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arXiv:1305.6044

A geometrical relation between symmetric operators and mutually unbiased operators

Amir Kalev

Published May 26, 2013• 3 citations
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Abstract

In this work we study the relation between the set of symmetric operators and the set of mutually unbiased operators from finite plane geometry point of view. Here symmetric operators are generalization of symmetric informationally complete probability-operator measurements (SIC POMs), while mutually unbiased operators are the operator generalization of mutually unbiased bases (MUB). We also discuss the implication of this relation to the particular cases of rank-1 SIC POMs and MUB.

DOI
10.48550/arxiv.1305.6044

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In this work we study the relation between the set of symmetric operators and the set of mutually unbiased operators from finite plane geometry point of view. Here symmetric operators are generalization of symmetric informationally complete probability-operator measurements (SIC POMs), while mutually unbiased operators are the operator generalization of mutually unbiased bases (MUB). We also discuss the implication of this relation to the particular cases of rank-1 SIC POMs and MUB.
We generalize the concept of unbiasedness from bases to measurements. We show that mutually unbiased (MU) measurements, which are not necessarily von Neumann measurements, exist in all finite-dimensional Hilbert spaces. We study the geometrical relation between these measurements and symmetric informationally complete measurements. In particular we find that these two kinds of measurements are related to each other as points and lines in finite affine plane geometries. This relation provides a natural way to define discrete phase-space functions based on MU measurements and symmetric informationally complete measurements.
The paper concerns Weyl-Heisenberg covariant SIC-POVMs (symmetric informationally complete positive operator valued measures) and full sets of MUBs (mutually unbiased bases) in prime dimension. When represented as vectors in generalized Bloch space a SIC-POVM forms a d^2-1 dimensional regular simplex (d being the Hilbert space dimension). By contrast, the generalized Bloch vectors representing a full set of MUBs form d+1 mutually orthogonal d-1 dimensional regular simplices. In this paper we show that, in the Weyl-Heisenberg case, there are some simple geometrical relationships between the single SIC-POVM simplex and the d+1 MUB simplices. We go on to give geometrical interpretations of the minimum uncertainty states introduced by Wootters and Sussman, and by Appleby, Dang and Fuchs, and of the fiduciality condition given by Appleby, Dang and Fuchs.
Quantum measurements play a fundamental role in quantum information. Therefore, increasing efforts are being made to construct symmetric measurement operators for qudit systems. A wide class of projective measurements corresponds to complex projective 2-designs, which include symmetric, informationally complete (SIC) POVMs and mutually unbiased bases (MUBs). In this paper, we establish a one-to-one correspondence between conical 2-designs and mutually unbiased generalized equiangular tight frames, both of which are common generalizations of SIC POVMs and MUBs to operators of arbitrary rank. It turns out that there exist rich families of operators that belong to only one of those two classes. This raises important questions about which symmetries have to be preserved for applicational prominence.
The paper concerns Weyl-Heisenberg covariant SIC-POVMs (symmetric informationally complete positive operator valued measures) and full sets of MUBs (mutually unbiased bases) in prime dimension. When represented as vectors in generalized Bloch space a SIC-POVM forms a d 2 − 1 dimensional regular simplex (d being the Hilbert space dimension). By contrast, the generalized Bloch vectors representing a full set of MUBs form d + 1 mutually orthogonal d − 1 dimensional regular simplices. In this paper we show that, in the Weyl-Heisenberg case, there are some simple geometrical relationships between the single SIC-POVM simplex and the d + 1 MUB simplices. We go on to give geometrical interpretations of the minimum uncertainty states introduced by Wootters and Sussman, and by Appleby, Dang and Fuchs, and of the fiduciality condition given by Appleby, Dang and Fuchs. 1
We generalize the concept of mutually unbiased bases (MUB) to measurements which are not necessarily described by rank one projectors. As such, these measurements can be a useful tool to study the long-standing problem of the existence of MUB. We derive their general form, and show that in a finite, d -dimensional Hilbert space, one can construct a complete set of mutually unbiased measurements. Besides their intrinsic link to MUB, we show that these measurements' statistics provide complete information about the state of the system. Moreover, they capture the physical essence of unbiasedness, and in particular, they satisfy a non-trivial entropic uncertainty relation similar to MUB.