800 research outputs found
Mutually unbiased phase states, phase uncertainties, and Gauss sums
Mutually unbiased bases (MUBs), which are such that the inner product between
two vectors in different orthogonal bases is a constant equal to 1/sqrt{d),
with d the dimension of the finite Hilbert space, are becoming more and more
studied for applications such as quantum tomography and cryptography, and in
relation to entangled states and to the Heisenberg-Weil group of quantum
optics. Complete sets of MUBs of cardinality d+1 have been derived for prime
power dimensions d=p^m using the tools of abstract algebra. Presumably, for non
prime dimensions the cardinality is much less. Here we reinterpret MUBs as
quantum phase states, i.e. as eigenvectors of Hermitean phase operators
generalizing those introduced by Pegg & Barnett in 1989. We relate MUB states
to additive characters of Galois fields (in odd characteristic p) and to Galois
rings (in characteristic 2). Quantum Fourier transforms of the components in
vectors of the bases define a more general class of MUBs with multiplicative
characters and additive ones altogether. We investigate the complementary
properties of the above phase operator with respect to the number operator. We
also study the phase probability distribution and variance for general pure
quantum electromagnetic states and find them to be related to the Gauss sums,
which are sums over all elements of the field (or of the ring) of the product
of multiplicative and additive characters. Finally, we relate the concepts of
mutual unbiasedness and maximal entanglement. This allows to use well studied
algebraic concepts as efficient tools in the study of entanglement and its
information aspectsComment: 16 pages, a few typos corrected, some references updated, note
acknowledging I. Shparlinski adde
Bistability for asymmetric discrete random walks
We show that asymmetric time-continuous discrete random walks can display
bistability for equal values of Jauslin's shifting parameters. The bistability
becomes more pronounced at increased asymmetry parameterComment: Follow-up note to hep-th/9411026[PRE 51, 5112 (May 95)], one fig.
include
MUBs: From finite projective geometry to quantum phase enciphering
This short note highlights the most prominent mathematical problems and
physical questions associated with the existence of the maximum sets of
mutually unbiased bases (MUBs) in the Hilbert space of a given dimensionComment: 5 pages, accepted for AIP Conf Book, QCMC 2004, Strathclyde, Glasgow,
minor correction
Supersymmetric Fokker-Planck strict isospectrality
I report a study of the nonstationary one-dimensional Fokker-Planck solutions
by means of the strictly isospectral method of supesymmetric quantum mechanics.
The main conclusion is that this technique can lead to a space-dependent
(modulational) damping of the spatial part of the nonstationary Fokker-Planck
solutions, which I call strictly isospectral damping. At the same time, using
an additive decomposition of the nonstationary solutions suggested by the
strictly isospectral procedure and by an argument of Englefield [J. Stat. Phys.
52, 369 (1988)], they can be normalized and thus turned into physical
solutions, i.e., Fokker-Planck probability densities. There might be
applications to many physical processes during their transient periodComment: revised version, scheduled for PRE 56 (1 August 1997) as a B
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