61 research outputs found
Informationally complete sets of Gaussian measurements
We prove necessary and sufficient conditions for the informational
completeness of an arbitrary set of Gaussian observables on continuous variable
systems with finite number of degrees of freedom. In particular, we show that
an informationally complete set either contains a single informationally
complete observable, or includes infinitely many observables. We show that for
a single informationally complete observable, the minimal outcome space is the
phase space, and the observable can always be obtained from the quantum optical
-function by linear postprocessing and Gaussian convolution, in a suitable
symplectic coordinatization of the phase space. In the case of projection
valued Gaussian observables, e.g., generalized field quadratures, we show that
an informationally complete set of observables is necessarily infinite.
Finally, we generalize the treatment to the case where the measurement coupling
is given by a general linear bosonic channel, and characterize informational
completeness for an arbitrary set of the associated observables.Comment: 10 pages, 2 figure
On measurements of the canonical phase observable
Measurements of single-mode phase observables are studied in the spirit of
the quantum theory of measurement. We determine the minimal measurement models
of phase observables and consider methods of measuring such observables by
using a double homodyne detector. We show that, in principle, the canonical
phase distribution of the signal state can be measured via double homodyne
detection by first processing the state using a two-mode unitary channel.Comment: 22 pages, 2 figure
Tilted phase space measurements
We show that the phase shift of {\pi}/2 is crucial for the phase space
translation covariance of the measured high-amplitude limit observable in
eight-port homodyne detection. However, for an arbitrary phase shift {\theta}
we construct explicitly a different nonequivalent projective representation of
R such that the observable is covariant with respect to this
representation. As a result we are able to determine the measured observable
for an arbitrary parameter field and phase shift. Geometrically the change in
the phase shift corresponds to the tilting of one axis in the phase space of
the system.Comment: 4 pages, 4 figure
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
Quantum Tomography with Phase Space Measurements
This thesis addresses the use of covariant phase space observables in quantum
tomography. Necessary and sufficient conditions for the informational completeness
of covariant phase space observables are proved, and some state
reconstruction formulae are derived. Different measurement schemes for measuring
phase space observables are considered. Special emphasis is given to the
quantum optical eight-port homodyne detection scheme and, in particular, on
the effect of non-unit detector efficiencies on the measured observable. It is
shown that the informational completeness of the observable does not depend
on the efficiencies.
As a related problem, the possibility of reconstructing the position and
momentum distributions from the marginal statistics of a phase space observable
is considered. It is shown that informational completeness for the phase space
observable is neither necessary nor sufficient for this procedure. Two methods
for determining the distributions from the marginal statistics are presented.
Finally, two alternative methods for determining the state are considered.
Some of their shortcomings when compared to the phase space method are
discussed.Siirretty Doriast
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
Maximally incompatible quantum observables
The existence of maximally incompatible quantum observables in the sense of a
minimal joint measurability region is investigated. Employing the universal
quantum cloning device it is argued that only infinite dimensional quantum
systems can accommodate maximal incompatibility. It is then shown that two of
the most common pairs of complementary observables (position and momentum;
number and phase) are maximally incompatible
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
- …