The n-Category Café

Blake wrote:

A point in $S$ is a quantum measurement—a POVM, possibly a noisy one.

Not quite, I think. We have $S = \mathbb{C}^2/\Gamma$, where $\Gamma$ is the binary icosahedral group. In this description we can think of it as the Hilbert space of a spin-1/2 particle, mod the double cover of the icosahedron’s rotational symmetries.

Last time I argued that $S$ is isomorphic to the space of regular icosahedra of arbitrary size, possibly even zero size, centered at the origin of $\mathbb{R}^3$.

If we eliminate the icosahedra of radius bigger than 1, we can think of these as icosahedra in the unit ball. We can reinterpret this ball as the ‘Bloch ball’: the space of mixed states of a spin-1/2 particle, with the center being the ‘state of maximal ignorance’.

So, we can also think of $S$ as the Bloch ball modulo the icosahedron’s rotational symmetry group. That is: mixed states of a spin-1/2 particle, but where we count two as the same if they differ by a rotation in the icosahedron’s rotation group.

Can you think of another nice way to think about this?

No matter how we think about it, $S$ is smooth except at one point in the middle, and $\widetilde{S}$ is $S$ with this singularity resolved in the nicest possible way.

Here’s what I had in mind. First, to establish some notation and terminology:

Any time we have a vector in $\mathbb{R}^3$ of length 1 or less, we can map it to a $2 \times 2$ 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 density matrix is a rank-1 projector that can be written as $\rho = |\psi\rangle\langle\psi|$ for some vector $|\psi\rangle$.

A POVM is a set of “effects” (positive semidefinite operators satisfying $0 \lt E \lt I$) that furnish a resolution of the identity:

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

for some density operators $\rho_i$ and weights $w_i$. Note that taking the trace of both sides gives a normalization constraint for the weights $w_i$ in terms of the dimension of the Hilbert space (for our purposes today, 2). 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

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

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.”

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 density operators (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 $r$ take the values $\pm 1$, and define

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

(The blog software is cutting my comment off here in the preview, so I’ll continue below.)

Continuing from previous comment:

To make the POVM effects, weight the rank-1 projectors $\rho_{s,r}$ equally. That is, take

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

Then, the four operators $E_{s,r}$ will sum to the identity. The icosahedron will work out basically the same way, but I don’t already have it in a notebook. Because I’m lazy tonight, I looked it up; the most detailed description I found in a couple minutes of checking is in section 10 of T. Decker, D. Janzing and T. Beth, “Quantum circuits for single-qubit measurements corresponding to Platonic solids,” Int. J. Quant. Info. 2 (2004), 353–77, arXiv:quant-ph/0308098.

So, we can think of $S$ in two ways: either as the Bloch ball modulo the icosahedron’s rotational symmetry group, or as the set of icosahedral POVMs.

(I figured out why the preview of my comment was stopping partway through: I had a square bracket that should have been a curly bracket.)

One more bit of terminology I will probably use later:

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

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

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 $d$. 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.

In many cases, we have both a displacement operator and its adjoint, e.g., when we conjugate $\Pi_0$:

$$\Pi_{l,\alpha} = D_{l\alpha} \Pi_0 D_{l\alpha}^\dagger.$$

So, we can often drop the phase factors and think of our SIC as being the orbit of a state under a group isomorphic to $\mathbb{Z}_d \times \mathbb{Z}_d$.

The only known exception to Weyl–Heisenberg covariance is a SIC that was discovered early on, by Stuart Hoggar. It lives in dimension 8, it is covariant under the tensor product of three copies of the Pauli group, and of course, it is secretly related to the octonions.

I wrote the following notes a few months ago. Looking at them again, I’m not sure why I didn’t post them here; no doubt I felt they were unfinished in some important way, but now they just look incomplete in an inevitable way.

Part 1: $\mathrm{E}_6$

In what follows, I will refer to H. S. M. Coxeter’s Regular Complex Polytopes, second edition (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. He calls the polyhedron “Hessian” because its nine diameters and twelve planes of symmetry interlock in a particular way. Their incidences reproduce the Hessian configuration, a set of nine points on twelve lines such that four lines pass through each point and three points lie on each line. (This configuration is also known as the discrete affine plane on nine points, and as the Steiner triple system of order 3. It is the solution to this interactive puzzle at Quanta Magazine.)

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 $\mathrm{E}_6$.

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:

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

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 $1/(d+1) = 1/4$.

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

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

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 states is known as the Hesse SIC.

Part 2: $\mathrm{E}_7$

We saw how to generate the Hesse SIC by taking the orbit of a fiducial state under the action of the $d = 3$ Weyl–Heisenberg group. Next, we will do something similar in $d = 8$. We start by defining the two states $$|\psi_0^\pm\rangle \propto (-1 \pm 2i, 1, 1, 1, 1, 1, 1, 1)^{\mathrm{T}}.$$ Here, we are taking the transpose to make our states column vectors, and we are leaving out the dull part, in which we normalize the states to satisfy $$\langle \psi_0^+ | \psi_0^+ \rangle = \langle \psi_0^- | \psi_0^- \rangle = 1.$$ First, we focus on $|\psi_0^+\rangle$. To create a SIC from the fiducial vector $|\psi_0^+\rangle$, we take the set of Pauli matrices, including the identity as an honorary member: $$\{ I, \sigma_x, \sigma_y, \sigma_z \}.$$ We turn this set of four elements into a set of sixty-four elements by taking all tensor products of three elements. This creates the Pauli operators on three qubits. By computing the orbit of $|\psi_0^+\rangle$ under multiplication (equivalently, the orbit of $\Pi_0^+ = |\psi_0^+\rangle\langle \psi_0^+|$ under conjugation), we find a set of 64 states that together form a SIC set.

The same construction works for the other choice of sign, $|\psi_0^-\rangle$, creating another SIC with the same symmetry group. We can call both of them SICs of Hoggar type.

Recall that when we invented SICs for a single qubit, they were tetrahedra in the Bloch ball, and we could fit together two tetrahedral SICs such that each vector in one SIC was orthogonal (in the Bloch picture, antipodal) to exactly one vector in the other. The two Hoggar-type SICs made from the fiducial states $\Pi_0^+$ and $\Pi_0^-$ satisfy the grown-up version of this relation: Each state in one is orthogonal to exactly twenty-eight states of the other.

Moreover, 28 = 28.

By that, I mean that these orthogonalities correspond to the antisymmetric elements of the three-qubit Pauli group. It is simplest to see why when we look for those elements of the $\Pi_0^-$ SIC that are orthogonal to the projector $\Pi_0^+$. These satisfy

$$\mathrm{tr}(\Pi_0^+ D \Pi_0^- D^\dagger) = 0$$

for some operator $D$ that is the tensor product of three Pauli matrices. Intuitively speaking, the product $\Pi_0^+ \Pi_0^-$ is a symmetric matrix, so if we want the trace to vanish, we ought to try introducing an asymmetry, but if we introduce too much, it will cancel out, on the “minus times a minus is a plus” principle. So, we want $D$ to comprise an odd number of factors of $\sigma_y$.

As explained in more detail in this old comment, these 28 antisymmetric matrices correspond exactly to the 28 bitangents of a quartic curve, and to pairs of opposite vertices of the Gosset polytope $3_{21}$.

Two fun things have happened here: First, we started with complex equiangular lines. By carefully considering the orthogonalities between two sets of complex equiangular lines, we arrived at the Gosset polytope $3_{21}$, which yields us a maximal set of real equiangular lines in $\mathbb{R}^7$. And since one cannot actually fit more equiangular lines into $\mathbb{R}^8$ than into $\mathbb{R}^7$, we have a connection between a maximal set of equiangular lines in $\mathbb{C}^8$ and a maximal set of them in $\mathbb{R}^8$.

Second, because we have made our way to the polytope $3_{21}$, we have arrived at $\mathrm{E}_7$.

Part 3: $\mathrm{E}_8$

I have already “spoiled” this part of the story, by hinting at a connection with the octonions. To spell it out more explicitly, we consider the stabilizer of the fiducial vector, i.e., the group of unitaries that map the SIC set to itself, leaving the fiducial where it is and permuting the other $d^2 - 1$ vectors. Huangjun Zhu observed that the stabilizer of any fiducial for a Hoggar-type SIC is isomorphic to the group $\mathrm{PSU}(3,3)$. In turn, this is up to a factor $\mathbb{Z}_2$ isomorphic to $G_2(2)$, the automorphism group of the octavians. And the unit octavians furnish the $\mathrm{E}_8$ lattice.

Great—thanks!

Now I’d like to transform this algebra into geometry. I’m less interested in proving things here than in ‘seeing’ them.

We’ve got the $n$-element cyclic group $\Gamma = \mathbb{Z}_n$ acting on the plane. A ‘typical’ point in the minimal resolution of $\mathbb{C}^2/\Gamma$ consists of $n$ distinct particles lying at the corners of a regular $n$-gon. Somehow there is a way for such a typical point to approach any point on any of the $\mathbb{P}^1$’s you describe. To do so, all $n$ particles need to approach the origin… but they need to do so in a carefully specified way!

The simplest thing is for the $n$-gon to shrink to a point without rotating at all. Which $\mathbb{P}^1$ does it approach then, and why?

Then: what’s the ‘second simplest thing’?

(Note: I’m using ‘particle’ as a name for a point in $\mathbb{C}^2$, and ‘point’ mainly as a name for a point in $S = \mathbb{C}^2/\Gamma$ or its minimal resolution $\widetilde{S}$. Also, where I’m saying ‘typical point’ above, the differential geometer in me wants to say ‘generic point’, but I’m afraid that could be confusing in algebraic geometry.)

This is very enlightening! It’ll take me a while to muster a good response, but it’s very nice to see the various ways a $\mathbb{Z}/n$-orbit in $\mathbb{C}^2$ can collapse to the origin.

What, in words, are the special geometrical features of those collapse processes that converge to nodes—points where two Riemann spheres in the exceptional fiber of $\pi : \widetilde{S} \to S$ intersect? This is one thing I’ve gotta know.

Edit: better to write the third equation as

$$|a|^2 = \sum_i a_i^2 = \mathrm{tr} A^2.$$

No problem—software will be software, it seems. I’ve been doing calculations relating to this off and on since I made my previous post, and I need to gather the results up in an intelligible form.

I finally got organized enough to gather these notes together, incorporating edits for clarity and recording one construction I haven’t found written in the literature anywhere.

• B. C. Stacey, “Sporadic SICs and Exceptional Lie Algebras”