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arXiv:1506.08283 · Physical Review A

Mutually unbiased bases with free parameters

Dardo Goyeneche, Santiago Gómez

Published Dec 14, 2015• 8 citations
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Abstract

We present a systematic method to introduce free parameters in sets of mutually unbiased bases. In particular, we demonstrate that any set of $m$ real mutually unbiased bases existing in dimension $N>2$ admits the introduction of $(m\ensuremath{-}1)N/2$ free parameters that cannot be absorbed by a global unitary operation. As consequence, there are $m=k+1$ mutually unbiased bases in every dimension $N={k}^{2}$ with ${k}^{3}/2$ free parameters, where $k$ 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.

DOI
10.1103/physreva.92.062325

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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(m\ensuremath{-}1)N/2(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.
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The study of Mutually Unbiased Bases continues to be developed vigorously, and presents several challenges in the Quantum Information Theory. Two orthonormal bases in Cd,BandB′\mathbb C^d, B {and} B'Cd,BandB′ are said mutually unbiased if ∀b∈B,b′∈B′\forall b\in B, b'\in B'∀b∈B,b′∈B′ the scalar product b⋅b′b\cdot b'b⋅b′ has modulus d−1/2d^{-1/2}d−1/2. In particular this property has been introduced in order to allow an optimization of the measurement-driven quantum evolution process of any state ψ∈Cdψ\in \mathbb C^dψ∈Cd when measured in the mutually unbiased bases B_jofCdB\_{j} {of} \mathbb C^dB_jofCd. At present it is an open problem to find the maximal umber of mutually Unbiased Bases when ddd is not a power of a prime number. \noindent In this article, we revisit the problem of finding Mutually Unbiased Bases (MUB's) in any dimension ddd. The method is very elementary, using the simple unitary matrices introduced by Schwinger in 1960, together with their diagonalizations. The Vandermonde matrix based on the ddd-th roots of unity plays a major role. This allows us to show the existence of a set of 3 MUB's in any dimension, to give conditions for existence of more than 3 MUB's for ddd even or odd number, and to recover the known result of existence of d+1d+1d+1 MUB's for ddd a prime number. Furthermore the construction of these MUB's is very explicit. As a by-product, we recover results about Gauss Sums, known in number theory, but which have apparently not been previously derived from MUB properties.
In this paper, we explore the concept of Mutually Unbiased Bases (MUBs) in discrete quantum systems. It is known that for dimensions ddd that are powers of prime numbers, there exists a set of up to d+1d+1d+1 bases that form an MUB set. However, the maximum number of MUBs in dimensions that are not powers of prime numbers is not known. To address this issue, we introduce three algorithms based on First-Order Logic that can determine the maximum number of bases in an MUB set without numerical approximation. Our algorithms can prove this result in finite time, although the required time is impractical. Additionally, we present a heuristic approach to solve the semi-decision problem of determining if there are kkk MUBs in a given dimension ddd. As a byproduct of our research, we demonstrate that the maximum number of MUBs in any dimension can be achieved with definable complex parameters, computable complex parameters, and other similar fields.
One of the essential features of quantum mechanics is that most pairs of observables cannot be measured simultaneously. This phenomenon manifests itself most strongly when observables are related to mutually unbiased bases. In this paper, we shed some light on the connection between mutually unbiased bases and another essential feature of quantum mechanics, quantum entanglement. It is shown that a complete set of mutually unbiased bases of a bipartite system contains a fixed amount of entanglement, independent of the choice of the set. This has implications for entanglement distribution among the states of a complete set. In prime-squared dimensions we present an explicit experiment-friendly construction of a complete set with a particularly simple entanglement distribution. Finally, we describe the basic properties of mutually unbiased bases composed of product states only. The constructions are illustrated with explicit examples in low dimensions. We believe that the properties of entanglement in mutually unbiased bases may be one of the ingredients to be taken into account to settle the question of the existence of complete sets. We also expect that they will be relevant to applications of bases in the experimental realization of quantum protocols in higher-dimensional Hilbert spaces.
For a system of NNN qubits, living in a Hilbert space of dimension d=2Nd={2}^{N}d=2N, it is known that there exists d+1d+1d+1 mutually unbiased bases. Different construction algorithms exist, and it is remarkable that different methods lead to sets of bases with different properties as far as separability is concerned. Here we derive four sets of nine bases for three qubits, and show how they are unitarily related. We also briefly discuss the four-qubit case, give the entanglement structure of 16 sets of bases, and show some of them and their interrelations, as examples. The extension of the method to the general case of NNN qubits is outlined.
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.