878 research outputs found
Entropic uncertainty relations and locking: tight bounds for mutually unbiased bases
We prove tight entropic uncertainty relations for a large number of mutually
unbiased measurements. In particular, we show that a bound derived from the
result by Maassen and Uffink for 2 such measurements can in fact be tight for
up to sqrt{d} measurements in mutually unbiased bases. We then show that using
more mutually unbiased bases does not always lead to a better locking effect.
We prove that the optimal bound for the accessible information using up to
sqrt{d} specific mutually unbiased bases is log d/2, which is the same as can
be achieved by using only two bases. Our result indicates that merely using
mutually unbiased bases is not sufficient to achieve a strong locking effect,
and we need to look for additional properties.Comment: 9 pages, RevTeX, v3: complete rewrite, new title, many new results,
v4: minor changes, published versio
An operational link between MUBs and SICs
We exhibit an operational connection between mutually unbiased bases and
symmetric infomationally complete positive operator-valued measures. Assuming
that the latter exists, we show that there is a strong link between these two
structures in all prime power dimensions. We also demonstrate that a similar
link cannot exists in dimension 6.Comment: 17 pages, 2 figure
Structure of the sets of mutually unbiased bases with cyclic symmetry
Mutually unbiased bases that can be cyclically generated by a single unitary
operator are of special interest, since they can be readily implemented in
practice. We show that, for a system of qubits, finding such a generator can be
cast as the problem of finding a symmetric matrix over the field
equipped with an irreducible characteristic polynomial of a given Fibonacci
index. The entanglement structure of the resulting complete sets is determined
by two additive matrices of the same size.Comment: 20 page
Solution to the Mean King's problem with mutually unbiased bases for arbitrary levels
The Mean King's problem with mutually unbiased bases is reconsidered for
arbitrary d-level systems. Hayashi, Horibe and Hashimoto [Phys. Rev. A 71,
052331 (2005)] related the problem to the existence of a maximal set of d-1
mutually orthogonal Latin squares, in their restricted setting that allows only
measurements of projection-valued measures. However, we then cannot find a
solution to the problem when e.g., d=6 or d=10. In contrast to their result, we
show that the King's problem always has a solution for arbitrary levels if we
also allow positive operator-valued measures. In constructing the solution, we
use orthogonal arrays in combinatorial design theory.Comment: REVTeX4, 4 page
Orthogonality for Quantum Latin Isometry Squares
Goyeneche et al recently proposed a notion of orthogonality for quantum Latin
squares, and showed that orthogonal quantum Latin squares yield quantum codes.
We give a simplified characterization of orthogonality for quantum Latin
squares, which we show is equivalent to the existing notion. We use this
simplified characterization to give an upper bound for the number of mutually
orthogonal quantum Latin squares of a given size, and to give the first
examples of orthogonal quantum Latin squares that do not arise from ordinary
Latin squares. We then discuss quantum Latin isometry squares, generalizations
of quantum Latin squares recently introduced by Benoist and Nechita, and define
a new orthogonality property for these objects, showing that it also allows the
construction of quantum codes. We give a new characterization of unitary error
bases using these structures.Comment: In Proceedings QPL 2018, arXiv:1901.0947
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