2,237 research outputs found
Quantum theory without Hilbert spaces
Quantum theory does not only predict probabilities, but also relative phases
for any experiment, that involves measurements of an ensemble of systems at
different moments of time. We argue, that any operational formulation of
quantum theory needs an algebra of observables and an object that incorporates
the information about relative phases and probabilities. The latter is the
(de)coherence functional, introduced by the consistent histories approach to
quantum theory. The acceptance of relative phases as a primitive ingredient of
any quantum theory, liberates us from the need to use a Hilbert space and
non-commutative observables. It is shown, that quantum phenomena are adequately
described by a theory of relative phases and non-additive probabilities on the
classical phase space. The only difference lies on the type of observables that
correspond to sharp measurements. This class of theories does not suffer from
the consequences of Bell's theorem (it is not a theory of Kolmogorov
probabilities) and Kochen- Specker's theorem (it has distributive "logic"). We
discuss its predictability properties, the meaning of the classical limit and
attempt to see if it can be experimentally distinguished from standard quantum
theory. Our construction is operational and statistical, in the spirit of
Kopenhagen, but makes plausible the existence of a realist, geometric theory
for individual quantum systems.Comment: 32 pages, Latex, 4 figures. Small changes in the revised version,
comments and references added; essentially the version to appear in Found.
Phy
A very brief introduction to quantum computing and quantum information theory for mathematicians
This is a very brief introduction to quantum computing and quantum
information theory, primarily aimed at geometers. Beyond basic definitions and
examples, I emphasize aspects of interest to geometers, especially connections
with asymptotic representation theory. Proofs of most statements can be found
in standard references
Quantum algorithms for highly non-linear Boolean functions
Attempts to separate the power of classical and quantum models of computation
have a long history. The ultimate goal is to find exponential separations for
computational problems. However, such separations do not come a dime a dozen:
while there were some early successes in the form of hidden subgroup problems
for abelian groups--which generalize Shor's factoring algorithm perhaps most
faithfully--only for a handful of non-abelian groups efficient quantum
algorithms were found. Recently, problems have gotten increased attention that
seek to identify hidden sub-structures of other combinatorial and algebraic
objects besides groups. In this paper we provide new examples for exponential
separations by considering hidden shift problems that are defined for several
classes of highly non-linear Boolean functions. These so-called bent functions
arise in cryptography, where their property of having perfectly flat Fourier
spectra on the Boolean hypercube gives them resilience against certain types of
attack. We present new quantum algorithms that solve the hidden shift problems
for several well-known classes of bent functions in polynomial time and with a
constant number of queries, while the classical query complexity is shown to be
exponential. Our approach uses a technique that exploits the duality between
bent functions and their Fourier transforms.Comment: 15 pages, 1 figure, to appear in Proceedings of the 21st Annual
ACM-SIAM Symposium on Discrete Algorithms (SODA'10). This updated version of
the paper contains a new exponential separation between classical and quantum
query complexit
Programming Telepathy: Implementing Quantum Non-Locality Games
Quantum pseudo-telepathy is an intriguing phenomenon which results from the
application of quantum information theory to communication complexity. To
demonstrate this phenomenon researchers in the field of quantum communication
complexity devised a number of quantum non-locality games. The setting of these
games is as follows: the players are separated so that no communication between
them is possible and are given a certain computational task. When the players
have access to a quantum resource called entanglement, they can accomplish the
task: something that is impossible in a classical setting. To an observer who
is unfamiliar with the laws of quantum mechanics it seems that the players
employ some sort of telepathy; that is, they somehow exchange information
without sharing a communication channel. This paper provides a formal framework
for specifying, implementing, and analysing quantum non-locality games
Pattern Recognition In Non-Kolmogorovian Structures
We present a generalization of the problem of pattern recognition to
arbitrary probabilistic models. This version deals with the problem of
recognizing an individual pattern among a family of different species or
classes of objects which obey probabilistic laws which do not comply with
Kolmogorov's axioms. We show that such a scenario accommodates many important
examples, and in particular, we provide a rigorous definition of the classical
and the quantum pattern recognition problems, respectively. Our framework
allows for the introduction of non-trivial correlations (as entanglement or
discord) between the different species involved, opening the door to a new way
of harnessing these physical resources for solving pattern recognition
problems. Finally, we present some examples and discuss the computational
complexity of the quantum pattern recognition problem, showing that the most
important quantum computation algorithms can be described as non-Kolmogorovian
pattern recognition problems
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