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Entropy

Equiangular Vectors Approach to Mutually Unbiased Bases

M. Kibler

Published May 8, 2013• 5 citations
Publisher

Abstract

Two orthonormal bases in the d-dimensional Hilbert space are said to be unbiased if the square modulus of the inner product of any vector of one basis with any vector of the other equals 1 d. The presence of a modulus in the problem of finding a set of mutually unbiased bases constitutes a source of complications from the numerical point of view. Therefore, we may ask the question: Is it possible to get rid of the modulus? After a short review of various constructions of mutually unbiased bases in Cd, we show how to transform the problem of finding d + 1 mutually unbiased bases in the d-dimensional space Cd (with a modulus for the inner product) into the one of finding d(d+1) vectors in the d2-dimensional space Cd2 (without a modulus for the inner product). The transformation from Cd to Cd2 corresponds to the passage from equiangular lines to equiangular vectors. The transformation formulas are discussed in the case where d is a prime number.

DOI
10.3390/e15051726

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Two orthonormal bases in the d-dimensional Hilbert space are said to be unbiased if the square modulus of the inner product of any vector of one basis with any vector of the other equals 1 d. The presence of a modulus in the problem of finding a set of mutually unbiased bases constitutes a source of complications from the numerical point of view. Therefore, we may ask the question: Is it possible to get rid of the modulus? After a short review of various constructions of mutually unbiased bases in Cd, we show how to transform the problem of finding d + 1 mutually unbiased bases in the d-dimensional space Cd (with a modulus for the inner product) into the one of finding d(d+1) vectors in the d2-dimensional space Cd2 (without a modulus for the inner product). The transformation from Cd to Cd2 corresponds to the passage from equiangular lines to equiangular vectors. The transformation formulas are discussed in the case where d is a prime number.
Two orthonormal bases in the complex space of dimension d, are said to be mutually unbiased if the square of the magnitude of the inner product of any vector from one basis with any vector in other basis is equal to the reciprocal of the dimension d. Mutually unbiased bases are used for optimal state determination of mixed quantum states. It is known that in any dimension d, the number of mutually unbiased bases is at most d+1. Ivanovic found a complete set of mutually unbiased bases for prime dimensions. His construction was generalized by Wootters and Fields for prime power dimensions. There is a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. Based on this connection, there exits a constructive proof of the existence of a complete set of mutually unbiased bases for prime power dimensions. This thesis is an exploration on construction of mutually unbiased bases.
A set of vectors of equal norm in Cd\mathbb{C}^dCd represents equiangular lines if the magnitudes of the Hermitian inner product of every pair of distinct vectors in the set are equal. The maximum size of such a set is d2d^2d2, and it is conjectured that sets of this maximum size exist in Cd\mathbb{C}^dCd for every d≥2d \geq 2d≥2. We take a combinatorial approach to this conjecture, using mutually unbiased bases (MUBs) in the following 3 constructions of equiangular lines: (1) adapting a set of ddd MUBs in Cd\mathbb{C}^dCd to obtain d2d^2d2 equiangular lines in Cd\mathbb{C}^dCd, (2) using a set of ddd MUBs in Cd\mathbb{C}^dCd to build (2d)2(2d)^2(2d)2 equiangular lines in C2d\mathbb{C}^{2d}C2d, (3) combining two copies of a set of ddd MUBs in Cd\mathbb{C}^dCd to build (2d)2(2d)^2(2d)2 equiangular lines in C2d\mathbb{C}^{2d}C2d. For each construction, we give the dimensions ddd for which we currently know that the construction produces a maximum-sized set of equiangular lines.
Two orthonormal bases B and B' of a d-dimensional complex inner-product space are called mutually unbiased if and only if ||^2=1/d holds for all b in B and b' in B'. The size of any set containing (pairwise) mutually unbiased bases of C^d cannot exceed d+1. If d is a power of a prime, then extremal sets containing d+1 mutually unbiased bases are known to exist. We give a simplified proof of this fact based on the estimation of exponential sums. We discuss conjectures and open problems concerning the maximal number of mutually unbiased bases for arbitrary dimensions.
In a quantum system having a finite number N of orthogonal states, two orthonormal bases {ai} and {bj} are called mutually unbiased if all inner products ⟨ai∣bj⟩ have the same modulus 1∕N. This concept appears in several quantum information problems. The number of pairwise mutually unbiased bases is at most N+1 and various constructions of such N+1 bases have been found when N is a power of a prime number. We study families of formulas that generalize these constructions to arbitrary dimensions using finite rings. We then prove that there exists a set of N+1 mutually unbiased bases described by such formulas, if and only if N is a power of a prime number.
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.