9,159 research outputs found
Discrete phase-space structure of -qubit mutually unbiased bases
We work out the phase-space structure for a system of qubits. We replace
the field of real numbers that label the axes of the continuous phase space by
the finite field \Gal{2^n} and investigate the geometrical structures
compatible with the notion of unbiasedness. These consist of bundles of
discrete curves intersecting only at the origin and satisfying certain
additional properties. We provide a simple classification of such curves and
study in detail the four- and eight-dimensional cases, analyzing also the
effect of local transformations. In this way, we provide a comprehensive
phase-space approach to the construction of mutually unbiased bases for
qubits.Comment: Title changed. Improved version. Accepted for publication in Annals
of Physic
Geometrical approach to mutually unbiased bases
We propose a unifying phase-space approach to the construction of mutually
unbiased bases for a two-qubit system. It is based on an explicit
classification of the geometrical structures compatible with the notion of
unbiasedness. These consist of bundles of discrete curves intersecting only at
the origin and satisfying certain additional properties. We also consider the
feasible transformations between different kinds of curves and show that they
correspond to local rotations around the Bloch-sphere principal axes. We
suggest how to generalize the method to systems in dimensions that are powers
of a prime.Comment: 10 pages. Some typos in the journal version have been correcte
Mutually unbiased bases and discrete Wigner functions
Mutually unbiased bases and discrete Wigner functions are closely, but not
uniquely related. Such a connection becomes more interesting when the Hilbert
space has a dimension that is a power of a prime , which describes a
composite system of qudits. Hence, entanglement naturally enters the
picture. Although our results are general, we concentrate on the simplest
nontrivial example of dimension . It is shown that the number of
fundamentally different Wigner functions is severely limited if one
simultaneously imposes translational covariance and that the generating
operators consist of rotations around two orthogonal axes, acting on the
individual qubits only.Comment: 9 pages, 6 tables, 6 figures. Accepted for publication in J. Opt.
Soc. Am. B, special issue on Optical Quantum Information Scienc
Group theoretical construction of mutually unbiased bases in Hilbert spaces of prime dimensions
Mutually unbiased bases in Hilbert spaces of finite dimensions are closely
related to the quantal notion of complementarity. An alternative proof of
existence of a maximal collection of N+1 mutually unbiased bases in Hilbert
spaces of prime dimension N is given by exploiting the finite Heisenberg group
(also called the Pauli group) and the action of SL(2,Z_N) on finite phase space
Z_N x Z_N implemented by unitary operators in the Hilbert space. Crucial for
the proof is that, for prime N, Z_N is also a finite field.Comment: 13 pages; accepted in J. Phys. A: Math. Theo
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
On the structure of the sets of mutually unbiased bases for N qubits
For a system of N qubits, spanning a Hilbert space of dimension d=2^N, it is
known that there exists d+1 mutually unbiased bases. Different construction
algorithms exist, and it is remarkable that different methods lead to sets of
bases with different properties as far as separability is concerned. Here we
derive the four sets of nine bases for three qubits, and show how they are
unitarily related. We also briefly discuss the four-qubit case, give the
entanglement structure of sixteen sets of bases,and show some of them, and
their interrelations, as examples. The extension of the method to the general
case of N qubits is outlined.Comment: 16 pages, 10 tables, 1 figur
Quantum phase uncertainty in mutually unbiased measurements and Gauss sums
Mutually unbiased bases (MUBs), which are such that the inner product between
two vectors in different orthogonal bases is constant equal to the inverse
, with the dimension of the finite Hilbert space, are becoming
more and more studied for applications such as quantum tomography and
cryptography, and in relation to entangled states and to the Heisenberg-Weil
group of quantum optics. Complete sets of MUBs of cardinality have been
derived for prime power dimensions using the tools of abstract algebra
(Wootters in 1989, Klappenecker in 2003). Presumably, for non prime dimensions
the cardinality is much less. The bases can be reinterpreted as quantum phase
states, i.e. as eigenvectors of Hermitean phase operators generalizing those
introduced by Pegg & Barnett in 1989. The MUB states are related to additive
characters of Galois fields (in odd characteristic p) and of Galois rings (in
characteristic 2). Quantum Fourier transforms of the components in vectors of
the bases define a more general class of MUBs with multiplicative characters
and additive ones altogether. We investigate the complementary properties of
the above phase operator with respect to the number operator. We also study the
phase probability distribution and variance for physical states and find them
related to the Gauss sums, which are sums over all elements of the field (or of
the ring) of the product of multiplicative and additive characters. Finally we
relate the concepts of mutual unbiasedness and maximal entanglement. This
allows to use well studied algebraic concepts as efficient tools in our quest
of minimal uncertainty in quantum information primitives.Comment: 11 page
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
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