The n-Category Café

Preliminaries

A set of equiangular lines is a set of unit vectors in a $d$-dimensional vector space such that the magnitude of the inner product of any pair is constant:

$$|\langle v_j, v_k\rangle| = \left\{\begin{array}{cc} 1, & j = k; \\ \alpha, & j \neq k. \end{array}\right.$$

The maximum number of equiangular lines in a space of dimension $d$ (the so-called Gerzon bound) is $d(d+1)/2$ for real vector spaces and $d^2$ for complex. In the real case, the Gerzon bound is only known to be attained in dimensions 2, 3, 7 and 23, and we know it can’t be attained in general. If you like peculiar alignments of mathematical topics, the appearance of 7 and 23 might make your ears prick up here. If you made the wild guess that the octonions and the Leech lattice are just around the corner… you’d be absolutely right. Meanwhile, the complex case is of interest for quantum information theory, because a set of $d^2$ equiangular lines in $\mathbb{C}^d$ specifies a measurement that can be performed upon a quantum-mechanical system. These measurements are highly symmetric, in that the lines which specify them are equiangular, and they are “informationally complete” in a sense that quantum theory makes precise. Thus, they are known as SICs. Unlike the real case, where we can only attain the Gerzon bound in a few sparse instances, it appears that a SIC exists for each dimension $d$, but nobody knows for sure yet.

Before SICs became a physics problem, constructions of $d^2$ complex equiangular lines were known for dimensions $d = 2$, 3 and 8. These arose from topics like higher-dimensional polytopes and generalizations thereof. Now, we have exact solutions for SICs in the following dimensions:

$$d = 2-21, 23, 24, 28, 30, 31, 35, 37, 39, 43, 48, 53, 120, 124, 195, 323.$$

Moreover, numerical solutions to high precision are known for the following cases:

$$d = 2-151, 168, 172, 199, 224, 228, 255, 259, 288, 292, 323, 327, 489, 528, 725, 844, 1155, 2208.$$

These lists have grown irregularly in the years since the quantum-information community first recognized the significance of SICs. (Andrew Scott’s contributions deserve particular mention, as he has found many solutions by solo work and in collaboration with Markus Grassl.) It is fair to say that researchers feel that SICs should exist for all integers $d \geq 2$, but we have no proof one way or the other. The attempts to resolve this question have extended into algebraic number theory, an intensely theoretical avenue of research with the surprisingly practical application of converting numerical solutions into exact ones. For additional (extensive) discussion, we refer to a review article and two textbooks, one slanted more to physics and the other towards mathematics.

In what follows, we will focus our attention mostly on the sporadic SICs, which comprise the SICs in dimensions 2 and 3, as well as one set of them in dimension 8. These SICs have been designated “sporadic” because they stand out in several ways, chiefly by residing outside the number-theoretic patterns observed for the rest of the known SICs. After laying down some preliminaries, we will establish a connection between the sporadic SICs and the exceptional Lie algebras $\mathrm{E}_6$, $\mathrm{E}_7$ and $\mathrm{E}_8$ by way of their root systems.

Quantum Measurements and Systems of Lines

The first key point to make is that we will be working with finite-dimensional Hilbert spaces. This is commonplace in quantum computation, where the dimension of your computer scales with your available budget, and is typically expressed in terms of the number of qubits you can operate. With each physical system of interest, we associate a complex Hilbert space; if the system is a collection of $N$ qubits, then the dimension is $d = 2^N$, but we will also consider dimensions that are not powers of 2. Ascribing a quantum state to a system encapsulates our expectations for how that system will behave in all possible experiments. For our purposes in this series, quantum states are positive semidefinite operators on the $d$-dimensional Hilbert space which have unit trace.

A positive-operator-valued measure (POVM) is a set of “effects” (Hermitian operators possessing eigenvalues in the unit interval) that furnish a resolution of the identity:

$$\sum_i E_i = \sum_i w_i \rho_i = I,$$

for some quantum states $\{\rho_i\}$ and weights $\{w_i\}$. Note that taking the trace of both sides gives a normalization constraint for the weights in terms of the dimension of the Hilbert space. In this context, the Born Rule says that when we perform the measurement described by this POVM, we obtain the $i$-th outcome with probability

$$p(i) = tr(\rho w_i \rho_i),$$

where $\rho$ without a subscript denotes our quantum state for the system. The weighting $w_i$ is, up to a constant, the probability we would assign to the $i$-th outcome if our state $\rho$ were the maximally mixed state $\frac{1}{d}I$, the “state of maximal ignorance.”

SICs are a special type of POVM. Given a set of $d^2$ equiangular unit vectors $\{|\pi_i\rangle\} \subset \mathbb{C}^d$, we can construct the operators which project onto them, and in turn we can rescale those projectors to form a set of effects:

$$E_i = \frac{1}{d}\Pi_i,$$

where each $\Pi_i$ is the operator that projects onto the vector $| \pi_i \rangle$. The equiangularity condition on the $\{|\pi_i \rangle\}$ turns out to imply that the $\{\Pi_i\}$ are linearly independent, and thus they span the space of Hermitian operators on $\mathbb{C}^d$. Because the SIC projectors $\{\Pi_i\}$ form a basis for the space of Hermitian operators, we can express any quantum state $\rho$ in terms of its (Hilbert–Schmidt) inner products with them. But, by the Born Rule, the inner product $tr(\rho \Pi_i)$ is, apart from a factor $1/d$, just the probability of obtaining the $i$-th outcome of the SIC measurement $\{E_i\}$. The formula for reconstructing $\rho$ given these probabilities is quite simple, thanks to the symmetry of the projectors:

$$\rho = \sum_i \left[(d+1)p(i) - \frac{1}{d}\right]\Pi_i,$$

where $p(i) = tr(\rho E_i)$ by the Born Rule. This furnishes us with a map from quantum state space into the probability simplex, a map that is one-to-one but not onto. In other words, we can fix a SIC as a “reference measurement” and then transform between quantum states and probability distributions without ambiguity, but the set of valid probability distributions for our reference measurement is a proper subset of the probability simplex.

Because we can treat quantum states as probability distributions, we can apply the concepts and methods of probability theory to them, including Shannon’s theory of information. The structures that I will discuss in the following sections came to my attention thanks to Shannon theory. In particular, the question of recurring interest is, “Out of all the extremal states of quantum state space — i.e., the ‘pure’ states $\rho = |\psi\rangle\langle\psi|$ — which minimize the Shannon entropy of their probabilistic representation?” I will focus on the cases of dimensions 2, 3 and 8, where the so-called sporadic SICs occur. In these cases, the information-theoretic question of minimizing Shannon entropy leads to intricate geometrical structures.

Any time we have a vector in $\mathbb{R}^3$ of length 1 or less, we can map it to a $2 \times 2$ Hermitian matrix by the formula

$$\rho = \frac{1}{2}\left(I + x\sigma_x + y\sigma_y + z\sigma_z\right),$$

where $(x,y,z)$ are the Cartesian components of the vector and $(\sigma_x, \sigma_y, \sigma_z)$ are the Pauli matrices. This yields a positive semidefinite matrix $\rho$ with trace equal to 1; when the vector has length 1, we have $\rho^2 = \rho$, and the matrix is a rank-1 projector that can be written as $\rho = |\psi\rangle\langle\psi|$ for some vector $|\psi\rangle$.

Given any polyhedron of unit radius or less in $\mathbb{R}^3$, we can feed its vertices into the Bloch representation and obtain a set of quantum states (which are pure states if they lie on the surface of the Bloch sphere). For a simple example, we can do a regular tetrahedron. Let $s$ and $s'$ take the values $\pm 1$, and define

$$\rho_{s,s'} = \frac{1}{2}\left(I + \frac{1}{\sqrt{3}}(s\sigma_x + s'\sigma_y + s s' \sigma_z)\right).$$

To make these quantum states into a POVM, scale them down by the dimension. That is, take

$$E_{s,s'} = \frac{1}{2} \rho_{s,s'}.$$

Then, the four operators $E_{s,s'}$ will sum to the identity. In fact, they comprise a SIC.

By introducing a sign change, we can define another SIC,

$$\tilde{\rho}_{s,s'} = \frac{1}{2}\left(I + \frac{1}{\sqrt{3}} (s\sigma_x + s'\sigma_y - s s' \sigma_z)\right).$$

Each state in the original SIC is orthogonal to exactly one state in the second. In the Bloch sphere representation, orthogonal states correspond to antipodal points, so taking the four points that are antipodal to the vertices of our original tetrahedron forms a second tetrahedron. Together, the states of the two SICs form a cube inscribed in the Bloch sphere.

Here we have our first appearance of Shannon theory entering the story. With respect to the original SIC, the states $\{|\tilde{\pi}_i \rangle\}$ of the antipodal SIC all minimize the Shannon entropy. The two interlocking tetrahedra are, entropically speaking, dual structures.

Next time, we will move up from dimension 2 to dimension 3, and we’ll see how $\mathrm{E}_6$ enters the story.