63 research outputs found
Dynamical Ambiguities in Singular Gravitational Field
We consider particle dynamics in singular gravitational field. In 2d
spacetime the system splits into two independent gravitational systems without
singularity. Dynamical integrals of each system define algebra, but
the corresponding symmetry transformations are not defined globally.
Quantization leads to ambiguity. By including singularity one can get the
global symmetry. Quantization in this case leads to unique quantum
theory.Comment: 7 pages, latex, no figures, submitted for publicatio
Information-disturbance tradeoff in estimating a maximally entangled state
We derive the amount of information retrieved by a quantum measurement in
estimating an unknown maximally entangled state, along with the pertaining
disturbance on the state itself. The optimal tradeoff between information and
disturbance is obtained, and a corresponding optimal measurement is provided.Comment: 4 pages. Accepted for publication on Physical Review Letter
On the fidelity of two pure states
The fidelity of two pure states (also known as transition probability) is a
symmetric function of two operators, and well-founded operationally as an event
probability in a certain preparation-test pair. Motivated by the idea that the
fidelity is the continuous quantum extension of the combinatorial equality
function, we enquire whether there exists a symmetric operational way of
obtaining the fidelity. It is shown that this is impossible. Finally, we
discuss the optimal universal approximation by a quantum operation.Comment: LaTeX2e, 8 pages, submitted to J. Phys. A: Math. and Ge
Weight bases of Gelfand-Tsetlin type for representations of classical Lie algebras
This paper completes a series devoted to explicit constructions of
finite-dimensional irreducible representations of the classical Lie algebras.
Here the case of odd orthogonal Lie algebras (of type B) is considered (two
previous papers dealt with C and D types). A weight basis for each
representation of the Lie algebra o(2n+1) is constructed. The basis vectors are
parametrized by Gelfand--Tsetlin-type patterns. Explicit formulas for the
matrix elements of generators of o(2n+1) in this basis are given. The
construction is based on the representation theory of the Yangians. A similar
approach is applied to the A type case where the well-known formulas due to
Gelfand and Tsetlin are reproduced.Comment: 29 pages, Late
Quantum state estimation and large deviations
In this paper we propose a method to estimate the density matrix \rho of a
d-level quantum system by measurements on the N-fold system. The scheme is
based on covariant observables and representation theory of unitary groups and
it extends previous results concerning the estimation of the spectrum of \rho.
We show that it is consistent (i.e. the original input state \rho is recovered
with certainty if N \to \infty), analyze its large deviation behavior, and
calculate explicitly the corresponding rate function which describes the
exponential decrease of error probabilities in the limit N \to \infty. Finally
we discuss the question whether the proposed scheme provides the fastest
possible decay of error probabilities.Comment: LaTex2e, 40 pages, 2 figures. Substantial changes in Section 4: one
new subsection (4.1) and another (4.2 was 4.1 in the previous version)
completely rewritten. Minor changes in Sect. 2 and 3. Typos corrected.
References added. Accepted for publication in Rev. Math. Phy
Covariant quantum measurements which maximize the likelihood
We derive the class of covariant measurements which are optimal according to
the maximum likelihood criterion. The optimization problem is fully resolved in
the case of pure input states, under the physically meaningful hypotheses of
unimodularity of the covariance group and measurability of the stability
subgroup. The general result is applied to the case of covariant state
estimation for finite dimension, and to the Weyl-Heisenberg displacement
estimation in infinite dimension. We also consider estimation with multiple
copies, and compare collective measurements on identical copies with the scheme
of independent measurements on each copy. A "continuous-variables" analogue of
the measurement of direction of the angular momentum with two anti-parallel
spins by Gisin and Popescu is given.Comment: 8 pages, RevTex style, submitted to Phys. Rev.
Differential Calculi on Some Quantum Prehomogeneous Vector Spaces
This paper is devoted to study of differential calculi over quadratic
algebras, which arise in the theory of quantum bounded symmetric domains. We
prove that in the quantum case dimensions of the homogeneous components of the
graded vector spaces of k-forms are the same as in the classical case. This
result is well-known for quantum matrices.
The quadratic algebras, which we consider in the present paper, are
q-analogues of the polynomial algebras on prehomogeneous vector spaces of
commutative parabolic type. This enables us to prove that the de Rham complex
is isomorphic to the dual of a quantum analogue of the generalized
Bernstein-Gelfand-Gelfand resolution.Comment: LaTeX2e, 51 pages; changed conten
Semi-infinite cohomology of W-algebras
We generalize some of the standard homological techniques to \cW-algebras,
and compute the semi-infinite cohomology of the \cW_3 algebra on a variety of
modules. These computations provide physical states in \cW_3 gravity coupled
to \cW_3 minimal models and to two free scalar fields.Comment: 15 page
Extremal projectors for contragredient Lie (super)symmetries (short review)
A brief review of the extremal projectors for contragredient Lie
(super)symmetries (finite-dimensional simple Lie algebras, basic classical Lie
superalgebras, infinite-dimensional affine Kac-Moody algebras and
superalgebras, as well as their quantum -analogs) is given. Some
bibliographic comments on the applications of extremal projectors are
presented.Comment: 21 pages, LaTeX; typos corrected, references adde
Triangulations and Severi varieties
We consider the problem of constructing triangulations of projective planes
over Hurwitz algebras with minimal numbers of vertices. We observe that the
numbers of faces of each dimension must be equal to the dimensions of certain
representations of the automorphism groups of the corresponding Severi
varieties. We construct a complex involving these representations, which should
be considered as a geometric version of the (putative) triangulations
- …