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March 14, 2019

Sporadic SICs and Exceptional Lie Algebras III

Posted by John Baez

guest post by Blake C. Stacey

On the fourteenth of March, 2016 — so, exactly three years ago — Maryna Viazovska published a proof that the E 8\mathrm{E}_8 lattice is the best way to pack hyperspheres in eight dimensions. Today, we’ll see how this relates to another packing problem that seems quite different: how to fit as many equiangular lines as possible into the complex space 8\mathbb{C}^8. The answer to this puzzle is an example of a SIC: a Symmetric Informationally Complete quantum measurement.

Last time, we saw how to build SICs by starting with a fiducial vector and taking the orbit of that vector under the action of a group, turning one line into d 2d^2. We said earlier that the Weyl–Heisenberg group was the group we use in call cases but one. Now, we take on that exception. It will lead us to the exceptional root systems E 7\mathrm{E}_7 and E 8\mathrm{E}_8. Actually, it will be a bit easier to tackle the latter first. Whence E 8\mathrm{E}_8 in the world of SICs?

We saw how to generate the Hesse SIC by taking the orbit of a fiducial state under the action of the d=3d = 3 Weyl–Heisenberg group. Today, we will do something similar in d=8d = 8. We start by defining the two states

|ψ 0 ±(1±2i,1,1,1,1,1,1,1) T. |\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 ψ 0 +|ψ 0 +=ψ 0 |ψ 0 =1\langle \psi_0^+ | \psi_0^+ \rangle = \langle \psi_0^- | \psi_0^- \rangle = 1.

First, we focus on |ψ 0 +|\psi_0^+\rangle. To create a SIC from the fiducial vector |ψ 0 +|\psi_0^+\rangle, we take the set of Pauli matrices, including the identity as an honorary member: {I,σ x,σ y,σ z}\{ 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 |ψ 0 +|\psi_0^+\rangle under multiplication (equivalently, the orbit of Π 0 +=|ψ 0 +ψ 0 +|\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 we used to make the |ψ 0 +|\psi_0^+\rangle SIC works for the other choice of sign, |ψ 0 |\psi_0^- \rangle, creating another SIC with the same symmetry group. We can call both of them SICs of Hoggar type, in honor of Stuart Hoggar.

To make the link with E 8\mathrm{E}_8, 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 21d^2 - 1 vectors. Huangjun Zhu observed that the stabilizer of any fiducial for a Hoggar-type SIC is isomorphic to the group of 3×33\times 3 unitary matrices over the finite field of order 9. This group is sometimes written U 3(3)U_3(3) or PSU(3,3)PSU(3,3). In turn, this group is up to a factor 2\mathbb{Z}_2 isomorphic to G 2(2)G_2(2), the automorphism group of the Cayley integers, a subset of the octonions also known as the octavians. Up to an overall scaling, the lattice of octavians is also the lattice known as E 8\mathrm{E}_8.

The octavian lattice contains a great deal of arithmetic structure. Of particular note is that it contains 240 elements of norm 1. In addition to the familiar +1+1 and 1-1, which have order 1 and 2 respectively, there are 56 units of order 3, 56 units of order 6 and 126 units of order 4. The odd-order units generate subrings of the octavians that are isomorphic to the Eisenstein integers and the Hurwitz integers, lattices in the complex numbers and the quaternions (as Conway and Smith describe in their book). From the symmetries of these lattices, we can in fact read off the stabilizer groups for fiducials of the qubit and Hesse SICs. It is as if the sporadic SICs are drawing their strength from the octonions.

Before moving on, we pause to note how peculiar it is that by trying to find a nice packing of complex unit vectors, we ended up talking about an optimal packing of Euclidean hyperspheres!

Now that we’ve met E 8\mathrm{E}_8, it’s time to visit the root system we skipped: Where does E 7\mathrm{E}_7 fit in?

With respect to the probabilistic representation furnished by the Π 0 +\Pi_0^+ SIC, the states of the Π 0 \Pi_0^- SIC minimize the Shannon entropy, and vice versa. Here are four of the states from the {Π i }\{\Pi_i^-\} Hoggar-type SIC, written in the probabilistic representation of three-qubit state space provided by the {Π i +}\{\Pi_i^+\} SIC. Up to an overall normalization by 1/361/36, these states are all binary sequences, i.e., they are uniform over their supports.

1110111011100001111011101110000111101110111000010001000100011110
1101110111010010110111011101001011011101110100100010001000101101
1011101110110100101110111011010010111011101101000100010001001011
0111011101111000011101110111100001110111011110001000100010000111

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 Π 0 +\Pi_0^+ and Π 0 \Pi_0^- satisfy the grown-up version of this relation: Each state in one is orthogonal to exactly twenty-eight states of the other.

We can understand these orthogonalities as corresponding to the antisymmetric elements of the three-qubit Pauli group. It is simplest to see why when we look for those elements of the Π 0 \Pi_0^- SIC that are orthogonal to the projector Π 0 +\Pi_0^+. These satisfy

tr(Π 0 +DΠ 0 D )=0 tr (\Pi_0^+ D \Pi_0^- D^\dagger) = 0

for some operator DD that is the tensor product of three Pauli matrices. For which such tensor-product operators will this expression vanish? Intuitively speaking, the product Π 0 +Π 0 \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. Recall the Pauli matrices:

σ x=(0 1 1 0),σ y=(0 i i 0),σ z=(1 0 0 1). \sigma_x = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right),\qquad \sigma_y = \left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right),\qquad \sigma_z = \left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right).

Note that of these three matrices, only σ y\sigma_y is antisymmetric, and also note that we have

σ zσ x=σ xσ z=iσ y. \sigma_z \sigma_x = -\sigma_x\sigma_z = i\sigma_y.

This much is familiar, though that minus sign gets around. For example, it is the fuel that makes the GHZ thought-experiment go, because it means that

σ xσ xσ x=(σ xσ zσ z)(σ zσ xσ z)(σ zσ zσ x). \sigma_x \otimes \sigma_x \otimes \sigma_x = -(\sigma_x \otimes \sigma_z \otimes \sigma_z)(\sigma_z \otimes \sigma_x \otimes \sigma_z)(\sigma_z \otimes \sigma_z \otimes \sigma_x).

Let’s consider the finite-dimensional Hilbert space made by composing three qubits. This state space is eight-dimensional, and we build the three-qubit Pauli group by taking tensor products of the Pauli matrices, considering the 2×22 \times 2 identity matrix to be the zeroth Pauli operator. There are 64 matrices in the three-qubit Pauligroup, and we can label them by six bits. The notation

(m 1 m 3 m 5 m 2 m 4 m 6) \left(\begin{array}{ccc} m_1 & m_3 & m_5 \\ m_2 & m_4 & m_6\end{array}\right)

means to take the tensor product

σ x m 1σ z m 2(i) m 3m 4σ x m 3σ z m 4(i) m 5m 6σ x m 5σ z m 6. \sigma_x^{m_1} \sigma_z^{m_2} \otimes (-i)^{m_3m_4} \sigma_x^{m_3} \sigma_z^{m_4} \otimes (-i)^{m_5m_6} \sigma_x^{m_5} \sigma_z^{m_6}.

Now, we ask: Of these 64 matrices, how many are symmetric and how many are antisymmetric? We can only get antisymmetry from σ y\sigma_y, and (speaking heuristically) if we include too much antisymmetry, it will cancel out. More carefully put: We need an odd number of factors of σ y\sigma_y in the tensor product to have the result be an antisymmetric matrix. Otherwise, it will come out symmetric. Consider the case where the first factor in the triple tensor product is σ y\sigma_y. Then we have (41) 2=9(4-1)^2 = 9 possibilities for the other two slots. The same holds true if we put the σ y\sigma_y in the second or the third position. Finally, σ yσ yσ y\sigma_y \otimes \sigma_y \otimes \sigma_y is antisymmetric, meaning that we have 93+1=289 \cdot 3 + 1 = 28 antisymmetric matrices in the three-qubit Pauli group. In the notation established above, they are the elements for which

m 1m 2+m 3m 4+m 5m 6=1mod2. m_1m_2 +m_3m_4 + m_5m_6 = 1 \mod 2.

Moreover, these 28 antisymmetric matrices correspond exactly to the 28 bitangents of a quartic curve, and to pairs of opposite vertices of the Gosset polytope 3 213_{21}. In order to make this connection, we need to dig into the octonions.

To recap: Each of the 64 vectors (or, equivalently, projectors) in the Hoggar SIC is naturally labeled by a displacement operator, which up to an overall phase is the tensor product of three Pauli operators. Recall that we can write the Pauli operator σ y\sigma_y as the product of σ x\sigma_x and σ z\sigma_z, up to a phase. Therefore, we can label each Hoggar-SIC vector by a pair of binary strings, each three bits in length. The bits indicate the power to which we raise the σ x\sigma_x and σ z\sigma_z generators on the respective qubits. The pair (010,101)(010,101), for example, means that on the three qubits, we act with σ x\sigma_x on the second, and we act with σ z\sigma_z on the first and third. Likewise, (000,111)(000, 111) stands for the displacement operator which has a factor of σ z\sigma_z on each qubit and no factors of σ x\sigma_x at all.

There is a natural mapping from pairs of this form to pairs of unit octonions. Simply turn each triplet of bits into an integer and pick the corresponding unit from the set {1,e 1,e 2,e 3,e 4,e 5,e 6,e 7}\{1, e_1, e_2, e_3, e_4, e_5, e_6, e_7\}, where each of the e je_j square to 1-1.

We can choose the labeling of the unit imaginary octonions so that the following nice property holds. Up to a sign, the product of two imaginary unit octonions is a third, whose index is the XOR of the indices of the units being multiplied. For example, in binary, 1=0011 = 001 and 4=1004 = 100; the XOR of these is 101=5101 = 5, and e 1e_1 times e 4e_4 is e 5e_5.

Translate the Cayley–Graves multiplication table here into binary if enlightenment has not yet struck:

e ie j 1 e 1 e 2 e 3 e 4 e 5 e 6 e 7 1 1 e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 1 e 1 1 e 3 e 2 e 5 e 4 e 7 e 6 e 2 e 2 e 3 1 e 1 e 6 e 7 e 4 e 5 e 3 e 3 e 2 e 1 1 e 7 e 6 e 5 e 4 e 4 e 4 e 5 e 6 e 7 1 e 1 e 2 e 3 e 5 e 5 e 4 e 7 e 6 e 1 1 e 3 e 2 e 6 e 6 e 7 e 4 e 5 e 2 e 3 1 e 1 e 7 e 7 e 6 e 5 e 4 e 3 e 2 e 1 1 \begin{array}{c|cccccccc} e_i e_j & 1 & e_1 & e_2 & e_3 & e_4 & e_5 & e_6 & e_7 \\ 1 & 1 & e_1 & e_2 & e_3 & e_4 & e_5 & e_6 & e_7 \\ e_1 & e_1 & -1 & e_3 & -e_2 & e_5 & -e_4 & -e_7 & e_6 \\ e_2 & e_2 & -e_3 & -1 & e_1 & e_6 & e_7 & -e_4 & -e_5 \\ e_3 & e_3 & e_2 & -e_1 & -1 & e_7 & -e_6 & e_5 & -e_4 \\ e_4 & e_4 & -e_5 & -e_6 & -e_7 & -1 & e_1 & e_2 & e_3 \\ e_5 & e_5 & e_4 & -e_7 & e_6 & -e_1 & -1 & -e_3 & e_2 \\ e_6 & e_6 & e_7 & e_4 & -e_5 & -e_2 & e_3 & -1 & -e_1 \\ e_7 & e_7 & -e_6 & e_5 & e_4 & -e_3 & -e_2 & e_1 & -1 \end{array}

So, each projector in the Hoggar SIC is labeled by a pair of octonions, and the group structure of the displacement operators is, almost, octonion multiplication. I say “almost” because we are neglecting the sign factors and only caring about which imaginary octonion a multiplication yields: All we need is the label. Rest assured, the sign factors will return soon.

Another way to express the Cayley–Graves multiplication table is with the Fano plane, a set of seven points grouped into seven lines that has been called “the combinatorialist’s coat of arms”. We can label the seven points with the imaginary octonions e 1e_1 through e 7e_7. When drawn on the page, a useful presentation of the Fano plane has the point e 4e_4 in the middle and, reading clockwise, the points e 1e_1, e 7e_7, e 2e_2, e 5e_5, e 3e_3 and e 6e_6 around it in a regular triangle.

Illustration of the Fano plane

The three sides and three altitudes of this triangle, along with the inscribed circle, provide the seven lines: (e 1,e 2,e 3)(e_1,e_2,e_3), (e 1,e 4,e 5)(e_1,e_4,e_5), (e 1,e 7,e 6)(e_1,e_7,e_6), (e 2,e 4,e 6)(e_2,e_4,e_6), (e 2,e 5,e 7)(e_2,e_5,e_7), (e 3,e 4,e 7)(e_3,e_4,e_7), (e 3,e 6,e 5)(e_3,e_6,e_5). It is apparent that each line contains three points, and it is easy to check that each point lies within three distinct lines, and that each pair of lines intersect at a single point. One consequence of this is that if we take the incidence matrix of the Fano plane, writing a 1 in the iji j-th entry if line ii contains point jj, then every two rows of the matrix have exactly the same overlap:

M=(1 1 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 1 1 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 1 0 0 1 0 0 1 0 1 1 0). M = \begin{pmatrix} 1 &1 &1 &0 &0 &0 &0 \\ 1 &0 &0 &1 &1 &0 &0 \\ 1 &0 &0 &0 &0 &1 &1 \\ 0 &1 &0 &1 &0 &1 &0 \\ 0 &1 &0 &0 &1 &0 &1 \\ 0 &0 &1 &1 &0 &0 &1 \\ 0 &0 &1 &0 &1 &1 &0 \end{pmatrix}.

The rows of the incidence matrix furnish us with seven equiangular lines in 7\mathbb{R}^7. We can build upon this by considering the signs in the Cayley–Graves multiplication table, which we can represent by adding orientations to the lines of the Fano plane. Start by taking the first row of the incidence matrix MM, which corresponds to the line (e 1,e 2,e 3)(e_1, e_2, e_3), and give it all possible choices of sign by multiplying by the elements not on that line. Multiplying by e 4e_4, e 5e_5, e 6e_6 and e 7e_7 respectively, we get

(+ + + 0 0 0 0 + 0 0 0 0 + 0 0 0 0 + 0 0 0 0). \begin{pmatrix} + &+ &+ &0 &0 &0 &0 \\ - &+ &- &0 &0 &0 &0 \\ - &- &+ &0 &0 &0 &0 \\ + &- &- &0 &0 &0 &0 \end{pmatrix}.

The sign we record here is simply the sign we find in the corresponding entry of the Cayley–Graves table. Doing this with all seven lines of the Fano plane, we obtain a set of 28 vectors, each one given by a choice of a line and a point not on that line. Moreover, all of these vectors are equiangular. This is easily checked: For any two vectors derived from the same Fano line, two of the terms in the inner product will cancel, leaving an overlap of magnitude 1. And for any two vectors derived from different Fano lines, the overlap always has magnitude 1 because each pair of lines always meets at exactly one point. Van Lint and Seidel noted that the incidence matrix of the Fano plane could be augmented into a full set of 28 equiangular lines, but to my knowledge, extracting the necessary choices of sign from octonion multiplication is not reported in the literature.

So, the 28 lines in a maximal equiangular set in 7\mathbb{R}^7 correspond to point-line pairs in the Fano plane, where the point and the line are not coincident. In discrete geometry, the combination of a line and a point not on that line is known as an anti-flag. It is straightforward to show from the above that the antisymmetric matrices in the three-qubit Pauli group also correspond to the anti-flags of the Fano plane: Simply take the powers of the σ x\sigma_x operators to specify a point and the powers of the σ z\sigma_z operators to specify a line.

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 a maximal set of real equiangular lines in 7\mathbb{R}^7. And it turns out that one cannot actually fit more equiangular lines into 8\mathbb{R}^8 than into 7\mathbb{R}^7, meaning that we have a connection between a maximal set of equiangular lines in 8\mathbb{C}^8 and a maximal set of them in 8\mathbb{R}^8.

Second, our equiangular lines in 7\mathbb{R}^7 are the diameters of the Gosset polytope 3 213_{21}. And because we have made our way to the polytope 3 213_{21}, we have arrived at E 7\mathrm{E}_7. 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 21n_{21} for n=2,3,4n = 2, 3, 4 and live in n+4n + 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 𝔢 n+4\mathfrak{e}_{n+4}, whose vertices are the weights of the minimal representations.

When we studied the Hesse SIC, we met the case n=2n = 2 and 𝔢 6\mathfrak{e}_6. The intricate orthogonalities between two conjugate SICs of Hoggar type have led us to the case n=3n = 3 and 𝔢 7\mathfrak{e}_7.

Puzzle: Another place that the number 28 arises when considering an 8-dimensional space is the free modular lattice on 3 generators. Is that 28 the same as this 28?

Posted at March 14, 2019 1:33 AM UTC

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Re: Sporadic SICs and Exceptional Lie Algebras III

It makes me happy to see Stuart Hoggar’s name. My first permanent job was at the University of Glasgow, and when I moved into my new office as a fresh young lecturer, the person moving out — and retiring — was none other than him.

Posted by: Tom Leinster on March 25, 2019 3:56 PM | Permalink | Reply to this

Re: Sporadic SICs and Exceptional Lie Algebras III

Here is an observation that I wasn’t quite able to place in the sequence of posts, but which ought to be mentioned somewhere. Recall that the condition trρ 2=1tr \rho^2 = 1, which is satisfied when ρ\rho is a pure state, becomes

ip(i) 2=2d(d+1) \sum_i p(i)^2 = \frac{2}{d(d+1)}

when we fix a SIC as a reference measurement and thereby map quantum state space into a probability simplex. Now, flipping this equation upside down makes both sides look like counting! The right-hand side becomes combinatorics: It’s just the binomial coefficient for choosing two things out of d+1d+1. Meanwhile, the left-hand side becomes the effective number of outcomes, which we are familiar with because it is a biodiversity index:

N eff=( ip(i) 2) 1. N_{eff} = \left(\sum_i p(i)^2 \right)^{-1} .

So, when we ascribe a pure state to a quantum system of Hilbert-space dimensionality dd, we are saying that the effective number of possible outcomes for a reference measurement is

N eff=(d+12). N_{eff} = \binom{d+1}{2}.

Consequently, ascribing a pure state means that we are effectively ruling out a number of outcomes equal to

d 2N eff=d(d1)2=(d2). d^2 - N_{eff} = \frac{d(d-1)}{2} = \binom{d}{2}.

This motivates the following question: What is the upper bound on the number of entries in p\vec{p} that can equal zero? A brief calculation with the Cauchy–Schwarz inequality reveals that the answer is, in fact, exactly d 2N effd^2 - N_{eff}. In other words, no vector p\vec{p} in the image of quantum state space can contain more than d(d1)/2d(d-1)/2 zeros. One reason why the sporadic SICs are distinguished from all the others is that they provide the examples where this bound is known to be saturated. We can attain it for qubit SICs (where it equals 1), the Hesse SIC (where it equals 3) and the Hoggar-type SICs (where it equals 28). The states which saturate this bound are also those that minimize the Shannon entropy, as we discussed in the main posts.

We can also deduce the corresponding bound for the case of real equiangular lines. In d\mathbb{R}^d, the Gerzon bound is d(d+1)/2d(d+1)/2, and in those cases where we have a complete set of equiangular lines, we can play the game of doing “real-vector-space quantum mechanics”, using our d(d+1)/2d(d+1)/2 equiangular lines to define a reference measurement. As in d\mathbb{C}^d, the Cauchy–Schwarz inequality gives an upper bound on the number of zero-valued entries in a probability distribution pp, which works out to be

N Z=d 213. N_Z = \frac{d^2 - 1}{3}.

Thinking about it for a moment, we realize that this is telling us about the maximum number of equiangular lines that can all simultaneously be orthogonal to a common vector. In particular, if we fix d=23d = 23, we have a “real-vector-space SIC” that we can derive from the Leech lattice, and we know that

N Z=23 213=176 N_Z = \frac{23^2 - 1}{3} = 176

of the elements of that set can be orthogonal to a “pure quantum state”, i.e., a vector in 23\mathbb{R}^{23}. All 176 such lines have to crowd together into a 22-dimensional subspace, while still being equiangular.

And that’s what biodiversity has to do with the classification of finite simple groups!

Posted by: Blake Stacey on April 7, 2019 1:05 AM | Permalink | Reply to this

Re: Sporadic SICs and Exceptional Lie Algebras III

Another observation that didn’t fit into my main sequence was how the regular icosahedron is the “real version” of the Hesse SIC.

Suppose that, just for the fun of it, we wished to build the Hesse SIC, but in real vector space. What might this even mean? It would entail finding a fiducial vector and an appropriate group, closely analogous to the qutrit Weyl–Heisenberg group, such that the orbit of said fiducial is a maximal set of equiangular lines. How big would such a set of lines be? Recall that the Gerzon bound is d 2d^2 for d\mathbb{C}^d, but only d(d+1)/2d(d+1)/2 in d\mathbb{R}^d. In both cases, this is essentially because those values are the dimensions of the appropriate operator spaces. It is not difficult to show that, if the Gerzon bound is attained, the magnitude of the inner product between the vectors is 1/d+11/\sqrt{d+1} in d\mathbb{C}^d and 1/d+21/\sqrt{d+2} in d\mathbb{R}^d.

We are familiar with the complex case, in which we define a shift operator XX and a phase operator ZZ that both have order dd. A cyclic shift is nice and simple, so we’d like to keep that idea, but the only “phase” we have to work with is the choice of positive or negative sign. So, let us consider the operators X=(0 0 1 1 0 0 0 1 0),andZ=(1 0 0 0 1 0 0 0 1). X = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}, \ and\ Z = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}. The shift operator XX still satisfies X 3=IX^3 = I, while for the phase operator ZZ, we now have Z 2=IZ^2 = I.

What group can we make from these operators? Note that (ZX) 3=I, (Z X)^3 = -I, and so (Z) 2=X 3=(ZX) 3=I, (-Z)^2 = X^3 = (-Z X)^3 = I, meaning that the operators XX and Z-Z generate the tetrahedral group, so designated because it is isomorphic to the rotational symmetry group of a regular tetrahedron.

Now, we want to take the orbit of a vector under this group! But what vector? It should not be an eigenvector of XX or of ZZ, for then we know we could never get a full set. Therefore, we don’t want a flat vector, nor do we want any of the basis vectors, so we go for the next simplest thing: v=(0 1 y), v = \begin{pmatrix} 0 \\ 1 \\ y \end{pmatrix}, where yy is a real number. We now have Zv=(0 1 y) Z v = \begin{pmatrix} 0 \\ -1 \\ y\end{pmatrix} and also X 2v=(1 y 0), X^2 v = \begin{pmatrix} 1 \\ y \\ 0 \end{pmatrix}, so if we want equality between the inner products, Zv,v=X 2v,v, \langle Z v, v \rangle = \langle{X^2 v, v \rangle}, then we need to have 1+y 2=y. -1 + y^2 = y. The positive solution to this quadratic equation is y=1+52, y = \frac{1 + \sqrt{5}}{2}, so we can in fact take our yy to be ϕ\phi, the golden ratio.

In the group we defined above, XX performs cyclic shifts, ZZ changes the relative phase of the components, and we have the freedom to flip all the signs. Therefore, the orbit of the fiducial vv is the set of twelve vectors (0 ±1 ±ϕ),(±1 ±ϕ 0),(±ϕ 0 ±1). \begin{pmatrix} 0 \\ \pm 1 \\ \pm \phi \end{pmatrix}, \ \begin{pmatrix} \pm 1 \\ \pm \phi \\ 0 \end{pmatrix}, \ \begin{pmatrix} \pm \phi \\ 0 \\ \pm 1 \end{pmatrix}. These are the vertices of a regular icosahedron, and the diagonals of that icosahedron are six equiangular lines. The inner products between these vectors are always ±ϕ\pm \phi. Since d(d+1)/2=6d(d+1)/2 = 6, there cannot be any larger set of equiangular lines in 3\mathbb{R}^3.

Recall that the reciprocal of the golden ratio ϕ\phi is ϕ 1=21+51515=22515=1+52=ϕ1. \phi^{-1} = \frac{2}{1 + \sqrt{5}} \frac{1 - \sqrt{5}}{1 - \sqrt{5}} = \frac{2 - 2\sqrt{5}}{1 - 5} = \frac{-1 + \sqrt{5}}{2} = \phi - 1. The golden ratio ϕ\phi is a root of the monic polynomial y 2y1y^2 - y - 1, and being a root of a monic polynomial with integer coefficients, it is consequently an algebraic integer. The same holds for its reciprocal, so ϕ 1\phi^{-1} is also an algebraic integer, making the two of them units in the number field (5)\mathbb{Q}(\sqrt{5}).

To summarize: For the diagonals of the regular icosahedron, the vector components are given by the units of the “golden field” (5)\mathbb{Q}(\sqrt{5}). But it has been discovered that the vector components for the Weyl–Heisenberg SICs in dimension four are derived from a unit in the ray class field over (5)\mathbb{Q}(\sqrt{5}). Ray class fields are generalizations of the idea of cyclotomic fields that arose from the study of Hilbert’s 12th problem. Recall that according to the Kronecker–Weber theorem, any abelian extension of the rationals is contained in a cyclotomic field. (This has a cameo in the movie Sneakers, where a mathematician character delivers a bombastic presentation about his cryptography research.) When we instead consider abelian extensions of real quadratic fields, the analogue of the cyclotomic fields are the ray class fields. The phases of the nonsporadic Weyl–Heisenberg SICs appear to be playing a role regarding ray class fields much like the role that roots of unity play with cyclotomic fields.

So, the icosahedron is the Euclidean version of the Hesse SIC, and the SICs in d=4d = 4 are a kind of number-theoretic extension of the icosahedron.

Futhermore, it has been observed empirically that in dimensions d k=ϕ 2k+ϕ 2k+1, d_k = \phi^{2k} + \phi^{-2k} + 1, there exist Weyl–Heisenberg SICs with additional group-theoretic properties that make their exact expressions easier to find. These are known as Fibonacci–Lucas SICs.

Posted by: Blake Stacey on April 15, 2019 6:55 PM | Permalink | Reply to this

Re: Sporadic SICs and Exceptional Lie Algebras III

Erica Klarreich has a new article in Quanta magazine about recent extensions of optimality proofs for the rmE 8{\rm E}_8 and Leech lattices. The result by Cohn, Kumar, Miller, Radchenko and Viazovska, posted as arXiv:1902.05438, shows that these lattices minimize any energy function that is completely monotonic in squared distance. This relates to real equiangular lines, as mentioned earlier, because the maximal set in 7\mathbb{R}^7 can be extracted from the rmE 8{\rm E}_8 lattice, and the maximal set in 23\mathbb{R}^{23} is extracted from the Leech lattice.

Intriguingly, the Hall–Janko sporadic simple group can be realized as the symmetries of a “quaternionic Leech lattice” (Tits (1980), Wilson (1986)), while it also arises if you play around a bit with the Hoggar-type SICs. In the original post, we explored a rather intricate pattern of orthogonalities between two conjugate Hoggar-type SICs in 8\mathbb{C}^8. Because each SIC provides a basis, we found that we can uniquely specify a vector in one SIC by listing the vectors in the other SIC with which it has nonzero overlap. A unitary symmetry of one SIC set corresponds to a permutation of the other. Using the second SIC to define a representation of the state space, each vector in the first SIC is essentially a binary string, and sending one vector to another permutes the 1’s and 0’s. In particular, a unitary that stabilizes a vector in one SIC must permute the vectors of the second SIC in such a way that the list of 1’s and 0’s remains the same. The 1’s can be permuted among themselves, and so can the 0’s, but the binary sequence as a whole does not change.

It would be nice to have a way of visualizing this with a more tangible structure than eight-dimensional complex Hilbert space — something like a graph. Thinking about the permutations of 36 vectors, we imagine a graph on 36 vertices, and we try to draw it in such a way that its group of symmetries is isomorphic to the stabilizer group of a Hoggar fiducial. Can this be done? Well, almost — that is, up to a “factor of two”: It’s called the U 3(3)U_3(3) graph.

Now, is there a way to illustrate the structure of both SICs together as a graph? We want to record the fact that each vector in one SIC is nonorthogonal to exactly 36 vectors of the other, that the stabilizer of each vector is PSU(3,3)PSU(3,3), and that a stabilizer unitary shuffles the nonorthogonal set within itself. So, we start with one vertex to represent a fiducial vector, then we add 63 more vertices to stand for the other vectors in the first SIC, and then we add 36 vertices to represent the vectors in the second SIC that are nonorthogonal to the fiducial of the first. We’d like to connect the vertices in such a way that the stabilizer of any vertex is isomorphic to PSU(3,3)PSU(3,3). In fact, because any vector in either SIC can be identified by the list of the 36 nonorthogonal vectors in the other, the graph should look locally like the U 3(3)U_3(3) graph everywhere! Is this possible?

In fact, it is, again up to a “factor of two”: It’s the Hall–Janko graph, whose automorphism group has the Hall–Janko group as an index-2 subgroup.

This is part of the final, most speculative section of my notes on sporadic SICs and related exceptional entities. It’s an odd confluence of structures! The notion that there might be a story tying it all together is still too vague for me to try our host’s patience by making it a fourth blog post, but seeing the rmE 8{\rm E}_8 and Leech lattices in the news brought it back to mind.

Posted by: Blake Stacey on May 20, 2019 7:43 PM | Permalink | Reply to this

Re: Sporadic SICs and Exceptional Lie Algebras III

And apparently I can’t get the markup right even with preview:

rmE 8E 8 rm E_8 \to \mathrm{E}_8

Posted by: Blake Stacey on May 20, 2019 8:35 PM | Permalink | Reply to this

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