This is a one-to-one translation of a German-written Ph.D. thesis from 1999. Quantum designs are sets of orthogonal projection matrices in finite(b)-dimensional Hilbert spaces. A fundamental differentiation is whether all projections have the same rank r, and furthermore the special case r = 1, which contains two important subclasses: Mutually unbiased bases (MUBs) were introduced prior to this thesis and solutions of b + 1 MUBs whenever b is a power of a prime were already given. Unaware of those papers, this concept was generalized here under the notation of regular affine quantum designs. Maximal solutions are given for the general case r ≥ 1, consisting of r(b2 - 1)/(b - r) so-called complete orthogonal classes whenever b is a power of a prime. For b = 6, an infinite family of MUB triples was constructed and it was — as already done in the author's master's thesis (1991) — conjectured that four MUBs do not exist in this dimension. Symmetric informationally complete positive operator-valued measures (SIC POVMs) in this paper are called regular quantum 2-designs with degree 1. The assigned vectors span b2 equiangular lines. These objects had been investigated since the 1960s, but only a few solutions were known in complex vector spaces. In this thesis further maximal analytic and numerical solutions were given (today a lot more solutions are known) and it was (probably for the first time) conjectured that solutions exist in any dimension b (generated by the Weyl–Heisenberg group and with a certain additional symmetry of order 3).
This is a one-to-one translation of a German-written Ph.D. thesis from 1999. Quantum designs are sets of orthogonal projection matrices in finite(b)-dimensional Hilbert spaces. A fundamental differentiation is whether all projections have the same rank r, and furthermore the special case r = 1, which contains two important subclasses: Mutually unbiased bases (MUBs) were introduced prior to this thesis and solutions of b + 1 MUBs whenever b is a power of a prime were already given. Unaware of those papers, this concept was generalized here under the notation of regular affine quantum designs. Maximal solutions are given for the general case r ≥ 1, consisting of r(b2 - 1)/(b - r) so-called complete orthogonal classes whenever b is a power of a prime. For b = 6, an infinite family of MUB triples was constructed and it was — as already done in the author's master's thesis (1991) — conjectured that four MUBs do not exist in this dimension. Symmetric informationally complete positive operator-valued measures (SIC POVMs) in this paper are called regular quantum 2-designs with degree 1. The assigned vectors span b2 equiangular lines. These objects had been investigated since the 1960s, but only a few solutions were known in complex vector spaces. In this thesis further maximal analytic and numerical solutions were given (today a lot more solutions are known) and it was (probably for the first time) conjectured that solutions exist in any dimension b (generated by the Weyl–Heisenberg group and with a certain additional symmetry of order 3).
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.
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.
Toric t -designs, or equivalently t -designs on the diagonal subgroup of the unitary group, are sets of points on the torus over which sums reproduce integrals of degree t monomials over the full torus. Motivated by the projective structure of quantum mechanics, we develop the notion of t -designs on the projective torus, which have a much more restricted structure than their counterparts on full tori. We provide various new constructions of toric and projective toric designs and prove bounds on their size. We draw connections between projective toric designs and a diverse set of mathematical objects, including difference and Sidon sets from the field of additive combinatorics, symmetric, informationally complete positive operator valued measures and complete sets of mutually unbiased bases (MUBs) from quantum information theory, and crystal ball sequences of certain root lattices. Using these connections, we prove bounds on the maximal size of dense Btmodm sets. We also use projective toric designs to construct families of quantum state designs. In particular, we construct families of (uniformly-weighted) quantum state 2 -designs in dimension d of size exactly d(d+1) that do not form complete sets of MUBs, thereby disproving a conjecture concerning the relationship between designs and MUBs (Zhu 2015). We then propose a modification of Zhu's conjecture and discuss potential paths towards proving this conjecture. We prove a fundamental distinction between complete sets of MUBs in prime-power dimensions versus in dimension 6 (and, we conjecture, in all non-prime-power dimensions), the distinction relating to group structure of the corresponding projective toric design. Finally, we discuss many open questions about the properties of these projective toric designs and how they relate to other questions in number theory, geometry, and quantum information.
Mutually Unbiased Bases (MUBs) constitute a fundamental geometric structure in quantum theory, known for providing an optimal measurement scheme for quantum state tomography. In prime and prime-power dimensions, analytical constructions of maximal sets of MUBs are well-known and standard construction relies on the Weyl-Heisenberg (WH) group and finite fields. In non-prime-power dimensions, on the other hand, the existence of such maximal sets remains an open question. We present a generalized numerical method of constructing MUBs without any reliance on \textit{a priori} group structure or specific algebraic frameworks. Formulating the problem at the level of Gram matrix, we reduce the search for complete sets of d+1 MUBs to a phase space optimisation problem. We use the fact that the MUB Gram matrix is a projection matrix, and the third- and fourth-order trace constraints are necessary and sufficient conditions for a projection matrix. We further develop a classification framework based on third-order Bargmann invariants and automorphism groups, allowing us to probe the underlying algebraic and geometric structure of the resulting configurations. Numerical applications of this method in dimensions 3, 4, and 5 demonstrate that all numerically constructed solutions are mutually isomorphic, are isolated points in phase space, and possess automorphism groups that coincide exactly with the Clifford group, the normalizer of the WH group. In the lowest non-prime-power dimension 6 where the existence of a complete set remains open, our numerical search yields no solution. However, for incomplete sets of solutions, our search finds no set of four MUBs, but it yields several inequivalent triples, which we classify and whose neighbourhood we then investigate.
Two orthonormal bases in the complex space of dimension d, are said to be mutually unbiased if the square of the magnitude of the inner product of any vector from one basis with any vector in other basis is equal to the reciprocal of the dimension d. Mutually unbiased bases are used for optimal state determination of mixed quantum states. It is known that in any dimension d, the number of mutually unbiased bases is at most d+1. Ivanovic found a complete set of mutually unbiased bases for prime dimensions. His construction was generalized by Wootters and Fields for prime power dimensions. There is a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. Based on this connection, there exits a constructive proof of the existence of a complete set of mutually unbiased bases for prime power dimensions. This thesis is an exploration on construction of mutually unbiased bases.
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.