Mutually unbiased bases (MUBs) are interesting for various reasons. The most attractive example of (a complete set of) MUBs is the one constructed by Ivanovi\ifmmode \acute{c}\else \'{c}\fi{} as well as Wootters and Fields, which is referred to as the canonical MUB. Nevertheless, little is known about anything that is unique to this MUB. We show that the canonical MUB in any prime power dimension is uniquely determined by an extremal orbit of the (restricted) Clifford group except in dimension 3, in which case the orbit defines a special symmetric informationally complete measurement (SIC), known as the Hesse SIC. Here the extremal orbit is the orbit with the smallest number of pure states. Quite surprisingly, this characterization does not rely on any concept that is related to bases or unbiasedness. As a corollary, the canonical MUB is the unique minimal 2-design covariant with respect to the Clifford group except in dimension 3. In addition, these MUBs provide an infinite family of highly symmetric frames and positive-operator-valued measures (POVMs), which are of independent interest.
Mutually unbiased bases (MUBs) are interesting for various reasons. The most attractive example of (a complete set of) MUBs is the one constructed by Ivanovi\ifmmode \acute{c}\else '{c}\fi{} as well as Wootters and Fields, which is referred to as the canonical MUB. Nevertheless, little is known about anything that is unique to this MUB. We show that the canonical MUB in any prime power dimension is uniquely determined by an extremal orbit of the (restricted) Clifford group except in dimension 3, in which case the orbit defines a special symmetric informationally complete measurement (SIC), known as the Hesse SIC. Here the extremal orbit is the orbit with the smallest number of pure states. Quite surprisingly, this characterization does not rely on any concept that is related to bases or unbiasedness. As a corollary, the canonical MUB is the unique minimal 2-design covariant with respect to the Clifford group except in dimension 3. In addition, these MUBs provide an infinite family of highly symmetric frames and positive-operator-valued measures (POVMs), which are of independent interest.
Mutually unbiased bases (MUB) are an elusive discrete structure in Hilbert spaces. Many (complete sets of) MUB are group covariant, but little is known whether they can be sharply covariant in the sense that the generating groups can have order equal to the total number of basis states, that is, d(d+1) for MUB in dimension d. Sharply covariant MUB, if they exist, would be most appealing from both theoretical and practical point of view. Since stabilizer MUB subsume almost all MUB that have ever been constructed, it is of fundamental interest to single out those candidates that are sharply covariant. We show that, quite surprisingly, only two stabilizer MUB are sharply covariant, and the conclusion still holds even if antiunitary transformations are taken into account. Our study provides valuable insight on the symmetry of stabilizer MUB, which may have implications for a number of research topics in quantum information and foundations. In addition, it exposes a sharp contrast between MUB and another elusive discrete structure known as symmetric informationally complete measurements (SICs), all known examples of which are sharply covariant.
Mutually unbiased bases (MUB) and symmetric informationally complete positive operator-valued measure (SIC-POVM) are both important objects in quantum information theory.While people do not know if there exists a complete MUB for non-prime-power dimension,several versions of approximately MUB have been considered by relaxed the inner product condition.So far there are only finite number of K such that SICPOVMs in C k have been found.As in the MUB case,several versions of approximately SIC-POVM have been considered by relaxed the inner product condition.In this paper,we use the definitions of approximate MUB and SIC-POVM given by Klappenecker et al.For prime power q,we present simple constructions of q approximately MUB (AMUB) for dimension q-1,q+1 AMUB for dimension q-1,which shows the number of orthonormal bases of an AMUB in C k can be more than K+1,and q AMUB for dimension q+1 by Gauss and Jacobi sums.We also present a construction of approximately SIC-POVM (ASIC-POVM) in dimension q-1 by Gauss sum.
Discrete structures in product Hilbert spaces are investigated. For monopartite systems of size d one relies on the Weyl--Heisenberg group WH(d), while in the case of composite Hilbert spaces with the local dimensions di we identify designs covariant with respect to the product group, ⨂iWH(di). In analogy with magic -- a quantity attaining its maximum for states fiducial with respect to WH(d) -- we introduce a similar quantifier of {\sl product magic}, defined with respect to the product group. The maximum of this quantity over all equimodular vectors yields fiducial states that generate d=∏idi \textit{a priori} isoentangled mutually unbiased bases (MUBs), which, when supplemented by the identity, form their complete set. Such fiducial states are explicitly constructed in all prime-power dimensions d=pn with p≥3. The result for p≥5 extends the construction of Klappenecker and R{ö}tteler, whereas for p=3 it is mathematically distinct and is based on Galois rings. The global maximum of the quantifier of product magic for d=23 yields fiducial states corresponding to the symmetric informationally complete (SIC) generalized measurement of Hoggar. Our approach feeds into a unifying perspective in which highly symmetric quantum designs emerge from fiducial states with extremal properties via structured group-orbit constructions.
In this work, the concept of mutually unbiased frames is introduced as the most general notion of unbiasedness for sets composed by linearly independent and normalized vectors. It encompasses the already existing notions of unbiasedness for orthonormal bases, regular simplices, equiangular tight frames, positive operator valued measure, and also includes symmetric informationally complete quantum measurements. After introducing the tool, its power is shown by finding the following results about the last mentioned class of constellations: (i) real fiducial states do not exist in any even dimension, and (ii) unknown d -dimensional fiducial states are parameterized, a priori, with roughly 3d/2 real variables only, without loss of generality. Furthermore, multi-parametric families of pure quantum states having minimum uncertainty with regard to several choices of d+1 orthonormal bases are shown, in every dimension d . These last families contain all existing fiducial states in every finite dimension, and the bases include maximal sets of d+1 mutually unbiased bases, when d is a prime number.
Mutually Unbiased bases has various application in quantum information procession and coding theory. There can be maximum d + 1 MUBs in C^d and d/2 +1 MUBs in R^d. But , over R^d MUBs are known to be non existent when d is odd and for most of the other even d there are mostly 3 Real MUBs. In case of C^d the construction for complete set of MUBs are known for only Prime Power dimension. Thus in general large set of MUBs are not known, particularly for composite dimensions which are not of the form of prime powers. Because of this, there are many constructions of Approximate version of MUBs. In this paper we make an attempt to define certain measures to characterise the AMUBs. Our construction of measures derives its inspiration from the applications of MUBs, and based on them, we define certain quantifiable measures, which are can be computed and gives estimates of how close the Approximate MUBs are to the MUBs. We use geometric interpretation, projective design features of MUBs and applications like Optimal State determination and Entropic Uncertainty of MUBs. We show generic relationship between these measures and show that it can be evaluated for APMUBs without known the exact construction details, thereby showing that definition of APMUB is sufficient completely characterise it. We also evaluate these measure for an interesting class of AMUBs called Weak MUBs and certain AMUBs constructed using RBDs.
Mutually unbiased bases (MUB) and symmetric informationally complete positive operator-valued measure (SIC-POVM) are both important objects in quantum information theory. While people do not know if there exists a complete MUB for non-prime-power dimension, several versions of approximately MUB have been considered by relaxed the inner product condition. So far there are only finite number of K such that SIC- POVMs in Ck have been found. As in the MUB case, several versions of approximately SIC-POVM have been considered by relaxed the inner product condition. In this paper, we use the definitions of approximate MUB and SIC-POVM given by Klappenecker et al. For prime power q, we present simple constructions of q approximately MUB (AMUB) for dimension q-1, q+1 AMUB for dimension q-1, which shows the number of orthonormal bases of an AMUB in Ck can be more than K+1, and q AMUB for dimension q+1 by Gauss and Jacobi sums. We also present a construction of approximately SIC-POVM (ASIC-POVM) in dimension q-1 by Gauss sum.