67 research outputs found
Super G-spaces
We review the basic theory of super -spaces. We prove a theorem relating
the action of a super Harish-Chandra pair on a
supermanifold to the action of the corresponding super Lie group . The
theorem was stated in [DM99] without proof. The proof given here does not use
Frobenius theorem but relies on Koszul realization of the structure sheaf of a
super Lie group (see [Kosz83]). We prove the representability of the stability
subgroup functor
Covariant mutually unbiased bases
The connection between maximal sets of mutually unbiased bases (MUBs) in a
prime-power dimensional Hilbert space and finite phase-space geometries is well
known. In this article we classify MUBs according to their degree of covariance
with respect to the natural symmetries of a finite phase-space, which are the
group of its affine symplectic transformations. We prove that there exist
maximal sets of MUBs that are covariant with respect to the full group only in
odd prime-power dimensional spaces, and in this case their equivalence class is
actually unique. Despite this limitation, we show that in even-prime power
dimension covariance can still be achieved by restricting to proper subgroups
of the symplectic group, that constitute the finite analogues of the oscillator
group. For these subgroups, we explicitly construct the unitary operators
yielding the covariance.Comment: 44 pages, some remarks and references added in v
Minimal covariant observables identifying all pure states
It has been recently shown that an observable that identifies all pure states
of a d-dimensional quantum system has minimally 4d-4 outcomes or slightly less
(the exact number depending on the dimension d). However, no simple
construction of this type of observable with minimal number of outcomes is
known. In this work we investigate the possibility to have a covariant
observable that identifies all pure states and has minimal number of outcomes
for this purpose. It is shown that the existence of these kind of observables
depends on the dimension of the Hilbert space. The fact that these kind of
observables fail to exist in some dimensions indicates that the dual pair of
observables -- pure states lacks the symmetry that the dual pair of observables
-- states has
Tasks and premises in quantum state determination
The purpose of quantum tomography is to determine an unknown quantum state
from measurement outcome statistics. There are two obvious ways to generalize
this setting. First, our task need not be the determination of any possible
input state but only some input states, for instance pure states. Second, we
may have some prior information, or premise, which guarantees that the input
state belongs to some subset of states, for instance the set of states with
rank less than half of the dimension of the Hilbert space. We investigate state
determination under these two supplemental features, concentrating on the cases
where the task and the premise are statements about the rank of the unknown
state. We characterize the structure of quantum observables (POVMs) that are
capable of fulfilling these type of determination tasks. After the general
treatment we focus on the class of covariant phase space observables, thus
providing physically relevant examples of observables both capable and
incapable of performing these tasks. In this context, the effect of noise is
discussed.Comment: minor changes in v
Probing quantum state space: does one have to learn everything to learn something?
Determining the state of a quantum system is a consuming procedure. For this
reason, whenever one is interested only in some particular property of a state,
it would be desirable to design a measurement setup that reveals this property
with as little effort as possible. Here we investigate whether, in order to
successfully complete a given task of this kind, one needs an informationally
complete measurement, or if something less demanding would suffice. The first
alternative means that in order to complete the task, one needs a measurement
which fully determines the state. We formulate the task as a membership problem
related to a partitioning of the quantum state space and, in doing so, connect
it to the geometry of the state space. For a general membership problem we
prove various sufficient criteria that force informational completeness, and we
explicitly treat several physically relevant examples. For the specific cases
that do not require informational completeness, we also determine bounds on the
minimal number of measurement outcomes needed to ensure success in the task.Comment: 23 pages, 4 figure
Commutative POVMs and Fuzzy Observables
In this paper we review some properties of fuzzy observables, mainly as
realized by commutative positive operator valued measures. In this context we
discuss two representation theorems for commutative positive operator valued
measures in terms of projection valued measures and describe, in some detail,
the general notion of fuzzification. We also make some related observations on
joint measurements.Comment: Contribution to the Pekka Lahti Festschrif
Extremal covariant positive operator valued measures: the case of a compact symmetry group
Given a unitary representation U of a compact group G and a transitive
G-space , we characterize the extremal elements of the convex set of
all U-covariant positive operator valued measures.Comment: minor corrections in version
Optimal covariant measurements: the case of a compact symmetry group and phase observables
We study various optimality criteria for quantum observables. Observables are
represented as covariant positive operator valued measures and we consider the
case when the symmetry group is compact. Phase observables are examined as an
example
Efficient Pure State Quantum Tomography from Five Orthonormal Bases
For any finite dimensional Hilbert space, we construct explicitly five
orthonormal bases such that the corresponding measurements allow for efficient
tomography of an arbitrary pure quantum state. This means that such
measurements can be used to distinguish an arbitrary pure state from any other
state, pure or mixed, and the pure state can be reconstructed from the outcome
distribution in a feasible way. The set of measurements we construct is
independent of the unknown state, and therefore our results provide a fixed
scheme for pure state tomography, as opposed to the adaptive (state dependent)
scheme proposed by Goyeneche et al. in [Phys. Rev. Lett. 115, 090401 (2015)].
We show that our scheme is robust with respect to noise in the sense that any
measurement scheme which approximates these measurements well enough is equally
suitable for pure state tomography. Finally, we present two convex programs
which can be used to reconstruct the unknown pure state from the measurement
outcome distributions.Comment: 5 pages, 2 figures, 1 page of supplemental materia
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