effective Pure States for Bulk Quantum Computation
In bulk quantum computation one can manipulate a large number of
indistinguishable quantum computers by parallel unitary operations and measure
expectation values of certain observables with limited sensitivity. The initial
state of each computer in the ensemble is known but not pure. Methods for
obtaining effective pure input states by a series of manipulations have been
described by Gershenfeld and Chuang (logical labeling) and Cory et al. (spatial
averaging) for the case of quantum computation with nuclear magnetic resonance.
We give a different technique called temporal averaging. This method is based
on classical randomization, requires no ancilla qubits and can be implemented
in nuclear magnetic resonance without using gradient fields. We introduce
several temporal averaging algorithms suitable for both high temperature and
low temperature bulk quantum computing and analyze the signal to noise behavior
of each.Comment: 24 pages in LaTex, 14 figures, the paper is also avalaible at
http://qso.lanl.gov/qc
Optimal estimation of multiple phases
We study the issue of simultaneous estimation of several phase shifts induced
by commuting operators on a quantum state. We derive the optimal positive
operator-valued measure corresponding to the multiple-phase estimation. In
particular, we discuss the explicit case of the optimal detection of double
phase for a system of identical qutrits and generalise these results to optimal
multiple phase detection for d-dimensional quantum states.Comment: 6 page
Information-theoretic interpretation of quantum error-correcting codes
Quantum error-correcting codes are analyzed from an information-theoretic
perspective centered on quantum conditional and mutual entropies. This approach
parallels the description of classical error correction in Shannon theory,
while clarifying the differences between classical and quantum codes. More
specifically, it is shown how quantum information theory accounts for the fact
that "redundant" information can be distributed over quantum bits even though
this does not violate the quantum "no-cloning" theorem. Such a remarkable
feature, which has no counterpart for classical codes, is related to the
property that the ternary mutual entropy vanishes for a tripartite system in a
pure state. This information-theoretic description of quantum coding is used to
derive the quantum analogue of the Singleton bound on the number of logical
bits that can be preserved by a code of fixed length which can recover a given
number of errors.Comment: 14 pages RevTeX, 8 Postscript figures. Added appendix. To appear in
Phys. Rev.
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