[Paper Review] The Mutually Unbiased Bases Revisited

M. Combescure|arXiv (Cornell University)|May 10, 2006

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper revisits the construction of Mutually Unbiased Bases (MUBs) in quantum information using elementary unitary matrices introduced by Schwinger, leveraging the Vandermonde matrix based on d-th roots of unity. It proves the existence of 3 MUBs in any dimension d, provides conditions for more than 3 MUBs when d is even or odd, and recovers the known result of d+1 MUBs for prime d, while also deriving Gauss sum identities from MUB structure.

ABSTRACT

The study of Mutually Unbiased Bases continues to be developed vigorously, and presents several challenges in the Quantum Information Theory. Two orthonormal bases in $\mathbb C^d, B {and} B'$ are said mutually unbiased if $\forall b\in B, b'\in B'$ the scalar product $b\cdot b'$ has modulus $d^{-1/2}$. In particular this property has been introduced in order to allow an optimization of the measurement-driven quantum evolution process of any state $ψ\in \mathbb C^d$ when measured in the mutually unbiased bases $B\_{j} {of} \mathbb C^d$. At present it is an open problem to find the maximal umber of mutually Unbiased Bases when $d$ is not a power of a prime number. oindent In this article, we revisit the problem of finding Mutually Unbiased Bases (MUB's) in any dimension $d$. 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 $d$-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 $d$ even or odd number, and to recover the known result of existence of $d+1$ MUB's for $d$ 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.

Motivation & Objective

• To provide an elementary, constructive approach to Mutually Unbiased Bases (MUBs) in arbitrary dimension d, avoiding advanced number theory.
• To establish the existence of at least 3 MUBs in any dimension d, regardless of whether d is prime or composite.
• To derive conditions under which more than 3 MUBs exist, particularly for even and odd d.
• To recover known Gauss sum identities from MUB construction, linking quantum information to number theory.
• To offer explicit, algorithmic constructions of MUBs using diagonalization of Schwinger matrices.

Proposed method

• Uses Schwinger’s unitary matrices U (diagonal with d-th roots of unity) and V (cyclic shift matrix) to define V_k = V U^k for k = 0, ..., d-1.
• Employs the Vandermonde matrix P_0 with entries (P_0)_{j,k} = d^{-1/2} q^{jk} where q = exp(2πi/d), which diagonalizes U via conjugation by V.
• Applies diagonalization of the matrices V_k to generate unitary matrices P_k that are unbiased and mutually unbiased.
• For composite dimensions d = 4m with m odd, constructs MUBs via tensor products of 4×4 and m×m MUB structures.
• Uses the property that if V_k = P_k D_k P_k^*, then all P_k are unbiased matrices, and P_j^* P_k is unbiased for j ≠ k.
• Derives Gauss sum identities such as |∑_{j=0}^{d-1} q^{k j(j+1)/2}| = √d for odd d and gcd(k,d)=1, from MUB unitarity and bias conditions.

Experimental results

Research questions

• RQ1 Can a simple, elementary construction of MUBs be achieved without relying on finite field theory or Gauss sums?
• RQ2 What is the minimal number of MUBs guaranteed to exist in any dimension d?
• RQ3 Under what conditions on d (even, odd, prime, composite) can more than 3 MUBs be constructed?
• RQ4 Can known number-theoretic identities like Gauss sums be derived directly from MUB properties?
• RQ5 Is it possible to construct explicit MUBs in non-prime-power dimensions such as d=6, 12, and 20?

Key findings

• The paper proves the existence of at least 3 mutually unbiased bases in any dimension d, regardless of primality.
• For d a prime number, the construction recovers the known maximal set of d+1 MUBs.
• In dimension d=6, the method constructs a set of 3 MUBs, supporting the conjecture that N(6)=3 is maximal.
• For d=12, the construction yields 4 MUBs using tensor products of 4×4 and 3×3 MUB structures.
• For d=20, the method produces 5 MUBs by combining 4×4 and 5×5 MUB constructions via tensor products.
• The paper derives the Gauss sum identity |∑_{j=0}^{d-1} q^{k j(j+1)/2}| = √d for odd d and gcd(k,d)=1 directly from MUB unitarity and bias conditions.