We study the properties of regular structures of lines, such as equiangular sets of lines and mutually unbiased bases (MUBs) in a general setting that includes real, complex and quaternionic spaces. We formulate a common generalization of several results in real and complex spaces that also hold in the quaternionic space.
A set of lines is called equiangular if the angle between each pair is the same. A set of MUBs is a collection of orthonormal bases such that the angle between vectors from different bases is constant. Regular structures of lines have been studied in several fields such as digital communication, quantum computing, discrete mathematics and analysis.
Our new concept of a multipartite equiangular set of lines is a common generalization of equiangular lines and MUBs. We prove a bound on the size of such set of lines, which generalizes the well-known absolute upper bounds.
The existence of d + 1 MUBs in Cd is only known for prime power dimensions. We study sets of d + 1 MUBs that are the union of a standard basis and an orbit of the Weyl-Heisenberg group. As an example, we construct such MUBs in prime power dimensions. We also show connections between spherical 2-designs and other structures of lines.
Fiducial vectors have been widely used to construct large sets of equiangular lines. A complex vector is fiducial if its orbit under a Weyl-Heisenberg group is an equiangular set of d2 lines. We give a new characterization of fiducial vectors, one that simplifies and significantly reduces the number of equations that must be solved to find a fiducial vector. We consider some possible classes of fiducial vectors and prove several nonexistence results. For example, using our new characterization we prove that the construction of fiducial vectors in small prime dimensions by Appleby (2005) essentially does not generalize.
Finally, we give some methods for constructing equiangular sets of lines in complex and quaternionic spaces. We also find numerical fiducial vectors with high precision in Cd , d ≤ 21.
We study the properties of regular structures of lines, such as equiangular sets of lines and mutually unbiased bases (MUBs) in a general setting that includes real, complex and quaternionic spaces. We formulate a common generalization of several results in real and complex spaces that also hold in the quaternionic space.
A set of lines is called equiangular if the angle between each pair is the same. A set of MUBs is a collection of orthonormal bases such that the angle between vectors from different bases is constant. Regular structures of lines have been studied in several fields such as digital communication, quantum computing, discrete mathematics and analysis.
Our new concept of a multipartite equiangular set of lines is a common generalization of equiangular lines and MUBs. We prove a bound on the size of such set of lines, which generalizes the well-known absolute upper bounds.
The existence of d + 1 MUBs in Cd is only known for prime power dimensions. We study sets of d + 1 MUBs that are the union of a standard basis and an orbit of the Weyl-Heisenberg group. As an example, we construct such MUBs in prime power dimensions. We also show connections between spherical 2-designs and other structures of lines.
Fiducial vectors have been widely used to construct large sets of equiangular lines. A complex vector is fiducial if its orbit under a Weyl-Heisenberg group is an equiangular set of d2 lines. We give a new characterization of fiducial vectors, one that simplifies and significantly reduces the number of equations that must be solved to find a fiducial vector. We consider some possible classes of fiducial vectors and prove several nonexistence results. For example, using our new characterization we prove that the construction of fiducial vectors in small prime dimensions by Appleby (2005) essentially does not generalize.
Finally, we give some methods for constructing equiangular sets of lines in complex and quaternionic spaces. We also find numerical fiducial vectors with high precision in Cd , d ≤ 21.
The principle of complementarity lies at the heart of quantum mechanics. In finite dimensional quantum systems this principle is captured by pairs of observables which are given by mutually unbiased bases (MUBs). Two orthonormal bases B and C of Cd are mutually unbiased if |<b|c>|2 = 1/d holds for all vectors b ∈ B and c ∈ C. This implies that whenever we are given a vector from one of these bases and perform a measurement with respect to any other of the bases, then there is no information gained from this measurement. A basic question about MUBs is how many of them can be found in a given dimension d. While constructions of maximal sets of d+1 such bases are known for system of prime power dimension d, it is unknown whether this bound can be achieved for any non-prime power dimension. We review the known constructions of MUBs and demonstrate that maximal sets of MUBs come with a rich combinatorial structure by showing that they actually are the same objects as the complex projective 2-designs with angle set {0,1/d}. Furthermore, we address the problem of constructing positive operator-valued measures (POVMs) in finite dimension d consisting of d2 operators of rank one which have an inner product equal to uniform or very close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. We also give a simple proof of the fact that symmetric informationally complete POVMs are complex projective 2-designs with angle set {1/(d+1)}. Moreover, we show that MUBs and SIC-POVMs form uniform tight frames in Cd.
Two orthonormal bases in the d-dimensional Hilbert space are said to be unbiased if the square modulus of the inner product of any vector of one basis with any vector of the other equals 1 d. The presence of a modulus in the problem of finding a set of mutually unbiased bases constitutes a source of complications from the numerical point of view. Therefore, we may ask the question: Is it possible to get rid of the modulus? After a short review of various constructions of mutually unbiased bases in Cd, we show how to transform the problem of finding d + 1 mutually unbiased bases in the d-dimensional space Cd (with a modulus for the inner product) into the one of finding d(d+1) vectors in the d2-dimensional space Cd2 (without a modulus for the inner product). The transformation from Cd to Cd2 corresponds to the passage from equiangular lines to equiangular vectors. The transformation formulas are discussed in the case where d is a prime number.
The notion of Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) arose in physics as a kind of optimal measurement basis for quantum systems. However the question of their existence is equivalent to that of the existence of a maximal set of \emph{complex equiangular lines}. That is to say, given a complex Hilbert space of dimension d, what is the maximal number of (complex) lines one can find which all make a common (real) angle with one another, in the sense that the inner products between unit vectors spanning those lines all have a common absolute value? A maximal set would consist of d2 lines all with a common angle of arccosd+11. The same question has been posed in the real case and some partial answers are known. But at the time of writing no unifying theoretical result has been found in the real or the complex case: some sporadic low-dimensional numerical constructions have been converted into algebraic solutions but beyond this very little is known. It is conjectured that such structures always arise as orbits of certain fiducial vectors under the action of the Weyl (or generalised Pauli) group. In this paper we point out some new construction methods in the lowest dimensions (d=2 and d=3). We should mention that the SIC-POVMs so constructed are all unitarily equivalent to previously known SIC-POVMs.
A set of vectors of equal norm in Cd represents equiangular lines if the magnitudes of the Hermitian inner product of every pair of distinct vectors in the set are equal. The maximum size of such a set is d2, and it is conjectured that sets of this maximum size exist in Cd for every d≥2. We take a combinatorial approach to this conjecture, using mutually unbiased bases (MUBs) in the following 3 constructions of equiangular lines: (1) adapting a set of d MUBs in Cd to obtain d2 equiangular lines in Cd, (2) using a set of d MUBs in Cd to build (2d)2 equiangular lines in C2d, (3) combining two copies of a set of d MUBs in Cd to build (2d)2 equiangular lines in C2d. For each construction, we give the dimensions d for which we currently know that the construction produces a maximum-sized set of equiangular lines.
We present a detailed computational and algebraic study of Mutually Unbiased Bases (MUBs) in finite-dimensional Hilbert spaces, with a particular focus on dimensions 2, 3, 4, and the challenging case of 6. Starting from the Hadamard-phase parametrization, we derive explicit analytical conditions for mutual unbiasedness in dimension 4, providing a tractable system of trigonometric constraints on the phase parameters. We then explore a tensor-product construction via Pauli operators, highlighting the algebraic and group-theoretical origin of MUBs in two-qubit systems, and demonstrating how these constructions yield a complete set of 5 MUBs in dimension 4. Extending our approach, we investigate the Fourier-family method in dimension 6, where the absence of a prime-power structure imposes strong rigidity constraints and limits the known constructions to sets of 3 MUBs. We provide a systematic computational framework for testing candidate phase vectors, bridging the gap between analytical insight and numerical exploration. Finally, we generalize the discussion to arbitrary prime-power dimensions, emphasizing the role of finite-field structures, Heisenberg-Weyl operators, and discrete symmetries in generating complete sets of MUBs. Our work offers a transparent, line-by-line verification methodology, highlighting both the geometric and algebraic richness of MUBs, and clarifying why certain dimensions resist full analytical constructions. This study serves as a comprehensive resource for researchers seeking both theoretical understanding and practical construction of MUBs in quantum information science.
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.