879 research outputs found
On the polar decomposition of right linear operators in quaternionic Hilbert spaces
In this article we prove the existence of the polar decomposition for densely
defined closed right linear operators in quaternionic Hilbert spaces: If is
a densely defined closed right linear operator in a quaternionic Hilbert space
, then there exists a partial isometry such that . In
fact is unique if . In particular, if is separable
and is a partial isometry with , then we prove that
if and only if either or .Comment: 17 page
Quantum Ballistic Evolution in Quantum Mechanics: Application to Quantum Computers
Quantum computers are important examples of processes whose evolution can be
described in terms of iterations of single step operators or their adjoints.
Based on this, Hamiltonian evolution of processes with associated step
operators is investigated here. The main limitation of this paper is to
processes which evolve quantum ballistically, i.e. motion restricted to a
collection of nonintersecting or distinct paths on an arbitrary basis. The main
goal of this paper is proof of a theorem which gives necessary and sufficient
conditions that T must satisfy so that there exists a Hamiltonian description
of quantum ballistic evolution for the process, namely, that T is a partial
isometry and is orthogonality preserving and stable on some basis. Simple
examples of quantum ballistic evolution for quantum Turing machines with one
and with more than one type of elementary step are discussed. It is seen that
for nondeterministic machines the basis set can be quite complex with much
entanglement present. It is also proved that, given a step operator T for an
arbitrary deterministic quantum Turing machine, it is decidable if T is stable
and orthogonality preserving, and if quantum ballistic evolution is possible.
The proof fails if T is a step operator for a nondeterministic machine. It is
an open question if such a decision procedure exists for nondeterministic
machines. This problem does not occur in classical mechanics.Comment: 37 pages Latexwith 2 postscript figures tar+gzip+uuencoded, to be
published in Phys. Rev.
Randomizations of models as metric structures
The notion of a randomization of a first order structure was introduced by
Keisler in the paper Randomizing a Model, Advances in Math. 1999. The idea was
to form a new structure whose elements are random elements of the original
first order structure. In this paper we treat randomizations as continuous
structures in the sense of Ben Yaacov and Usvyatsov. In this setting, the
earlier results show that the randomization of a complete first order theory is
a complete theory in continuous logic that admits elimination of quantifiers
and has a natural set of axioms. We show that the randomization operation
preserves the properties of being omega-categorical, omega-stable, and stable
Multipartite entanglement in fermionic systems via a geometric measure
We study multipartite entanglement in a system consisting of
indistinguishable fermions. Specifically, we have proposed a geometric
entanglement measure for N spin-1/2 fermions distributed over 2L modes (single
particle states). The measure is defined on the 2L qubit space isomorphic to
the Fock space for 2L single particle states. This entanglement measure is
defined for a given partition of 2L modes containing m >= 2 subsets. Thus this
measure applies to m <= 2L partite fermionic system where L is any finite
number, giving the number of sites. The Hilbert spaces associated with these
subsets may have different dimensions. Further, we have defined the local
quantum operations with respect to a given partition of modes. This definition
is generic and unifies different ways of dividing a fermionic system into
subsystems. We have shown, using a representative case, that the geometric
measure is invariant under local unitaries corresponding to a given partition.
We explicitly demonstrate the use of the measure to calculate multipartite
entanglement in some correlated electron systems. To the best of our knowledge,
there is no usable entanglement measure of m > 3 partite fermionic systems in
the literature, so that this is the first measure of multipartite entanglement
for fermionic systems going beyond the bipartite and tripartite cases.Comment: 25 pages, 8 figure
Quantum Homodyne Tomography as an Informationally Complete Positive Operator Valued Measure
We define a positive operator valued measure on
describing the measurement of randomly sampled quadratures in quantum homodyne
tomography, and we study its probabilistic properties. Moreover, we give a
mathematical analysis of the relation between the description of a state in
terms of and the description provided by its Wigner transform.Comment: 9 page
Deterministic and Unambiguous Dense Coding
Optimal dense coding using a partially-entangled pure state of Schmidt rank
and a noiseless quantum channel of dimension is studied both in
the deterministic case where at most messages can be transmitted with
perfect fidelity, and in the unambiguous case where when the protocol succeeds
(probability ) Bob knows for sure that Alice sent message , and when
it fails (probability ) he knows it has failed. Alice is allowed any
single-shot (one use) encoding procedure, and Bob any single-shot measurement.
For a bound is obtained for in terms of the largest
Schmidt coefficient of the entangled state, and is compared with published
results by Mozes et al. For it is shown that is strictly
less than unless is an integer multiple of , in which case
uniform (maximal) entanglement is not needed to achieve the optimal protocol.
The unambiguous case is studied for , assuming for a
set of messages, and a bound is obtained for the average
\lgl1/\tau\rgl. A bound on the average \lgl\tau\rgl requires an additional
assumption of encoding by isometries (unitaries when ) that are
orthogonal for different messages. Both bounds are saturated when is a
constant independent of , by a protocol based on one-shot entanglement
concentration. For it is shown that (at least) messages can
be sent unambiguously. Whether unitary (isometric) encoding suffices for
optimal protocols remains a major unanswered question, both for our work and
for previous studies of dense coding using partially-entangled states,
including noisy (mixed) states.Comment: Short new section VII added. Latex 23 pages, 1 PSTricks figure in
tex
Dimension minimization of a quantum automaton
A new model of a Quantum Automaton (QA), working with qubits is proposed. The
quantum states of the automaton can be pure or mixed and are represented by
density operators. This is the appropriated approach to deal with measurements
and dechorence. The linearity of a QA and of the partial trace super-operator,
combined with the properties of invariant subspaces under unitary
transformations, are used to minimize the dimension of the automaton and,
consequently, the number of its working qubits. The results here developed are
valid wether the state set of the QA is finite or not. There are two main
results in this paper: 1) We show that the dimension reduction is possible
whenever the unitary transformations, associated to each letter of the input
alphabet, obey a set of conditions. 2) We develop an algorithm to find out the
equivalent minimal QA and prove that its complexity is polynomial in its
dimension and in the size of the input alphabet.Comment: 26 page
There exist non orthogonal quantum measurements that are perfectly repeatable
We show that, contrarily to the widespread belief, in quantum mechanics
repeatable measurements are not necessarily described by orthogonal
projectors--the customary paradigm of "observable". Nonorthogonal
repeatability, however, occurs only for infinite dimensions. We also show that
when a non orthogonal repeatable measurement is performed, the measured system
retains some "memory" of the number of times that the measurement has been
performed.Comment: 4 pages, 1 figure, revtex4, minor change
Semispectral measures as convolutions and their moment operators
The moment operators of a semispectral measure having the structure of the
convolution of a positive measure and a semispectral measure are studied, with
paying attention to the natural domains of these unbounded operators. The
results are then applied to conveniently determine the moment operators of the
Cartesian margins of the phase space observables.Comment: 7 page
Economical ontological models for discrete quantum systems
I use the recently proposed framework of ontological models [Harrigan et al.,
arXiv:0709.1149v2] to obtain economical models for results of tomographically
complete sets of measurements on finite-dimensional quantum systems. I describe
a procedure that simplifies the models by decreasing the number of necessary
ontic states, and present an explicit model with just 33 ontic states for a
qutrit.Comment: 6 pages, 2 figures. v2: erroneous claim removed, compression method
improved, reference adde
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