Vol.3 No.4
July 1,
2003
Researches:
Infinitely entangled states
(pp281306)
M. Keyl,
D. Schlingemann and R.F. Werner
For states in infinite dimensional Hilbert spaces
entanglement quantities like the entanglement of distillation can become
infinite. This leads naturally to the question, whether one system in
such an infinitely entangled state can serve as a resource for tasks
like the teleportation of arbitrarily many qubits. We show that
appropriate states cannot be obtained by density operators in an
infinite dimensional Hilbert space. However, using techniques for the
description of infinitely many degrees of freedom from field theory and
statistical mechanics, such states can nevertheless be constructed
rigorously. We explore two related possibilities, namely an extended
notion of algebras of observables, and the use of singular states on the
algebra of bounded operators. As applications we construct the
essentially unique infinite analogue of maximally entangled states, and
the singular state used heuristically in the fundamental paper of
Einstein, Rosen and Podolsky.
Nonempty quantum dot as a spinentangler (pp307316)
C.L. Chou
We consider a threeport singlelevel quantum dot system
with one input and two output leads. Instead of considering an empty
dot, we study the situations that two input electrons cotunnel through
the quantum dot occupied by one or two dot electrons. We show that
electron entanglement can be generated via the cotunneling processes
when the dot is occupied by two electrons, yielding nonlocal
spinsinglet states at the output leads. When the dot is occupied by a
single electron, we show that by carefully selecting model parameters
nonlocal spintriplet electrons can also be obtained at the output
leads if the final dot electron has the same spin as that of the initial
dot electron.
Shor's discrete logarithm quantum
algorithm for elliptic curves (pp317344)
J. Proos and Ch. Zalka
We show in some detail how to implement Shor's efficient
quantum algorithm for discrete logarithms for the particular case of
elliptic curve groups. It turns out that for this problem a smaller
quantum computer can solve problems further beyond current computing
than for integer factorisation. A 160 bit elliptic curve cryptographic
key could be broken on a quantum computer using around 1000 qubits while
factoring the securitywise equivalent 1024 bit RSA modulus would
require about 2000 qubits. In this paper we only consider elliptic
curves over GF(p) and not yet the equally important ones over GF(2^n)
or other finite fields. The main technical difficulty is to implement
Euclid's gcd algorithm to compute multiplicative inverses modulo p. As
the runtime of Euclid's algorithm depends on the input, one difficulty
encountered is the ``quantum halting problem''.
Tight Bell inequality for doutcome
measurements correlations (pp345358)
Ll. Masanes
In this paper we prove that the inequality introduced by
Collins, Gisin, Linden, Massar and Popescu is tight, or in other words,
it is a facet of the convex polytope generated by all localrealistic
joint probabilities of doutcomes. This means that this inequality is
optimal. We also show that, for correlation functions generalized to
deal with threeoutcome measurements, the satisfyability of this
inequality is a necessary and sufficient condition for the existence of
a localrealistic model accounting for them.
Universal compression of ergodic quantum sources (pp359375)
A Kaltchenko and EH Yang
For a real number r>0, let F(r) be the family of all stationary ergodic quantum sources with von Neumann entropy rates less than
r. We
prove that, for any r>0, there exists a blind, sourceindependent
block compression scheme which compresses every source from F(r) to rn qubits per input block length~n with arbitrarily high fidelity for
all large n.}As our second result, we show that the stationarity and
the ergodicity of a quantum source \rho_m_{m=1}^{\infty} are
preserved by any tracepreserving completely positive linear map of the
tensor product form {\cal E}^{\otimes m}, where a copy of {\cal E}
acts locally on each spin lattice site. We also establish ergodicity
criteria for so
called classicallycorrelated quantum sources.
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