21,706 research outputs found
Lower-dimensional pure-spinor superstrings
We study to what extent it is possible to generalise Berkovits' pure-spinor
construction in d=10 to lower dimensions. Using a suitable definition of a
``pure'' spinor in d=4,6, we propose models analogous to the d=10 pure-spinor
superstring in these dimensions. Similar models in d=2,3 are also briefly
discussed.Comment: 17 page
The Kadison-Singer Problem in Mathematics and Engineering
We will show that the famous, intractible 1959 Kadison-Singer problem in
-algebras is equivalent to fundamental unsolved problems in a dozen
areas of research in pure mathematics, applied mathematics and Engineering.
This gives all these areas common ground on which to interact as well as
explaining why each of these areas has volumes of literature on their
respective problems without a satisfactory resolution. In each of these areas
we will reduce the problem to the minimum which needs to be proved to solve
their version of Kadison-Singer. In some areas we will prove what we believe
will be the strongest results ever available in the case that Kadison-Singer
fails. Finally, we will give some directions for constructing a counter-example
to Kadison-Singer
Conditional Density Operators and the Subjectivity of Quantum Operations
Assuming that quantum states, including pure states, represent subjective
degrees of belief rather than objective properties of systems, the question of
what other elements of the quantum formalism must also be taken as subjective
is addressed. In particular, we ask this of the dynamical aspects of the
formalism, such as Hamiltonians and unitary operators. Whilst some operations,
such as the update maps corresponding to a complete projective measurement,
must be subjective, the situation is not so clear in other cases. Here, it is
argued that all trace preserving completely positive maps, including unitary
operators, should be regarded as subjective, in the same sense as a classical
conditional probability distribution. The argument is based on a reworking of
the Choi-Jamiolkowski isomorphism in terms of "conditional" density operators
and trace preserving completely positive maps, which mimics the relationship
between conditional probabilities and stochastic maps in classical probability.Comment: 10 Pages, Work presented at "Foundations of Probability and
Physics-4", Vaxjo University, June 4-9 200
Rank one perturbations and singular integral operators
We consider rank one perturbations
of a self-adjoint operator with cyclic vector on a Hilbert space . The spectral representation of the
perturbed operator is given by a singular integral operator of
special form. Such operators exhibit what we call 'rigidity' and are connected
with two weight estimates for the Hilbert transform.
Also, some results about two weight estimates of Cauchy (Hilbert) transforms
are proved. In particular, it is proved that the regularized Cauchy transforms
are uniformly (in ) bounded operators from
to , where and are the spectral
measures of and , respectively.
As an application, a sufficient condition for to have a pure
absolutely continuous spectrum on a closed interval is given in terms of the
density of the spectral measure of with respect to . Some
examples, like Jacobi matrices and Schr\"odinger operators with
potentials are considered.Comment: 24 page
Uniquely determined pure quantum states need not be unique ground states of quasi-local Hamiltonians
We consider the problem of characterizing states of a multipartite quantum
system from restricted, quasi-local information, with emphasis on uniquely
determined pure states. By leveraging tools from dissipative quantum control
theory, we show how the search for states consistent with an assigned list of
reduced density matrices may be restricted to a proper subspace, which is
determined solely by their supports. The existence of a quasi-local observable
which attains its unique minimum over such a subspace further provides a
sufficient criterion for a pure state to be uniquely determined by its reduced
states. While the condition that a pure state is uniquely determined is
necessary for it to arise as a non-degenerate ground state of a quasi-local
Hamiltonian, we prove the opposite implication to be false in general, by
exhibiting an explicit analytic counterexample. We show how the problem of
determining whether a quasi-local parent Hamiltonian admitting a given pure
state as its unique ground state is dual, in the sense of semidefinite
programming, to the one of determining whether such a state is uniquely
determined by the quasi-local information. Failure of this dual program to
attain its optimal value is what prevents these two classes of states to
coincide.Comment: 17 pages, 1 figur
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