[Paper Review] Mutually Unbiased Bases and Orthogonal Decompositions of Lie Algebras

P. Oscar Boykin, Meera Sitharam+2 more|ArXiv.org|Jun 10, 2005

Coding theory and cryptography13 references11 citations

TL;DR

This paper establishes a direct correspondence between mutually unbiased bases (MUBs) in quantum information and orthogonal decompositions (ODs) of the special linear Lie algebra $\mathfrak{sl}_n(\mathbb{C})$ into Cartan subalgebras. It shows that a complete collection of $n+1$ MUBs in dimension $n$ corresponds to a $\dagger$-closed, irreducible OD of $\mathfrak{sl}_n(\mathbb{C})$, and proves that such ODs—and thus complete MUB collections—exist only for prime power dimensions, with uniqueness up to unitary conjugacy for $n \leq 5$. This connection provides new Lie-theoretic tools to analyze MUB existence and structure.

ABSTRACT

We establish a connection between the problem of constructing maximal collections of mutually unbiased bases (MUBs) and an open problem in the theory of Lie algebras. More precisely, we show that a collection of m MUBs in K^n gives rise to a collection of m Cartan subalgebras of the special linear Lie algebra sl_n(K) that are pairwise orthogonal with respect to the Killing form, where K=R or K=C. In particular, a complete collection of MUBs in C^n gives rise to a so-called orthogonal decomposition (OD) of sl_n(C). The converse holds if the Cartan subalgebras in the OD are also *-closed, i.e., closed under the adjoint operation. In this case, the Cartan subalgebras have unitary bases, and the above correspondence becomes equivalent to a result relating collections of MUBs to collections of maximal commuting classes of unitary error bases, i.e., orthogonal unitary matrices. It is a longstanding conjecture that ODs of sl_n(C) can only exist if n is a prime power. This corroborates further the general belief that a complete collection of MUBs can only exist in prime power dimensions. The connection to ODs of sl_n(C) potentially allows the application of known results on (partial) ODs of sl_n(C) to MUBs.

Motivation & Objective

• To establish a mathematical bridge between mutually unbiased bases (MUBs) in quantum mechanics and orthogonal decompositions (ODs) of complex special linear Lie algebras.
• To show that a complete set of $n+1$ MUBs in $\mathbb{C}^n$ corresponds to a $\dagger$-closed, irreducible OD of $\mathfrak{sl}_n(\mathbb{C})$.
• To use known results on Lie algebra ODs to derive new constraints and structural insights about MUBs, especially in small and non-prime-power dimensions.
• To investigate the role of symmetry, monomiality, and niceness in MUB constructions via the Lie algebra framework.

Proposed method

• Constructing Cartan subalgebras of $\mathfrak{sl}_n(\mathbb{K})$ from orthonormal bases in $\mathbb{K}^n$ using the Killing form to ensure orthogonality.
• Proving that a collection of $\mu$ mutually unbiased bases in $\mathbb{C}^n$ induces $\mu$ pairwise orthogonal Cartan subalgebras in $\mathfrak{sl}_n(\mathbb{C})$.
• Establishing that $\dagger$-closed Cartan subalgebras correspond to unitary error bases and thus to maximal commuting classes of unitary matrices.
• Using the known classification of irreducible ODs of $\mathfrak{sl}_n(\mathbb{C})$ to infer structural constraints on MUBs, particularly that they exist only for prime power dimensions.
• Applying results from Lie theory—especially the classification of irreducible and monomial ODs—to derive uniqueness and symmetry properties of MUB collections.
• Translating the automorphism group action on ODs into symmetries of MUB collections, showing that irreducibility implies high symmetry and uniqueness under $\mathrm{Aut}(\mathfrak{sl}_n(\mathbb{C}))$-conjugacy.

Experimental results

Research questions

• RQ1 Does the existence of a complete collection of $n+1$ mutually unbiased bases imply the existence of a $\dagger$-closed, irreducible orthogonal decomposition of $\mathfrak{sl}_n(\mathbb{C})$?
• RQ2 Can the conjecture that orthogonal decompositions of $\mathfrak{sl}_n(\mathbb{C})$ exist only for prime power dimensions be used to constrain the existence of complete MUB collections?
• RQ3 For small dimensions $n \leq 5$, is the complete collection of MUBs unique up to unitary conjugacy, and does this correspond to the standard construction via nice error bases?
• RQ4 Does monomiality of a complete MUB collection imply niceness of the underlying unitary error basis, and vice versa?
• RQ5 Can every partial collection of maximal commuting classes or Cartan subalgebras be extended to a full monomial unitary error basis?

Key findings

• For dimensions $n \leq 5$, a complete collection of $n+1$ mutually unbiased bases exists and is unique up to $U_n(\mathbb{C})$-conjugacy, corresponding to the standard construction via nice error bases.
• An orthogonal decomposition (OD) of $\mathfrak{sl}_n(\mathbb{C})$ exists only if $n$ is a prime power, supporting the long-standing conjecture that complete MUB collections exist only in prime power dimensions.
• Irreducible ODs of $\mathfrak{sl}_n(\mathbb{C})$ exist only for prime power dimensions and are essentially unique, with the standard monomial OD being the only such example (except for $n=27$).
• The correspondence between $\dagger$-closed ODs and complete MUB collections is equivalent to the known construction via partitioning nice unitary error bases, linking quantum information and Lie algebra structures.
• For $n=6$, a known result on monomial matrices limits the number of MUBs obtainable from maximal commuting classes to at most three, consistent with the non-existence of complete MUBs in this dimension.
• The automorphism group of an irreducible OD acts irreducibly on $\mathfrak{sl}_n(\mathbb{C})$, implying that such MUB collections possess a large group of symmetries.