130 research outputs found
Mutually Unbiased Bases and Semi-definite Programming
A complex Hilbert space of dimension six supports at least three but not more
than seven mutually unbiased bases. Two computer-aided analytical methods to
tighten these bounds are reviewed, based on a discretization of parameter space
and on Grobner bases. A third algorithmic approach is presented: the
non-existence of more than three mutually unbiased bases in composite
dimensions can be decided by a global optimization method known as semidefinite
programming. The method is used to confirm that the spectral matrix cannot be
part of a complete set of seven mutually unbiased bases in dimension six.Comment: 11 pages
On properties of Karlsson Hadamards and sets of Mutually Unbiased Bases in dimension six
The complete classification of all 6x6 complex Hadamard matrices is an open
problem. The 3-parameter Karlsson family encapsulates all Hadamards that have
been parametrised explicitly. We prove that such matrices satisfy a non-trivial
constraint conjectured to hold for (almost) all 6x6 Hadamard matrices. Our
result imposes additional conditions in the linear programming approach to the
mutually unbiased bases problem recently proposed by Matolcsi et al.
Unfortunately running the linear programs we were unable to conclude that a
complete set of mutually unbiased bases cannot be constructed from Karlsson
Hadamards alone.Comment: As published versio
Genuinely multi-point temporal quantum correlations and universal measurement-based quantum computing
We introduce a constructive procedure that maps all spatial correlations of a
broad class of states into temporal correlations between general quantum
measurements. This allows us to present temporal phenomena analogous to
genuinely multipartite nonlocal phenomena, such as Greenberger-Horne-Zeilinger
correlations, which do not exist if only projective measurements on qubits are
considered. The map is applied to certain lattice systems in order to replace
one spatial dimension with a temporal one, without affecting measured
correlations. We use this map to show how repeated application of a
1d-cluster-gate leads to universal one-way quantum computing when supplemented
with the general measurements.Comment: New presentation of relations between temporal quantum correlations
and measurement based quantum computin
Non-classicality of temporal correlations
The results of space-like separated measurements are independent of distant
measurement settings, a property one might call two-way no-signalling. In
contrast, time-like separated measurements are only one-way no-signalling since
the past is independent of the future but not vice-versa. For this reason
temporal correlations that are formally identical to non-classical spatial
correlations can still be modelled classically. We define non-classical
temporal correlations as the ones which cannot be simulated by propagating in
time a classical information content of a quantum system. We first show that
temporal correlations between results of any projective quantum measurements on
a qubit can be simulated classically. Then we present a sequence of POVM
measurements on a single -level quantum system that cannot be explained by
propagating in time -level classical system and using classical computers
with unlimited memory.Comment: 6 pages, 1 figur
- …