221 research outputs found
Multiple Qubits as Symplectic Polar Spaces of Order Two
It is surmised that the algebra of the Pauli operators on the Hilbert space
of N-qubits is embodied in the geometry of the symplectic polar space of rank N
and order two, W_{2N - 1}(2). The operators (discarding the identity) answer to
the points of W_{2N - 1}(2), their partitionings into maximally commuting
subsets correspond to spreads of the space, a maximally commuting subset has
its representative in a maximal totally isotropic subspace of W_{2N - 1}(2)
and, finally, "commuting" translates into "collinear" (or "perpendicular").Comment: 2 pages, no figur
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
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
Classicality in discrete Wigner functions
Gibbons et al. [Phys. Rev. A 70, 062101(2004)] have recently defined a class
of discrete Wigner functions W to represent quantum states in a Hilbert space
with finite dimension. We show that the only pure states having non-negative W
for all such functions are stabilizer states, as conjectured by one of us
[Phys. Rev. A 71, 042302 (2005)]. We also show that the unitaries preserving
non-negativity of W for all definitions of W form a subgroup of the Clifford
group. This means pure states with non-negative W and their associated unitary
dynamics are classical in the sense of admitting an efficient classical
simulation scheme using the stabilizer formalism.Comment: 10 pages, 1 figur
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