149 research outputs found
Negativity Bounds for Weyl-Heisenberg Quasiprobability Representations
The appearance of negative terms in quasiprobability representations of
quantum theory is known to be inevitable, and, due to its equivalence with the
onset of contextuality, of central interest in quantum computation and
information. Until recently, however, nothing has been known about how much
negativity is necessary in a quasiprobability representation. Zhu proved that
the upper and lower bounds with respect to one type of negativity measure are
saturated by quasiprobability representations which are in one-to-one
correspondence with the elusive symmetric informationally complete quantum
measurements (SICs). We define a family of negativity measures which includes
Zhu's as a special case and consider another member of the family which we call
"sum negativity." We prove a sufficient condition for local maxima in sum
negativity and find exact global maxima in dimensions and . Notably, we
find that Zhu's result on the SICs does not generally extend to sum negativity,
although the analogous result does hold in dimension . Finally, the Hoggar
lines in dimension make an appearance in a conjecture on sum negativity.Comment: 21 pages. v2: journal version, added reference
A Short Note on the Frame Set of Odd Functions
In this work we derive a simple argument which shows that Gabor systems
consisting of odd functions of variables and symplectic lattices of density
cannot constitute a Gabor frame. In the 1--dimensional, separable case,
this is a special case of a result proved by Lyubarskii and Nes, however, we
use a different approach in this work exploiting the algebraic relation between
the ambiguity function and the Wigner distribution as well as their relation
given by the (symplectic) Fourier transform. Also, we do not need the
assumption that the lattice is separable and, hence, new restrictions are added
to the full frame set of odd functions.Comment: accepted: Bulletin of the Australian Mathematical Society; 12 pages;
Version 3 makes use of symmetric time-frequency shifts. In this case the
appearing phase factors are easier to handle. Also, the main result is
extended to higher dimensions. [In Version 2 a mistake in the assumptions was
corrected. The windows should be chosen from Feichtinger's algebra rather
than from the Hilbert space L2.
Interpolation in Wavelet Spaces and the HRT-Conjecture
We investigate the wavelet spaces arising from square integrable representations of a locally compact group . We show that
the wavelet spaces are rigid in the sense that non-trivial intersection between
them imposes strong conditions. Moreover, we use this to derive consequences
for wavelet transforms related to convexity and functions of positive type.
Motivated by the reproducing kernel Hilbert space structure of wavelet spaces
we examine an interpolation problem. In the setting of time-frequency analysis,
this problem turns out to be equivalent to the HRT-Conjecture. Finally, we
consider the problem of whether all the wavelet spaces
of a locally compact group
collectively exhaust the ambient space . We show that the answer is
affirmative for compact groups, while negative for the reduced Heisenberg
group.Comment: Added a relevant citation and made minor modifications to the
expositio
SIC~POVMs and Clifford groups in prime dimensions
We show that in prime dimensions not equal to three, each group covariant
symmetric informationally complete positive operator valued measure (SIC~POVM)
is covariant with respect to a unique Heisenberg--Weyl (HW) group. Moreover,
the symmetry group of the SIC~POVM is a subgroup of the Clifford group. Hence,
two SIC~POVMs covariant with respect to the HW group are unitarily or
antiunitarily equivalent if and only if they are on the same orbit of the
extended Clifford group. In dimension three, each group covariant SIC~POVM may
be covariant with respect to three or nine HW groups, and the symmetry group of
the SIC~POVM is a subgroup of at least one of the Clifford groups of these HW
groups respectively. There may exist two or three orbits of equivalent
SIC~POVMs for each group covariant SIC~POVM, depending on the order of its
symmetry group. We then establish a complete equivalence relation among group
covariant SIC~POVMs in dimension three, and classify inequivalent ones
according to the geometric phases associated with fiducial vectors. Finally, we
uncover additional SIC~POVMs by regrouping of the fiducial vectors from
different SIC~POVMs which may or may not be on the same orbit of the extended
Clifford group.Comment: 30 pages, 1 figure, section 4 revised and extended, published in J.
Phys. A: Math. Theor. 43, 305305 (2010
Sampling of operators
Sampling and reconstruction of functions is a central tool in science. A key
result is given by the sampling theorem for bandlimited functions attributed to
Whittaker, Shannon, Nyquist, and Kotelnikov. We develop an analogous sampling
theory for operators which we call bandlimited if their Kohn-Nirenberg symbols
are bandlimited. We prove sampling theorems for such operators and show that
they are extensions of the classical sampling theorem
- …