Unbiased nonorthogonal bases for tomographic reconstruction

Abstract

We have developed a general method for constructing a set of nonorthogonal bases with equal separations between all different basis states in prime dimensions. The results are that the corresponding biorthogonal counterparts are pairwise unbiased with the components of the original bases. Using these bases, we derive an explicit expression for the optimal tomography in nonorthogonal bases. A special two-dimensional case is analyzed separately.

Equidistant-state preparation

“…This is directly reflected in the quality of the quantum state reconstruction when such bases are used for quantum tomography purposes. We will show that efficiency of mutually unbiased bases (MUB) tomographic protocol [8] rapidly decays when nonorthogonal bases are used (as is intuitively expected) by applying the Fisher information approach [11,12].…”

“…The advantage of such bases consists in the possibility to use methods similar to the orthonormal case to obtain the explicit form of physically relevant algebraic structures. In the particular case of prime dimension such important tools as optimal unbiased tomography [8] and phase-space representation [9] were recently developed.…”

“…In this article we present a scheme for preparing all nonorthogonal bases required for the quantum state tomography [8] in a system with p energy levels, where p is a prime number. We will show that starting with a single entangled system + ancilla state and applying local transformations one can obtain different sets of equidistant bases after performing appropriate von Neumann measurements on the ancillary subsystem.…”

Error estimation in the direct state tomography

“…The bases (4) form an informationally complete set for |λ| < 1 [15], and the density matrix of the system can be reconstructed in terms of the probabilities (3) according to [11] …”

“…In this letter we show that conveniently reformulating the approach [6] as a Mutually Unbiased Bases (MUB)-like reconstruction scheme in non-orthogonal bases [11,12] one can carry out the mean square error (MSE) analysis of DST , including the weak measurement limit, in the framework of measurement statistics [8,9]. In particular, we exemplify on the single qubit case that non-orthogonal bases appear as effective projective states, in such a way that a weak coupling corresponds to projection into near-parallel bases.…”