The connection between maximal sets of mutually unbiased bases (MUBs) in a prime-power dimensional Hilbert space and finite phase-space geometries is well known. In this article, we classify MUBs according to their degree of covariance with respect to the natural symmetries of a finite phase-space, which are the group of its affine symplectic transformations. We prove that there exist maximal sets of MUBs that are covariant with respect to the full group only in odd prime-power dimensional spaces, and in this case, their equivalence class is actually unique. Despite this limitation, we show that in dimension [Formula: see text] covariance can still be achieved by restricting to proper subgroups of the symplectic group, that constitute the finite analogues of the oscillator group. For these subgroups, we explicitly construct the unitary operators yielding the covariance.
The connection between maximal sets of mutually unbiased bases (MUBs) in a prime-power dimensional Hilbert space and finite phase-space geometries is well known. In this article, we classify MUBs according to their degree of covariance with respect to the natural symmetries of a finite phase-space, which are the group of its affine symplectic transformations. We prove that there exist maximal sets of MUBs that are covariant with respect to the full group only in odd prime-power dimensional spaces, and in this case, their equivalence class is actually unique. Despite this limitation, we show that in dimension [Formula: see text] covariance can still be achieved by restricting to proper subgroups of the symplectic group, that constitute the finite analogues of the oscillator group. For these subgroups, we explicitly construct the unitary operators yielding the covariance.
One way to construct a maximal set of mutually unbiased bases (MUBs) in a prime-power dimensional Hilbert space is by means of finite phase-space methods. MUBs obtained in this way are covariant with respect to some subgroup of the group of all affine symplectic phase-space transformations. However, this construction is not canonical: as a consequence, many different choices of covariance subgroups are possible. In particular, when the Hilbert space is 2n dimensional, it is known that covariance with respect to the full group of affine symplectic phase-space transformations can never be achieved. Here we show that in this case there exist two essentially different choices of maximal subgroups admitting covariant MUBs. For both of them, we explicitly construct a family of 2n covariant MUBs. We thus prove that, contrary to the odd dimensional case, maximally covariant MUBs are very far from being unique in even prime-power dimensions.
Mutually unbiased bases (MUB) are useful in a number of research areas. The symmetry of MUB is an elusive and interesting subject. A (complete set of) MUB in dimension d is sharply covariant if it can be generated by a group of order d(d+1) from a basis state. Such MUB, if they exist, would be most appealing to theoretical studies and practical applications. Unfortunately, they seem to be quite rare. Here we prove that no MUB in odd prime dimensions is sharply covariant, by virtue of clever applications of Mersenne primes, Galois fields, and Frobenius groups. This conclusion provides valuable insight about the symmetry of MUB and the geometry of quantum state space. It complements and strengthens the earlier result of the author that only two stabilizer MUB are sharply covariant. Our study leads to the conjecture that no MUB other than those in dimensions 2 and 4 is sharply covariant.
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.
Mutually unbiased bases for quantum degrees of freedom are central to all theoretical investigations and practical exploitations of complementary properties. Much is known about mutually unbiased bases, but there are also a fair number of important questions that have not been answered in full as yet. In particular, one can find maximal sets of N + 1 mutually unbiased bases in Hilbert spaces of prime-power dimension N = p M , with p prime and M a positive integer, and there is a continuum of mutually unbiased bases for a continuous degree of freedom, such as motion along a line. But not a single example of a maximal set is known if the dimension is another composite number (N = 6, 10, 12,…). In this review, we present a unified approach in which the basis states are labeled by numbers 0, 1, 2, …, N - 1 that are both elements of a Galois field and ordinary integers. This dual nature permits a compact systematic construction of maximal sets of mutually unbiased bases when they are known to exist but throws no light on the open existence problem in other cases. We show how to use the thus constructed mutually unbiased bases in quantum-informatics applications, including dense coding, teleportation, entanglement swapping, covariant cloning, and state tomography, all of which rely on an explicit set of maximally entangled states (generalizations of the familiar two–q-bit Bell states) that are related to the mutually unbiased bases. There is a link to the mathematics of finite affine planes. We also exploit the one-to-one correspondence between unbiased bases and the complex Hadamard matrices that turn the bases into each other. The ultimate hope, not yet fulfilled, is that open questions about mutually unbiased bases can be related to open questions about Hadamard matrices or affine planes, in particular the notorious existence problem for dimensions that are not a power of a prime. The Hadamard-matrix approach is instrumental in the very recent advance, surveyed here, of our understanding of the N = 6 situation. All evidence indicates that a maximal set of seven mutually unbiased bases does not exist — one can find no more than three pairwise unbiased bases — although there is currently no clear-cut demonstration of the case.
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.