## February 28, 2019

### Sporadic SICs and Exceptional Lie Algebras II

#### Posted by John Baez

guest post by Blake C. Stacey

Today, we carry forward with the project we began last week: exploring Symmetric Informationally Complete quantum measurements, otherwise known as SICs. They’re really just maximal sets of equiangular lines in a complex vector space!

In our first post, we laid the groundwork and studied one example: a set of four equiangular lines in $\mathbb{C}^2$. Now, we move up a dimension and investigate a set of 9 equiangular lines in $\mathbb{C}^3$. This will bring the exceptional Lie algebras into the narrative, and we’ll also get a chance to greet a biodiversity measure and a polytope known as the 24-cell.

Last time, we introduced the idea of a “reference measurement”: a measurement that lets us write our quantum states for a system as probability distributions. Mathematically speaking, such a measurement is a positive-operator-valued measure whose elements form a basis for the space of Hermitian operators on a finite-dimensional Hilbert space. Because the elements form a basis, we can represent any Hermitian operator as a list of its inner products with the basis elements. The textbook rules of quantum theory tell us that, if we ascribe the quantum state $\rho$ to our system of interest, then the probability of obtaining the $i$-th outcome of the measurement $\{E _i \}$ is $p(i) = tr(\rho E_i)$. So, if our operators $\{E_i\}$ form a basis, then we can replace the matrix of complex numbers $\rho$ with the vector of nonnegative real numbers $p$. In order to be “informationally complete” like this, a POVM must have at least $d^2$ elements, where $d$ is the dimension of our system’s Hilbert space. We focused our attention on the symmetric informationally complete POVMs, or SICs, which have exactly $d^2$ elements. We defined a SIC as a set of $d^2$ unit vectors $|\pi_j\rangle$ that have pairwise inner products of equal magnitude:

$|\langle \pi_j | \pi_k \rangle|^2 = \frac{d \delta_{j k} + 1}{d+1}.$

The projectors onto these lines comprise a POVM when scaled appropriately:

$E_j = \frac{1}{d} | \pi_j \rangle\langle \pi_j |.$

Fixing a SIC thus establishes a mapping from the set of quantum states into the ($d^2 - 1)$-dimensional probability simplex. If $\rho$ and $\sigma$ are two quantum states whose probabilistic representations are $p_\rho$ and $p_\sigma$, then the Hilbert–Schmidt inner product of these states is simply related to the Euclidean inner product of the probability vectors:

$tr(\rho \sigma) = d(d+1)\sum_{i=1}^{d^2} p_\rho(i) p_\sigma(i) - 1.$

From this, it follows that

$\frac{1}{d(d+1)} \leq p_\rho \cdot p_\sigma \leq \frac{2}{d(d+1)}.$

The upper bound is attained if and only if $\rho = \sigma$ and $tr \rho^2 = 1$, which happens when $\rho$ is a “pure state”, i.e., $\rho = | \psi\rangle\langle \psi |$ for some vector $|\psi\rangle$. The set of all quantum states for Hilbert-space dimension $d$ is the convex hull of the pure states. Intriguingly, once we use a SIC to write quantum states as probabilities, then the question of how close a state is to the boundary of state space is a matter of diversity! The effective number of possible outcomes for a SIC measurement given a pure state $p_\psi$ is the pleasingly combinatorial quantity

$\left( p_\psi \cdot p_\psi \right)^{-1} = \frac{d(d+1)}{2}.$

Today, we will bring the Weyl group of the $\mathrm{E}_6$ root system into the picture. There will be a slight amount of nomenclatural turbulence, since Hermann Weyl’s name will be associated with two different kinds of group, but that’s just the risk you run in this business.

Our primary reference for the next bit will be H. S. M. Coxeter’s Regular Complex Polytopes (Cambridge University Press, 1991). Coxeter devotes a goodly portion of chapter 12 to the Hessian polyhedron, which lives in $\mathbb{C}^3$ and has 27 vertices. These 27 vertices lie on nine diameters in sets of three apiece. (In a real vector space, only two vertices of a convex polyhedron can lie on a diameter. But in a complex vector space, where a diameter is a complex line through the center of the polyhedron, we can have more.) He calls the polyhedron “Hessian” because its nine diameters and twelve planes of symmetry interlock in a particular way. Their incidences reproduce the Hesse configuration, a set of nine points on twelve lines such that four lines pass through each point and three points lie on each line.

Coxeter writes the 27 vertices of the Hessian polyhedron explicitly, in the following way. First, let $\omega$ be a cube root of unity, $\omega = e^{2\pi i / 3}$. Then, construct the complex vectors

$(0, \omega^\mu, -\omega^\nu),\ (-\omega^\nu, 0, \omega^\mu), \ (\omega^\mu, -\omega^\nu, 0),$

where $\mu$ and $\nu$ range over the values 0, 1 and 2. As Coxeter notes, we could just as well let $\mu$ and $\nu$ range over 1, 2 and 3. He prefers this latter choice, because it invites a nice notation: We can write the vectors above as

$0\mu\nu,\ \nu0\mu,\ \mu\nu0.$

For example,

$230 = (\omega^2, -1, 0),$

and

$103 = (-\omega, 0, 1).$

Coxeter then points out that this notation was first introduced by Beniamino Segre, “as a notation for the 27 lines on a general cubic surface in complex projective 3-space. In that notation, two of the lines intersect if their symbols agree in just one place, but two of the lines are skew if their symbols agree in two places or nowhere.” Consequently, the 27 vertices of the Hessian polyhedron correspond to the 27 lines on a cubic surface “in such a way that two of the lines are intersecting or skew according as the corresponding vertices are non-adjacent or adjacent.”

Casting the Hessian polyhedron into the real space $\mathbb{R}^6$, we obtain the polytope known as $2_{21}$, which is related to $\mathrm{E}_6$, since the Coxeter group of $2_{21}$ is the Weyl group of $\mathrm{E}_6$. The Weyl group of $\mathrm{E}_6$ can also be thought of as the automorphism group of the 27 lines on a cubic surface.

We make the connection to symmetric quantum measurements by following the trick that Coxeter uses in his Eq. (12.39). We transition from the space $\mathbb{C}^3$ to the complex projective plane by collecting the 27 vertices into equivalence classes, which we can write in homogeneous coordinates as follows:

$\begin{array}{ccc} (0, 1, -1), & (-1, 0, 1), & (1, -1, 0) \\ (0, 1, -\omega), & (-\omega, 0, 1), & (1, -\omega, 0) \\ (0, 1, -\omega^2), & (-\omega^2, 0, 1), & (1, -\omega^2, 0) \end{array}$

Let $u$ and $v$ be any two of these vectors. We find that

$|\langle u, u\rangle|^2 = 4$

when the vectors coincide, and

$|\langle u, v\rangle|^2 = 1$

when $u$ and $v$ are distinct. We can normalize these vectors to be quantum states on a three-dimensional Hilbert space by dividing each vector by $\sqrt{2}$.

We have found a SIC for $d = 3$. When properly normalized, Coxeter’s vectors furnish a set of $d^2 = 9$ pure quantum states, such that the magnitude squared of the inner product between any two distinct states is $1/(d+1) = 1/4$.

Every known SIC has a group covariance property. Talking in terms of projectors, a SIC is a set of $d^2$ rank-1 projectors $\{\Pi_j\}$ on a $d$-dimensional Hilbert space that satisfy the Hilbert–Schmidt inner product condition

$tr (\Pi_j \Pi_k) = \frac{d\delta_{j k} + 1}{d+1}.$

These form a POVM if we rescale them by $1/d$. In every known case, we can compute all the projectors $\{\Pi_j\}$ by starting with one projector, call it $\Pi_0$, and then taking the orbit of $\Pi_0$ under the action of some group. The projector $\Pi_0$ is known as the fiducial state. (I don’t know who picked the word “fiducial”; I think it was something Carl Caves decided on, way back.)

In all known cases but one, the group is the Weyl–Heisenberg group in dimension $d$. (Don’t worry, octonion fans! We’ll be getting to the exception soon.) To define this group, fix an orthonormal basis $\{|n\rangle\}$ and define the operators $X$ and $Z$ such that

$X|n\rangle = |n+1\rangle,$

interpreting addition modulo $d$, and

$Z|n\rangle = e^{2\pi i n / d} |n\rangle.$

The Weyl–Heisenberg displacement operators are

$D_{l\alpha} = (-e^{i\pi / d})^{l\alpha} X^l Z^\alpha.$

Because the product of two displacement operators is another displacement operator, up to a phase factor, we can make them into a group by inventing group elements that are displacement operators multiplied by phase factors. This group has Weyl’s name attached to it, because he invented $X$ and $Z$ back in 1925, while trying to figure out what the analogue of the canonical commutation relation would be for quantum mechanics on finite-dimensional Hilbert spaces. It is also called the generalized Pauli group, because $X$ and $Z$ generalize the Pauli matrices $\sigma_x$ and $\sigma_z$ to higher dimensions (at the expense of no longer being Hermitian).

To relate this with the Coxeter construction we discussed earlier, turn the first of Coxeter’s vectors into a column vector:

$\left(\begin{array}{r} 0 \\ 1 \\ -1\end{array}\right).$

Apply the $X$ operator twice in succession to get the other two vectors in Coxeter’s table (converted to column-vector format). Then, apply $Z$ twice in succession to recover the right-hand column of Coxeter’s table. Finally, apply $X$ to these vectors again to effect cyclic shifts and fill out the table. This set of nine vectors is known as the Hesse SIC.

Each of the 27 lines corresponds to a weight in the minimal representation of $\mathrm{E}_6$, and so each element in the Hesse SIC corresponds to three weights of $\mathrm{E}_6$.

In dimension $d = 3$, we encounter a veritable cat’s cradle of vectors. First, there’s the Hesse SIC. Like all informationally complete POVMs, it defines a probabilistic representation of quantum state space, in this case mapping from $3 \times 3$ density matrices to the probability simplex for 9-outcome experiments. As we did last time for the qubit case, we can look for the pure states whose probabilistic representations minimize the Shannon entropy. The result is a set of twelve states, which sort themselves into four orthonormal bases of three states apiece. What’s more, these bases are mutually unbiased: The Hilbert–Schmidt inner product of a state from one basis with any state from another is always constant. In a sense, the Hesse SIC has a “dual” structure, and that dual is a set of Mutually Unbiased Bases (MUB). This duality relation is rather intricate: Each of the 9 SIC states is orthogonal to exactly 4 of the MUB states, and each of the MUB states is orthogonal to exactly 3 SIC states.

An easy way to remember these relationships is to consider the Hesse configuration that we mentioned earlier. This configuration is also known as the discrete affine plane on nine points, and as the Steiner triple system of order 3. That’s a lot of different names for something which is pretty easy to put together! To construct it, first draw a $3 \times 3$ grid of points, and label them consecutively:

$\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}$

These will be the points of our discrete geometry. To obtain the lines, we read along the horizontals, the verticals and the leftward and rightward diagonals:

$\begin{array}{ccc} (123) & (456) & (789) \\ (147) & (258) & (369) \\ (159) & (267) & (348) \\ (168) & (249) & (357) \end{array}$

Each point lies on four lines, and every two lines intersect in exactly one point. For our purposes today, each of the points corresponds to a SIC vector, and each of the lines correponds to a MUB vector, with point-line incidence implying orthogonality. The four bases are the four ways of carving up the plane into parallel lines (horizontals, verticals, diagonals and other diagonals).

To construct a MUB vector, pick one of the 12 lines we constructed above, and insert zeroes into those slots of a 9-entry probability distribution, filling in the rest uniformly. For example, picking the line $(123)$, we construct the probability distribution

$\left(0,0,0,\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6} \right).$

This represents a pure quantum state that is orthogonal to the quantum state

$\left(\frac{1}{6},\frac{1}{6},\frac{1}{6},0,0,0,\frac{1}{6},\frac{1}{6},\frac{1}{6} \right)$

and to

$\left(\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},0,0,0 \right),$

while all three of these have the same Hilbert–Schmidt inner product with the quantum state represented by

$\left(0,\frac{1}{6},\frac{1}{6},0,\frac{1}{6},\frac{1}{6},0,\frac{1}{6},\frac{1}{6} \right),$

for example.

Considering all the lines in the original structure that are orthogonal to a given line in the dual yields a maximal set of real equiangular lines in one fewer dimensions. (Oddly, I noticed this happening up in dimension 8 before I thought to check in dimension 3, but we’ll get to that soon.) To visualize the step from $\mathbb{C}^3$ to $\mathbb{R}^2$, we can use the Bloch sphere representation for two-dimensional quantum state space. Pick a state in the dual structure, i.e., one of the twelve MUB vectors. All the SIC vectors that are orthogonal to it must crowd into a 2-dimensional subspace. In other words, they all fit into a qubit-sized state space, and we can draw them on the Bloch sphere. When we do so, they are coplanar and lie at equal intervals around a great circle, a configuration sometimes called a trine. This configuration is a maximal equiangular set of lines in the plane $\mathbb{R}^2$.

What happens if, starting with the Hesse SIC, you instead consider all the lines in the dual structure that are orthogonal to a given vector in the original? This yields a SIC in dimension 2. I don’t know where in the literature that is written, but if feels like something Coxeter would have known.

All of this ends up being significant for quantum computation and for a no-go theorem about hidden variables, but right now, the pertinent point is how the study of exceptional Lie groups is starting to enter our tale of quantum measurements.

And we’ve barely scratched the surface! Here’s something else to ponder. Let’s consider the stabilizer group of the Hesse fiducial. That is, let’s take the vector

$\frac{1}{\sqrt{2}} \left(\begin{array}{r} 0 \\ 1 \\ -1 \end{array}\right)$

and consider those unitary operations which hold it fixed while permuting the other 8 vectors in the Hesse SIC among themselves. This is a subgroup of the symmetry group of the Hesse SIC, and it is known to be isomorphic to $SL(2, \mathbb{F}_3)$, the group of unit-determinant $2 \times 2$ matrices over the three-element finite field. But this group has other names, too: It is isomorphic to the binary tetrahedral group, which we get by taking the vertices of the 24-cell and multiplying them together as quaternions.

Another path from the sporadic SICs to $\mathrm{E}_6$ starts with the qubit SICs, i.e., regular tetrahedra inscribed in the Bloch sphere. Shrinking a tetrahedron, pulling its vertices closer to the origin, yields a type of quantum measurement (sometimes designated a SIM) that has more intrinsic noise. Apparently, $\mathrm{E}_6$ is part of the story of what happens when the noise level becomes maximal and the four outcomes of the measurement merge into a single degenerate case. This corresponds to a singularity in the space of all rotated and scaled tetrahedra centered at the origin. Resolving this singularity turns out to involve the Dynkin diagram of $\mathrm{E}_6$: We invent a smooth manifold that maps to the space of tetrahedra, by a mapping that is one-to-one and onto everywhere except the origin. The pre-image of the origin in this smooth manifold is a set of six spheres, and two spheres intersect if and only if the corresponding vertices in the Dynkin diagram are connected.

This observation prompted this series of blog posts, but I’m still not sure how it’s connected to the other things we’re exploring!

Posted at February 28, 2019 10:00 AM UTC

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

Cool stuff!

I don’t know who picked the word “fiducial”; I think it was something Carl Caves decided on, way back.

In general relativity there are lots of “gauge symmetries” in the physical sense: symmetries that permute different mathematical ways of describing the same physical thing. Often you need to pick a representative of an orbit under these symmetries: that is, pick a particular mathematical description, even though there are plenty of others that are equally good at describing the same physical thing. And people often call this arbitrarily chosen representative “fiducial”. I especially think of Ashtekar when I hear this word, because he used it more than most folks I know in this business.

Posted by: John Baez on February 28, 2019 9:28 PM | Permalink | Reply to this

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

Interesting! Caves started in gravity (his PhD advisor was Kip Thorne), so maybe that’s the tradition where the term comes from. It is rather analogous: Thanks to the group covariance, any vector in a SIC can be used as the representative from which all the others are generated, although when we speak of “the fiducial” we generally mean the one that is distinguished in some informal way, like being easiest to remember.

Posted by: Blake Stacey on March 1, 2019 5:44 PM | Permalink | Reply to this

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

Maybe this was stated explicitly, but I want to check: if we take these 27 vectors in C^3, or let me say R^6, can we identify their Z-span with E_6’s weight lattice, and the vectors themselves with the weights of the 27-rep?

Posted by: Allen Knutson on March 1, 2019 12:37 AM | Permalink | Reply to this

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

That sounds spot-on. To quote a fascinating paper by Manivel,

Gosset seems to have been the first, at the very beginning of the 20th century, to understand that the lines on the cubic surface can be interpreted as the vertices of a polytope, whose symmetry group is precisely the automorphism group of the configuration. Coxeter extended this observation to the 28 bitangents, and Todd to the 120 tritangent planes. Du Val and Coxeter provided systematic ways to construct the polytopes, which are denoted $n_{21}$ for $n = 2, 3, 4$ and live in $n + 4$ dimensions. They have the characteristic property of being semiregular, which means that the automorphism group acts transitively on the vertices, and the faces are regular polytopes. In terms of Lie theory they are best understood as the polytopes in the weight lattices of the exceptional simple Lie algebras $\mathfrak{e}_{n+4}$, whose vertices are the weights of the minimal representations.

Here, we’ve taken $n = 2$.

Posted by: Blake Stacey on March 1, 2019 5:41 PM | Permalink | Reply to this

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

Nice post! Not having perused your links about the 27 lines on a cubic surface stuff, I’m not sure if this is a meaningful question, but you say that the 27 vertices of the Hessian polyhedron correspond to the 27 lines on a cubic surface in the way that Coxeter wrote. Is this a unique cubic surface? Which one? Does it “look” like something?

Posted by: John DeBrota on March 1, 2019 5:13 PM | Permalink | Reply to this

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

Good question! Every smooth cubic surface in the complex projective space $\mathbb{C}P^3$ has exactly 27 lines that can be drawn on it. For a special case, the Clebsch surface, we can actually get a real surface (in $\mathbb{R}P^3$) that we can mold in plaster:

Or animate on a computer:

Apparently — and this is not at all a thing I understand deeply — the coefficients for the lines on the Clebsch surface live in the “golden field” $\mathbb{Q}(\sqrt{5})$, which we will meet again in this series.

Posted by: Blake Stacey on March 1, 2019 6:45 PM | Permalink | Reply to this

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

I finally made good on that remark about $\mathbb{Q}(\sqrt{5})$, here.

Posted by: Blake Stacey on April 15, 2019 11:07 PM | Permalink | Reply to this

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

Excellent.

For E6(-26) I found it quite elegant to define octonionic qutrits and have E6(-26) act as a SLOCC symmetry group (i.e., invertible local transformations on the qutrits). F4 would provide the ordinary unitary transformations.

It appears your MUB analysis can be generalized for these octonionic qutrits. The caveat is that O^3 states must first be mapped to rank 1 projectors (points of OP^2 whose matrices satisfy P x P=0), due to nonassociativity, as you are well aware. After mapping to rank 1 projectors one can take these as eigenmatrices for the Jordan algebraic eigenvalue problem (of T. Dray et al.). It then can be proved that the corresponding eigenvalues are real using the Frobenius norm induced by the inner product tr(XoY), for X,Y in J3O.

I revisited this recently in considering the dyonic states that can be built from FTS elements (over J3O) where the quantum computation becomes topological.

Posted by: Metatron on March 2, 2019 7:37 AM | Permalink | Reply to this

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

That’s intriguing. It is possible to construct the analogue of a SIC for an “octonionic qutrit”, i.e., a set of equiangular lines in the octonionic projective plane that saturate the Gerzon bound there (which equals 27). Cohn, Kumar and Minton give a non-constructive proof that such an equiangular set exists. In their supplemental code, they present the numerical solution they found, which is a mess of coefficients. They do, however, give a solution for a complete set of octonionic MUB that generalizes the qutrit set in a fairly straightforward way. I spent an evening trying to follow that lead and make an exact octonionic version of the Hesse SIC, but without any luck.

Posted by: Blake Stacey on March 2, 2019 4:06 PM | Permalink | Reply to this

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

Thanks for the reference. After Theorem 5.9 the 819-point tight 5-design is mentioned. On pg. 14 it is stated that such a 819-point universal optimum can be thought of informally as the 196560 minimal Leech lattice vectors modulo the 240 roots of $E_8$.

This ties in with Wilson’s construction of the Leech lattice with integral octonions

Conway’s group and octonions

where the Conway group $Co_0$ is shown to be generated by 3x3 (invertible) matrices over the integral octonions.

Indeed the 196560 norm four vectors, as Wilson demonstrates, can be constructed from 3x240x(1+16+16x16)=196560 $\mathbb{O}^3$ elements. Modulo the roots of $E_8$, 3x(1+16+16x16)=3*273=819, as required by the 5-design.

How can we interpret this? As remarked, there are three affine charts for $\mathbb{OP}^2$. The 196560 states are mapped to each chart as 3x65520. This can be imagined as a mapping from $S^{23}$ to (integral) $\mathbb{OP}^2$, that must be carefully performed chart wise as $S^{15}\rightarrow \mathbb{OP}^1\sim S^8$ Hopf fibration, due to nonassociativity. The Conway group $Co_0=2Co_1$ acts on these “octonionic quantum states”, but not projectively. To see the projective action one must go up to the Monster group that acts on (vertex operators assigned to) such vectors modulo antipodal equivalence, i.e., $v\sim (-v)$, hence on 196560/2=98280 projective states.

The Monster also acts on states assigned to 3x32768=98304, where 32768 encodes 15-dimensional cohomology (from $S^{15}$), and 300 remaining vectors from the space of 24x24 symmetric matrices or equivalently, the adjoint of $so(25)$. Together one recovers 98280+98304+300=196884, as required for the Griess algebra as the degree two piece of the monster vertex algebra.

In a sense, the Monster appears to exist as a symmetry of (integral) octonionic qutrits, due to its projective action on 196560 states. It would thus be able to “see” the 819-point tight 5-design, projectively. Maybe you have some ideas on how to elaborate on this correspondence.

Posted by: Metatron on March 3, 2019 5:29 PM | Permalink | Reply to this

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

I had hoped to touch on the Conway groups, if not the Monster, by the end of this series, so I’ll definitely be thinking about all this. It might take me a while, though — I’m at the APS March Meeting this week. On Thursday afternoon, I’ll be talking about SICs.

Posted by: Blake Stacey on March 4, 2019 1:16 PM | Permalink | Reply to this

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

I am being even slower than expected in replying to this question and preparing the next installment in this series, because after the APS March Meeting, I participated in a philosophy-of-physics workshop over the weekend, and it is very draining to watch people go into a room and talk past each other.

It occurred to me that my mention of the Conway groups in my previous comment was perhaps needlessly cryptic, so I can at least elaborate upon that. The story begins, as one might guess, with the Leech lattice. Conway constructed the maximal set of equiangular lines in $\mathbb{R}^{23}$ by starting from the Leech lattice in $\mathbb{R}^{24}$ and extracting a set of diagonals that saturates the Gerzon bound. In other words, you can get $23 \times 24 / 2 = 276$ equiangular lines by studying the Leech lattice sufficiently carefully. Cohn and Kumar (2007) explain how: “Choose any $w \in \Lambda_{24}$ [the Leech lattice] with $|w|^2 = 6$. Then the 552 points are the vectors $v \in \Lambda_{24}$ satisfying $|v|^2 = |w-v|^2 = 4$. These points all lie on a hyperplane, but it does not pass through the origin, so subtract $w/2$ from each vector to obtain points on a sphere centered at the origin.” Pairs of opposing points then yield the desired equiangular lines. The automorphism group of this set of lines is Conway’s third group, $Co_3$. The stabilizer of a line in this set — that is, the subset of the symmetry group which holds a chosen line fixed and permutes the others — is the McLaughlin group. Furthermore, you can find a maximal set of equiangular lines one dimension down, in $\mathbb{R}^{22}$, by taking a subset of the 276 lines in $\mathbb{R}^{23}$ that are all orthogonal to a common vector. This set contains 176 lines, and its automorphism group is the Higman–Sims group. See Lemmens and Seidel (1973).

Moreover, Gillespie (2018) gives a method for selecting a subset of the 276 lines that yields the Mathieu group $M_{23}$.

Note that all of this has been done with real equiangular lines, not with the complex version of the problem, i.e., the SICs. The question then arises rather naturally of whether equiangular lines in $\mathbb{C}^d$ also relate to the sporadic simple groups. The answer, I think, is potentially. A close investigation of the next sporadic SIC in our journey, the “Hoggar lines” in $\mathbb{C}^8$, turns out to involve common themes with the Hall–Janko and Rudvalis groups. I frankly confess a vast ignorance of this subject; physicist school did not prepare me for the classification of finite simple groups, and it is hard to learn out of books the tacit knowledge of how things are actually investigated and proven. My sense is that the sporadic SICs, the Leech lattice, the Hall–Janko group and the Rudvalis pariah all belong in the same chapter of some unwritten textbook, even though I have no specific theorems to provide the detailed connections.

On that note, a question. You write,

The Conway group $Co_0=2Co_1$ acts on these “octonionic quantum states”, but not projectively. To see the projective action one must go up to the Monster group that acts on (vertex operators assigned to) such vectors modulo antipodal equivalence, i.e., $v\sim (-v)$, hence on 196560/2=98280 projective states.

Is it clear why it’s the Monster group to which we must go up? Is that just because the Monster is “next in line”, as it were, one step above $Co_1$ in the Happy Family?

Posted by: Blake Stacey on March 11, 2019 8:31 PM | Permalink | Reply to this

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

I participated in a philosophy-of-physics workshop over the weekend, and it is very draining to watch people go into a room and talk past each other.

This is exactly what I’ve always assumed a philosophy-of-X workshop would be like, for any value of X.

Posted by: Mark Meckes on March 12, 2019 3:02 PM | Permalink | Reply to this

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

Just for the fun of it, here is a bit from Broom and Waldron (2013), where they bring together the Monster group and symmetric sets of lines:

Let $\rho$ be the irreducible representation of $\mathbb{M}$ in $d = 196\,883$ (the smallest nontrivial dimension). There is a largest conjugacy class of elements of order $2, 3, 4$ in $\mathbb{M}$, which in the ATLAS notation are labelled $2A, 3A, 4A$. For any element $a$ in one of these three classes the centraliser of $a$ in $\mathbb{M}$ fixes a unique vector $v_a$ (called the axis of $a$) under the action of the $196\,883$-representation (see [9, §14]). Thus the orbit of under the action given by $\rho$ is a highly symmetric tight frame of $n = |\mathbb{M}|/|C_G(a)|$ vectors for $\mathbb{C}^{196\,883}$.

A “tight frame” in frame theory is a POVM in quantum information. That is, finding the projector $|v_i\rangle\langle v_i|$ onto each vector $v_i$ in a tight frame and adding them up, we get an operator that is proportional to the identity. The smallest of the three frames that Broome and Waldron define has

$n_{2A} = 97\, 239\, 461\, 142\, 009\, 186\, 000$

vectors! As they say,

One can only speculate how many angles these frames might have.

Posted by: Blake Stacey on April 25, 2020 6:25 PM | Permalink | Reply to this

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

I’m not sure whether trying to learn about the sporadic simple groups while under quarantine is the best idea or the worst, but I’m giving it a shot.

I still don’t really know the answer to my previous question, but possibly, the way to think about it is a theorem that if $G$ is a finite group that has a centralizer $2^{1+24} \cdot Co_1$, then $G$ is the Monster (Smith (1979); quoted in SPLAG, p. 564).

Interestingly (to me), we can also relate the Leech lattice to “octonionic qutrits” in a different way than sketched above. We can embed the Leech lattice in the traceless part of the exceptional Jordan algebra. Instead of making the lattice vectors into quantum states, this is like making them into Hamiltonians, because we’re turning each one into a $3 \times 3$ self-adjoint matrix with zero trace. In ordinary quantum mechanics, the generators of time evolutions are self-adjoint operators (or skew-adjoint, depending on convention), but that only works when we try do to quantum mechanics over $\mathbb{C}$. However, we can avail ourselves of a decomposition for the derivations of the Jordan algebra $\mathfrak{h}_3(\mathbb{K})$:

$\mathfrak{der}(\mathfrak{h}_3(\mathbb{K})) \simeq \mathfrak{der}(\mathbb{K}) \oplus \mathfrak{sa}_3(\mathbb{K}),$

which for the case we care about now gives

$\mathfrak{f}_4 \simeq \mathfrak{g}_2 \oplus \mathfrak{sa}_3(\mathbb{O}).$

So, we don’t exhaust the derivations of the exceptional Jordan algebra by looking for $3 \times 3$ Hamiltonians, but they do qualify.

If each point in the Leech lattice is a generator for octonionic-qutrit time evolution, does that shake any new ideas loose?

Posted by: Blake Stacey on May 2, 2020 9:37 PM | Permalink | Reply to this

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

Yes, that embedding is what I used recently, then identified $F_4$ fundamental rep 26 with the 26 vector of so(25,1). This allows $F_4$ to act on the 26 vector of D=25+1 string theory (under Leech lattice $\mathbb{R}^{24}/\Lambda_{24}$ orbifold), and we know by Wilson’s results that the Conway group $Co_0$ is generated by 3x3 unitary matrices over the octonions, which form $F_4$. Thus, we get the desired result, as the Monster $\mathbb{M}$ is the symmetry of the CFT of the orbifolded string theory.

Posted by: Metatron on May 9, 2020 4:02 PM | Permalink | Reply to this

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

Ah, OK. Slowly, things start to become incrementally more clear.

To lay my own motivations out on the table: Something feels off about having to invoke string theory. We go to all the trouble to bring in this exotic and elaborate physics model, and then we orbifold almost all of it away, leaving behind “the smallest possible nontrivial string theory” (Lepowsky). It makes me wonder if there might be a simpler physics-inspired route to the same object.

The “octonionic qutrit” naturally suggests itself as a setting in which to obtain physics motivations for mathematical developments. We know that we can write the Leech lattice as points in $\mathbb{O}^3$. But quantum mechanics wants unit vectors, while the different norms that Leech vectors come in are really quite important, so those ideas seem like a bad fit. The mapping that seems less obvious or more indirect — into octonionic qutrit “Hamiltonians” — also seems like it would stand a better chance of respecting the relevant information.

Posted by: Blake Stacey on May 20, 2020 11:46 PM | Permalink | Reply to this

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

I wouldn’t say we “orbifold it almost all away”: to get a theory with the Monster as symmetries we’ve got a bosonic string moving in 26-dimensional Minkowski spacetime mod an even unimodular lattice and mod $x \mapsto -x$. That’s a fairly hefty (and highly symmetrical) 26-dimensional orbifold.

And somehow string theory, with its Riemann surface worldsheets, seems somehow right to get a theory deeply connected to the $j$-function.

But of course it’d be great to get something simpler!

If you can get your hands on the Griess algebra in a physically inspired way without invoking string theory, that’d be interesting.

Posted by: John Baez on May 21, 2020 12:05 AM | Permalink | Reply to this

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

It’s probably the incoherent dream of a quarantine-addled brain, much like my joke that the discrete affine plane on 9 points appears in the study of qutrit states and in the study of the Moonshine module, so they have to be related!

What seems more plausible is that the Hall–Janko and possibly Rudvalis groups come into play (see pages 15–16 of the arXiv version of my guest posts). If true, that would be interesting, because we’d have an octonion-flavored connection between a group in the Happy Family and a Pariah. Moreover, it would imply that the Leech lattice enters the topic of equiangular lines in two different ways. It matters for real equiangular lines because the Leech lattice in $\mathbb{R}^{24}$ is the source for the Gerzon-bound-saturating set in $\mathbb{R}^{23}$. Granting the connection between the Hoggar lines and the Hall–Janko group, the Leech lattice — in its quaternionic guise — would also relate to complex equiangular lines, in a more subtle fashion. By starting with two conjugate SICs in $\mathbb{C}^8$ and representing the patterns of orthogonalities between them in graph form, we would eventually come back to the Leech lattice.

I say that’s more plausible, but I don’t get it yet, and the idea stemmed from looking up every instance of the group variously called $U_3(3)$, $PSU(3,3)$ and $G_2(2)'$ that I could find, so take it for what it seems to be worth.

Posted by: Blake Stacey on May 22, 2020 1:21 AM | Permalink | Reply to this

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