## February 20, 2019

### Sporadic SICs and Exceptional Lie Algebras I

#### Posted by John Baez guest post by Blake C. Stacey

Sometimes, mathematical oddities crowd in upon one another, and the exceptions to one classification scheme reveal themselves as fellow-travelers with the exceptions to a quite different taxonomy. I am grateful to John for giving me the opportunity to discuss one such confluence, where quantum information theory comes together with geometry, root systems and even the octonions.

In what follows, I will be divvying up these notes into blog posts. The first step is to introduce the geometrical statement of the SIC problem. Then, we’ll establish some basics about quantum theory, which may be fairly standard if you learned out of Mike and Ike while being a little un-standard outside of quantum information.

### 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.

Posted at February 20, 2019 5:21 AM UTC

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### Re: Sporadic SICs and Exceptional Lie Algebras I

Nice! I didn’t get what, exactly, is so good about getting exactly $n^2$ equiangular lines in $n$ dimensions. But I can guess: if we have $n^2$ equiangular lines in $\mathbb{C}^n$, and we pick unit vectors $\psi_i$, one in each line, we can hope to know any $n \times n$ self-adjoint operator $A : \mathbb{C}^n \to \mathbb{C}^n$ if we know all the numbers

$\langle \psi_i, A \psi_i \rangle$

because the space of such operators has dimension $n^2$.

Is that the idea? Is that what ‘informationally complete’ means? If we have $n^2$ equiangular lines, can we always recover any self-adjoint operator from the numbers $\langle \psi_i, A \psi_i \rangle$?

Posted by: John Baez on February 21, 2019 10:18 PM | Permalink | Reply to this

### Re: Sporadic SICs and Exceptional Lie Algebras I

I guess maybe you said this in the post, implicitly. For those of us who are too dense for density operators, my explanation might help.

Posted by: John Baez on February 21, 2019 10:20 PM | Permalink | Reply to this

### Re: Sporadic SICs and Exceptional Lie Algebras I

Yes, if you have $n^2$ lines, you can reconstruct an arbitrary self-adjoint operator $A$ from the overlaps $tr(A | \psi_i \rangle \langle \psi_i |) = \langle \psi_i | A | \psi_i \rangle$. You don’t need equiangularity for informational completeness, just that the operators which project onto your lines are linearly independent in the space of operators. But equiangularity implies the linear independence of those operators, and it makes the formula for reconstructing $A$ from the overlaps particularly clean.

Posted by: Blake Stacey on February 22, 2019 12:53 AM | Permalink | Reply to this

### Re: Sporadic SICs and Exceptional Lie Algebras I

It looks like the list of dimensions I wrote in the post above is already obsolete!

WH SICs have been constructed numerically in every dimension up to 181, and in many other dimensions up to 2208, while solutions have been constructed in every dimension up to 21 and in many other dimensions up to 1299 (in both cases, numerical and exact, this listing includes dimensions still unpublished).

Posted by: Blake Stacey on March 19, 2019 3:33 PM | Permalink | Reply to this

### Re: Sporadic SICs and Exceptional Lie Algebras I

I was going through my archive of references, and I found that one of them was a Google+ post that John had made back in May 2013, and which isn’t available any more now that Google+ is defunct. It’s not even on the Internet Archive, as far as I can tell! Since I never meta-joke that I didn’t like, it seemed like a good idea to put that text back online here as a guest comment on the guest post that John very kindly let me write.

Here’s the hecatonicosihexapentacosiheptacontihexaexon, more commonly called the 7-dimensional Gosset polytope. A polytope is a higher-dimensional generalization of a polyhedron. This is the most symmetrical one in 7 dimensions!

It has 56 corners, but you can only see 55 here, since it’s been projected down to a plane, and the orange one in the middle is directly in front of another. If you draw lines between the opposite corners you get 28 equiangular lines: they’re all at equal angles to each other. That’s the most equiangular lines you can get in 7 dimensions… and also in 8, 9, 10, 11, 12, or 13 dimensions! So, it’s very nice collection of lines.

I’m happy because I asked two hard questions on G+ yesterday, and I got good answers to both. First: why are there 28 equiangular lines in 7 dimensions? +Philip Gibbs gave me the crucial clue I needed to get this nice answer. Second: is it worthwhile making a day trip from Urumqi to Turpan if you don’t have much time to explore the far west of China? +Dalibor Smid settled that one: yes!

This 7-dimensional polytope was discovered by Thorold Gosset. Gosset was a student at Pembroke College in Cambridge, and then went on to get a law degree in 1895. When he started he had no clients and so - being a very shrewd and practical fellow - he decided to classify all the regular polytopes in higher dimensions. After succeeding in this (which people had already done), he tried to classify the semiregular polytopes, which have regular polytopes as faces and are so symmetrical that every corner looks alike. In 3 dimensions these are well-known and beautiful things, studied already by Archimedes. But Gosset discovered that in 6, 7 and 8 dimensions there are “exceptional” semiregular polytopes that you’d never expect from lower dimensions. There are none of these in any higher dimension.

Gosset couldn’t get any mathematicians interested in his work, so he quit this hobby, started practicing law… and lived until 1962.

The polytopes he discovered turned out to be related to symmetry groups called E6, E7 and E8. E8 is the “king” - the biggest and best - and it contains the other two inside it. I have wasted many hours happily studying these symmetry groups and related structures. It’s fun, because at first it seems impossible to understand them… but in fact you can.

To mathematicians, the nicest description of the 7-dimensional Gosset polytope is that its corners are the weights of the smallest irreducible representation of E7. But if that jargon means nothing to you, just take the vectors

$(3, 3, -1, -1, -1, -1, -1, -1)$

and

$(-3, -3, 1, 1, 1, 1, 1, 1)$

and permute their coordinates in all possible ways. You’ll get 56 vectors in 8 dimensions, but they all point at right angles to the vector

$(1, 1, 1, 1, 1, 1, 1, 1)$

so they live in a 7-dimensional space… and if you project them down to a pathetic little 2-dimensional plane, you get this shape here!

You can take this 7-dimensional polytope and truncate it in various ways to get 127 different polytopes that are all just as symmetrical. Tom Ruen has drawn pictures of ten of them, here:

http://en.wikipedia.org/wiki/List_of_E7_polytopes

For more on this particular one, see:

http://en.wikipedia.org/wiki/3_21_polytope

Coxeter called it the $3_{21}$ polytope, but someone has more whimsically called it the hecatonicosihexapentacosiheptacontihexaexon. Richard Green

The polytope $3_{21}$ is a fascinating object, and I devoted a huge chunk of my book to it. It is also intimately connected to the 28 bitangents to a plane quartic curve.

John Baez

WOW! When Tobias Fritz asked me why there were 28 equiangular lines in 7 dimensions, the numerologist in me instantly thought about those 28 bitangents. However, since I don’t understand those 28 bitangents, and I don’t see why they’re related to 7-dimensional geometry this didn’t help me. It took a kick in the butt from Philip Gibbs to make me notice that 28 lines means 56 unit vectors… and since I knew E7’s smallest rep is 56-dimensional, everything fell into place.

What’s your book about it, and (briefly) how is $3_{21}$ related to those 28 bitangents? I need to understand those, and also the 27 lines on the cubic surface. Algebraic geometry is one of my weaker areas….

I should add that the numerologist in me also instantly thought about the 28 smooth structures on the 7-sphere, but this seemed even less useful.

Richard Green

+John Baez, my book is about minuscule representations. I consider the jewel in the crown of these to be the 56 dimensional representation you mentioned, and I say so in the introduction to the book.

There isn’t any substantial algebraic geometry in my book, but the basic idea is as follows. It’s clear from the coordinates you give (which are also the ones I use; I found them in a paper of du Val) that the weights are closed under negation. The 28 positive-negative pairs can be identified with the 28 bitangents. The Weyl group of type E7 acts as a rank 4 permutation group on the 56 weights, and acts doubly transitively on the 28 bitangents. It has two orbits on triples of bitangents, which are called syzygetic and azygetic or asyzygetic in the literature. I could go on, but it would take two whole chapters.

[…]

Oh, and another thing: all this is very closely tied in with the combinatorics of Del Pezzo surfaces. The work of du Val that I mentioned above is very interested in this aspect of the polytope.

[…]

There’s a lot about the 27 lines too. The basic idea here is to take the point stabilizer of the action of W(E7) on the 28 bitangents. The azygetic/syzygetic split I mentioned then corresponds to the relations of being skew or incident among the 27 lines.

Richard Green’s book is Combinatorics of Minuscule Representations (Cambridge University Press, 2013).

Posted by: Blake Stacey on January 2, 2020 3:20 AM | Permalink | Reply to this

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