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arXiv:1404.3035 · Journal of Physics A Mathematical and Theoretical

Structure of the sets of mutually unbiased bases with cyclic symmetry

Ulrich Seyfarth, Luis Lorenzo Sánchez-Soto, Gerd Leuchs

Published Oct 29, 2014• 8 citations
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Abstract

Mutually unbiased bases that can be cyclically generated by a single unitary operator are of special interest, for they can be readily implemented in practice. We show that, for a system of qubits, finding such a generator can be cast as the problem of finding a symmetric matrix over the field 2 equipped with an irreducible characteristic polynomial of a given Fibonacci index. The entanglement structure of the resulting complete sets is determined by two additive matrices of the same size.

DOI
10.1088/1751-8113/47/45/455303

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Mutually unbiased bases that can be cyclically generated by a single unitary operator are of special interest, for they can be readily implemented in practice. We show that, for a system of qubits, finding such a generator can be cast as the problem of finding a symmetric matrix over the field 2 equipped with an irreducible characteristic polynomial of a given Fibonacci index. The entanglement structure of the resulting complete sets is determined by two additive matrices of the same size.
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.
The number of measurements necessary to perform the quantum state reconstruction of a system of qubits grows exponentially with the number of constituents, creating a major obstacle for the design of scalable tomographic schemes. We work out a simple and efficient method based on cyclic generation of mutually unbiased bases. The basic generator requires only Hadamard and controlled-phase gates, which are available in most practical realizations of these systems. We show how complete sets of mutually unbiased bases with different entanglement structures can be realized for three and four qubits. We also analyze the quantum circuits implementing the various entanglement classes.
We propose a systematic procedure to construct all the possible bases with a definite factorization structure (eigenstates of n commuting monomials constructed as products of Pauli operators) for an n -qubit system, as well as the possibility of collecting them into mutually unbiased sets. We also discuss an algorithm for the determination of basis separability and propose a criteria for complementarity between such bases. The results are applied to generate non-isomorphic complete sets of mutually unbiased bases.
We present a systematic method to introduce free parameters in sets of mutually unbiased bases. In particular, we demonstrate that any set of mmm real mutually unbiased bases existing in dimension N>2N>2N>2 admits the introduction of $(m\ensuremath{-}1)N/2$ free parameters that cannot be absorbed by a global unitary operation. As consequence, there are m=k+1m=k+1m=k+1 mutually unbiased bases in every dimension N=k2N={k}^{2}N=k2 with k3/2{k}^{3}/2k3/2 free parameters, where kkk is even. We explicitly construct the maximal set of triplets of mutually unbiased bases for two-qubit systems and triplets, quadruplets, and quintuplets of mutually unbiased bases with free parameters for three-qubit systems. Furthermore, we study the richness of the entanglement structure of such bases and provide the quantum circuits required to implement all these bases with free parameters in the laboratory. We also show that the free parameters introduced can be controlled by a single party of the system. Finally, we find the upper bound for the maximal number of real and complex mutually unbiased bases existing in every dimension. This proof is simple, short, and considers basic matrix algebra.
A simple recipe for generating a complete set of mutually unbiased bases in dimension 2j + 1, with 2j integer and 2j + 1 prime, is developed from a single matrix Va acting on a space of constant angular momentum j and defined in terms of the irreducible characters of the cyclic group C2j + 1. This recipe yields an (apparently new) compact formula for the vectors spanning the various mutually unbiased bases. In dimension (2j + 1)e, with 2j integer, 2j + 1 prime and e positive integer, the use of direct products of matrices of type Va makes it possible to generate mutually unbiased bases. As two pending results, the matrix Va is used in the derivation of a polar decomposition of SU (2) and of a FFZ algebra.