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arXiv:0708.4114 · Journal of Physics A Mathematical and Theoretical

Group theoretical construction of mutually unbiased bases in Hilbert spaces of prime dimensions

Petr Šulc, J Tolar

Published Nov 28, 2007• 32 citations
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Abstract

Mutually unbiased bases in Hilbert spaces of finite dimensions are closely related to the quantal notion of complementarity. An alternative proof of existence of a maximal collection of ( N + 1) mutually unbiased bases in Hilbert spaces of prime dimension N is given by exploiting the finite Heisenberg group (also called the Pauli group) and the action of on finite phase space implemented by unitary operators in the Hilbert space. Crucial for the proof is that, for prime is also a finite field.

DOI
10.1088/1751-8113/40/50/013

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Mutually unbiased bases in Hilbert spaces of finite dimensions are closely related to the quantal notion of complementarity. An alternative proof of existence of a maximal collection of ( N + 1) mutually unbiased bases in Hilbert spaces of prime dimension N is given by exploiting the finite Heisenberg group (also called the Pauli group) and the action of on finite phase space implemented by unitary operators in the Hilbert space. Crucial for the proof is that, for prime is also a finite field.
We present and generalize the basic ideas underlying recent work aimed at the construction of mutually unbiased bases in finite dimensional Hilbert spaces with the help of group and graph theoretical concepts. In this approach finite groups are used to construct maximal sets of mutually unbiased bases. Thus the prime number restrictions of previous approaches are circumvented and this construction principle sheds new light onto the intricate relation between mutually unbiased bases and characteristic geometrical structures of Hilbert spaces.
Analogous to the notion of mutually unbiased bases for Hilbert spaces, we consider mutually unbiased unitary bases (MUUBs) for the space of operators, Mðd; CÞ, acting on such Hilbert spaces. The notion of MUUB re°ects the equiprobable guesses of unitary operators in one basis of Mðd; CÞ when estimating a unitary operator in another. Though, for prime dimension d, the maximal number of MUUBs is known to be d2 1, there is no known recipe for constructing them, assuming they exist. However, one can always construct a minimum of three MUUBs, and the maximal number is approached for very large values of d. MUUBs can also exist for some d-dimensional subspace of Mðd; CÞ with the maximal number being d.
Abstract In this contribution the main ideas underlying recent work aiming at the construction of mutually unbiased bases in finite dimensional Hilbert spaces are discussed. This approach relies on a systematic use of group and graph theoretical concepts announced by Charnes and Beth (2005 ERATO Conf. on Quantum Information Science ) and extended significantly by Charnes (2018 in preparation) recently. A principal feature of this method is its independence of prime number restrictions thus distinguishing it from almost all previous constructions which have relied on finite fields and related concepts of finite geometry. This group and graph theoretical approach offers the possibility to gain new insight into the intricate relation between quantum theoretical complementarity as encoded in mutually unbiased bases and characteristic geometrical structures of the Hilbert space involved.
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.
Analogous to the notion of mutually unbiased bases for Hilbert spaces, we consider mutually unbiased unitary bases (MUUBs) for the space of operators, [Formula: see text], acting on such Hilbert spaces. The notion of MUUB reflects the equiprobable guesses of unitary operators in one basis of [Formula: see text] when estimating a unitary operator in another. Though, for prime dimension [Formula: see text], the maximal number of MUUBs is known to be [Formula: see text], there is no known recipe for constructing them, assuming they exist. However, one can always construct a minimum of three MUUBs, and the maximal number is approached for very large values of [Formula: see text]. MUUBs can also exist for some [Formula: see text]-dimensional subspace of [Formula: see text] with the maximal number being [Formula: see text].
A complete set of d + 1 mutually unbiased bases exists in a Hilbert space of dimension d whenever d is power of a prime. We discuss a simple construction of d + 1 disjoint classes (each one having d − 1 commuting operators) such that the corresponding eigenstates form sets of unbiased bases. One of these classes is diagonal and can be mapped to “ladder” operators by means of the finite Fourier transform. Using this idea, we naturally introduce the notion of quantum phase as complementary to the inversion. Relevant examples involving qubits and qutrits are discussed.