Mutually Unbiased Bases and the Complementarity Polytope

Abstract

A complete set of N + 1 mutually unbiased bases (MUBs) forms a convex polytope in the N 2 − 1 dimensional space of N × N Hermitian matrices of unit trace. As a geometrical object such a polytope exists for all values of N , while it is unknown whether it can be made to lie within the body of density matrices unless N = p k , where p is prime. We investigate the polytope in order to see if some values of N are geometrically singled out. One such feature is found: It is possible to select N 2 facets in such a way that their centers form a regular simplex if and only if there exists an affine plane of order N . Affine planes of order N are known to exist if N = p k ; perhaps they do not exist otherwise. However, the link to the existence of MUBs-if any-remains to be found.

Symmetric informationally complete–positive operator valued measures and the extended Clifford group

“…Wootters [18], Bengtsson and Ericsson [19,20] and Grassl [21] have made further contributions. There appear to be some intimate connections with the theory of mutually unbiased bases [18,22,23], finite affine planes [18,19,20], and polytopes [19,20]. If SIC-POVMs existed in every finite dimension (or, failing that, in a sufficiently large set of finite dimensions) they would constitute a naturally distinguished class of POVMs which might be expected to have many interesting applications to quantum tomography, cryptography and information theory generally.…”

SIC-POVMS and MUBS: Geometrical Relationships in Prime Dimension

“…We also give a geometrical interpretation of the minimum uncertainty states introduced by Wootters and Sussman [48,49] and Appleby, Dang and Fuchs [30], and of the related fiduciality conditions given by Appleby, Dang and Fuchs [30]. Our analysis builds on previous work by Bengtsson and Ericcson [10,12] and Appleby [27].…”

“…There has been much interest in recent years in SIC-POVMs [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34] (symmetric informationally complete positive operator valued measures; citations in order of first appearance online or in print). SIC-POVMs have been constructed analytically in Hilbert space dimension d = 2-13, 15 and 19 (existence of analytic solutions for d = 11, 15 communicated to author privately [35]; for d = 15 also see ref.…”

“…MUBs [6,9,10,11,12,37,38,39,40,41,42,43,44,45,46,47] (mutually unbiased bases; only a representative selection of papers cited) have been the subject of intense investigation for a rather longer period of time. It is known that the number of MUBs in dimension d cannot exceed d+1, and that the maximum number of d+1 MUBs exist whenever d is a power of a prime number.…”

“…We will work in generalized Bloch space [10,12,27,50,51,52,53,54,55,56,57]. If d = 2 the Bloch representation of an arbitrary density matrix ρ is defined by…”