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Physical Review A

Unitarily inequivalent mutually unbiased bases for n qubits

Arun Sehrawat, A. B. Klimov

Published Dec 2, 2014• 5 citations
Publisher

Abstract

The standard construction of complete sets of mutually unbiased bases (MUBs) in prime power dimensions is based on the quadratic Gauss sums. We introduce complete MUB sets for three, four, and five qubits that are unitarily inequivalent to all existing MUB sets. These sets are constructed by using certain exponential sums, where the degree of the polynomial appearing in the exponent can be higher than 2. Every basis of these MUBs (except the computational) consists of two disjoint blocks of vectors with different factorization structures and associated with a unique hypergraph (or graph) that represents an interaction between the qubits.

DOI
10.1103/physreva.90.062308

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The standard construction of complete sets of mutually unbiased bases (MUBs) in prime power dimensions is based on the quadratic Gauss sums. We introduce complete MUB sets for three, four, and five qubits that are unitarily inequivalent to all existing MUB sets. These sets are constructed by using certain exponential sums, where the degree of the polynomial appearing in the exponent can be higher than 2. Every basis of these MUBs (except the computational) consists of two disjoint blocks of vectors with different factorization structures and associated with a unique hypergraph (or graph) that represents an interaction between the qubits.
A pair of orthonormal bases is called mutually unbiased if all mutual overlaps between any element of one basis and an arbitrary element of the other basis coincide. In case the dimension, ddd, of the considered Hilbert space is a power of a prime number, complete sets of d+1d+1d+1 mutually unbiased bases (MUBs) exist. Here we present a method based on the graph-state formalism to construct such sets of MUBs. We show that for nnn ppp-level systems, with ppp being prime, one particular graph suffices to easily construct a set of pn+1{p}^{n}+1pn+1 MUBs. In fact, we show that a single nnn-dimensional vector, which is associated with this graph, can be used to generate a complete set of MUBs and demonstrate that this vector can be easily determined. Finally, we discuss some advantages of our formalism regarding the analysis of entanglement structures in MUBs, as well as experimental realizations.
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 (MUBs) are important in quantum information theory. While constructions of complete sets of d + 1 MUBs in are known when d is a prime power, it is unknown if such complete sets exist in non-prime power dimensions. It has been conjectured that complete sets of MUBs only exist in if a maximal set of mutually orthogonal Latin squares (MOLS) of side length d also exists. There are several constructions (Roy and Scott 2007 J. Math. Phys. 48 072110; Paterek, Dakić and Brukner 2009 Phys. Rev. A 79 012109) of complete sets of MUBs from specific types of MOLS, which use Galois fields to construct the vectors of the MUBs. In this paper, two known constructions of MUBs (Alltop 1980 IEEE Trans. Inf. Theory 26 350–354; Wootters and Fields 1989 Ann. Phys. 191 363–381), both of which use polynomials over a Galois field, are used to construct complete sets of MOLS in the odd prime case. The MOLS come from the inner products of pairs of vectors in the MUBs.
There are fairly large families of unitarily inequivalent complete sets of N + 1 mutually unbiased bases (MUBs) known in C\mathbb{C}CCN for various prime powers N. The number of such sets is not bounded above by any polynomial as a function of N. While it is standard that there is a superficial similarity between complete sets of MUBs and finite affine planes, there is an intimate relationship between these large families and affine planes. This note briefly summarizes “old” results that do not appear to be well known concerning known families of complete sets of MUBs and their associated planes.
Mutually unbiased bases (MUBs) describe quantum measurements for which certainty in one basis gives uniform outcomes in the others. Their maximum number remains unknown in dimensions that are not powers of a prime. We introduce an ansatz that, in many dimensions, allows us to construct more MUBs than have previously been found. Each basis in the ansatz consists of a diagonal phase matrix applied to a fixed unitary, which is a tensor product of Fourier matrices and, optionally, any real Hadamard matrix. This construction yields five MUBs in dimension 12, six in dimensions 48, 96, and 192, seven in dimensions 36 and 108, and ten in dimension 648. These bases can then be extended to exceed the tensor product bound, asymptotically, in every eighteenth dimension. Moreover, we show that Paley's real Hadamard matrix can be used to construct q+1q+1q+1 bases in dimension d=q(q+1)d = q(q+1)d=q(q+1) for every prime power q≡3(mod4)q\equiv3\pmod4q≡3(mod4). This count grows as d\sqrt dd​ and cannot be exceeded by tensor products of smaller sets.
We present a construction method for complete sets of cyclic mutually unbiased bases (MUBs) in Hilbert spaces of even prime-power dimensions. In comparison to the usual complete sets of MUBs, complete cyclic sets possess the additional property of being generated by a single unitary operator. The construction method is based on the idea of obtaining a partition of multi-qubit Pauli operators into maximal commuting sets of operators with the help of a suitable element of the Clifford group. As a consequence, we explicitly obtain complete sets of cyclic MUBs generated by a single element of the Clifford group in dimensions 2 m for m = 1, 2, …, 24.