Discrete phase-space structure of n-qubit mutually unbiased bases

Abstract

We work out the phase-space structure for a system of n qubits. We replace the field of real numbers that label the axes of the continuous phase space by the finite field GF(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 n qubits.

Tomography from collective measurements

“…The advantage of the present approach with respect to previously discussed (and experimentally verified) methods is given by the explicit expressions (19) for the reconstructed density matrix from experimental data. In addition, we have shown that restricting ourselves to fully symmetric states, the tomographic protocol is reduced to projections from an overcomplete set of pure states (32), which still allows to obtain an explicit reconstruction expression (35). Such a set of states has been worked out from the first principles of state reconstruction in an 2 N -dimensional Hilbert space.…”

Coarse graining the phase space ofNqubits

“…Note that we still have to fix the sign of the phase (α,β). We choose the phase as On the phase-space grid one can introduce a variety of geometrical structures with many of the same properties as in the continuous case [27][28][29]. The simplest are the straight lines passing through the origin (also called rays), with equations…”

Discrete phase-space structures and Wigner functions for N qubits

“…Otherwise, the curve are called degenerate [33]. In that case, both α and β do not take some values in F 2 N and they are multivalued for some other values.…”