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 $m$-level quantum system that cannot be explained by propagating in time $m$-level classical system and using classical computers with unlimited memory.Comment: 6 pages, 1 figur