A Curious Connection between Quantum Information and Finite Geometry

Mutually Unbiased Bases

Mutually unbiased bases (MUBs) are motivated by simple quantum measurement procedures. Don’t worry if you’re not familiar with quantum mechanics – we’ll only need some basic linear algebra. To start, a pure quantum state is represented by a complex unit vector $\psi \in \mathbb{C}^d$, for some dimension $d$. For each orthonormal basis $\mathcal{B}= \{ b_1, \dots, b_d \} \in \mathbb{C}^d$, there exists a quantum measurement probing with respect to that basis, such that $$\Pr[\psi \text{ observed in state } b_i \text{ after measurement w.r.t. } \mathcal{B}] = | \langle \psi | b_i \rangle |^2,$$ where $\langle \cdot | \cdot \rangle$ denotes the standard inner product (conjugate-linear in the first argument and linear in the second¹). If this quantity is constant for all $i$, then $$\begin{align*} | \langle \psi | b_1 \rangle |^2 = \dots = | \langle \psi | b_d \rangle |^2 &= \frac{1}{d} \sum_{j = 1}^d | \langle \psi| b_j \rangle |^2\\ &= \frac{1}{d} \bigg\langle \psi \,\Bigg| \sum_{j = 1}^d \langle \psi | b_j \rangle b_j \bigg\rangle\\ &= \frac{1}{d} \langle \psi | \psi \rangle = \frac{1}{d}, \end{align*}$$ where the second to last inequality follows because $\mathcal{B}$ is orthonormal. In this case, measurement with respect to $\mathcal{B}$ reveals no information about $\psi$. Mutual unbiasedness extends this notion to describe a relationship between different bases.

That is, if we prepare a state from one basis and perform a measurement with respect to the other, all outcomes are equally likely. MUBs arise naturally in several quantum information protocols for key distribution, error correction, and more.

Example

Here is a diagram of two MUBs in $\mathbb{C}^2$ with no complex components.

2 MUBs in C2

These correspond to, say, $$\color{#5b9bd5}{\mathcal{B}_1} = \left\{ \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \end{pmatrix} \right\} \quad \text{and} \quad \color{#ed7d31}{\mathcal{B}_2} = \left\{ \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \frac{1}{\sqrt{2}} \begin{pmatrix} -1 \\ 1 \end{pmatrix} \right\}.$$ If we allow complex components, $$\mathcal{B}_3 = \left\{ \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ i \end{pmatrix}, \frac{1}{\sqrt{2}} \begin{pmatrix} -1 \\ i \end{pmatrix} \right\}$$ can be added as a third mutually unbiased basis².

This example raises a natural question: for a fixed dimension $d$, what is the maximum number of MUBs in $\mathbb{C}^d$? Denoting this count by $\mathfrak{M}_1(d)$ (the reason for the subscript will be revealed soon), it is relatively straightforward to prove that $\mathfrak{M}_1(d) \leq d + 1$. Hence, collections of $d+1$ MUBs in $\mathbb{C}^d$ are said to be complete. The following is conjectured.

Even the case of $d = 6$ is unresolved, and this remains a primary open problem in quantum information theory. Personally, I find it quite interesting that such a condition might arise in this geometric setting.

Combinatorial $k$-Nets

Next, we’ll consider a class of combinatorial designs called $k$-nets, of which affine planes are a special case. We begin with a base set of $d^2$ points. Each subset of $d$ points is called a line, and a partition of the points into $d$ disjoint lines is called a parallel class. Then, we define the following.

These definitions are slightly non-standard³, but I wanted to emphasize a connection to MUBs which will be detailed soon.

Example

Below is a $2$-net of order 2 and its extension to a 3-net.

2- and 3-nets of order 2

As before, this raises a question: for a fixed order $d$, what is the maximum $k$ for which a $k$-net of order $d$ exists? Denoting this count by $\mathfrak{M}_2(d)$, a simple counting argument reveals that $\mathfrak{M}_2(d) \leq d + 1$. Complete $k$-nets of order $d+1$ are called affine planes and have many applications in finite geometry and coding theory. Again, we have an existence conjecture.

A bit more progress has been made towards this hypothesis; by the Bruck-Ryser-Chowla Theorem, we know that affine planes of order $d$ cannot exist if $d \equiv 1 \text{ or } 2 \pmod{4}$ and is not the sum of two squares (e.g. $d=6$ is forbidden).

A Common Framework

As these similar definitions and conjectures suggest, researchers have discovered many connections between affine planes and complete collections of MUBs (although it’s not clear that these similarities are sufficient to translate existence results)⁴. In our paper, we introduce a common framework so that we can work with MUBs and $k$-nets simultaneously, the basis for which is a finite dimensional $C^*$-algebra. First, we recall the following.

The canonical example of a *-algebra is the field of complex numbers $\mathbb{C}$ under standard multiplication and conjugation. The details of the definition aren’t so important, as we will soon focus on two concrete cases.

A $C^*$-algebra is a *-algebra equipped with a norm obeying certain properties, but we will only use the following.

More details are provided on the Wikipedia page. Now, let’s get to the relevant examples.

Next, we recall a final definition.

This translates cleanly into our two settings.

Finally, we are prepared to introduce our common framework. Let $\mathfrak{U}$ be a $d^2$-dimensional $C^*$-algebra with canonical normalized trace $\tau$. As I’ve hinted at so far, $\mathfrak{U}= M_d(\mathbb{C})$ will correspond to the case of MUBs, and $\mathfrak{U}= \mathbb{C}^X$ will correspond to combinatorial $k$-nets with point set $X$. For a fixed order $d$, we construct our generalization using a collection of familiar components. The first component is a (generalized) line, defined to be an orthogonal projection $P = P^* = P^2 \in \mathfrak{U}$ with $\tau(P) = \frac{1}{d}$.

Next, we define a parallel class to be a set of lines $\mathcal{C} = P_1, \dots, P_d$ with $P_iP_j = 0$ for $i \neq j$.

Then, we say that two parallel classes $\mathcal{C}_1, \mathcal{C}_2$ are mutually unbiased if, for each $P \in \mathcal{C}_1$ and $Q \in \mathcal{C}_2$, $\tau(PQ) = \frac{1}{d^2}$.

Finally, our full generalization is just a collection of mutually unbiased parallel classes.

By our previous examples, it’s clear that systems of MUBs and combinatorial $k$-nets can both be realized in this new setting, and it’s not difficult to show that all generalized $k$-nets over $M_d(\mathbb{C})$ and $\mathbb{C}^X$, $|X|=d^2$, are of this form⁵. Below, we summarize these correspondences.

Shared Properties

One might worry that this framework lacks the power to prove facts of interest. Indeed, if we are seeking to establish a result about $k$-nets over algebras, then we can’t refer to the points of lines (which exist in the commutative case, but not with MUBs), and we can’t exploit the non-commutativity of matrix multiplication. Fortunately, we are still able to reach some interesting conclusions. As a first example, we have the following familiar fact.

It turns out that we can prove even stronger statements. Since complete $k$-nets are of mathematical interest, one motivating question for our paper was the following: how many ways can a $k$-net of order $d$ be extended to a complete $d+1$-net of order $d$? It turns out that for large enough $k$, such a completion is unique, if it exists.

We call this a rigidity result because it implies that any two distinct $(d+1)$-nets of order $d$ must differ by more than $\sqrt{d}$ parallel classes. This mirrors a classical result for combinatorial nets due to Bruck⁶, but with an entirely new proof, and is novel for MUBs. I encourage interested readers to examine its proof in Section 3 of the paper. We later extend this rigidity result to a special class of algebraically constructed MUBs and use this to show that certain large systems of MUBs cannot be completed.

In general, we hope that this framework will help to develop a better understanding of the connections between MUBs and affine planes, perhaps shedding some light onto their respective existence conjectures.