G. S. Kopp (2019) “SIC-POVMs and the Stark Conjectures,”International Mathematics Research Notices, Vol. 00, No. 0, pp. 1–27
doi:10.1093/imrn/rnz153

Gene S. Kopp∗
Heilbronn Institute for Mathematical Research, School of Mathematics,
University of Bristol, Bristol, UK∗ Correspondence to be sent to: e-mail: gene.kopp@bristol.ac.uk

The existence of d2 pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in d-dimensional
Hilbert space is known only for finitely many dimensions d. We prove that, if there
exists a set of real units in a certain ray class field (depending on d) satisfying certain
algebraic properties, a SIC-POVM exists, when d is an odd prime congruent to 2 modulo
3. We give an explicit analytic formula that we expect to yield such a set of units.
Our construction uses values of derivatives of zeta functions at s = 0 and is closely
connected to the Stark conjectures over real quadratic fields. We verify numerically that
our construction yields SIC-POVMs in dimensions 5, 11, 17, and 23, and we give the firstexact SIC-POVM in dimension 23.

1 Introduction
A set of m complex equiangular lines in d dimensions is a set of one-dimensional|vi ,vj |subspaces Cv1 , Cv2 , . . . , Cvm in Cd having equal angles arccos v ·v
 = θ for all i = j.
ijThese geometric configurations were first studied in the context of design theory in the
1970s, as a complex analogue of sets of real equiangular lines. The maximal cardinalityof a set of complex equiangular lines was shown to be bounded above by d2 by Delsarte
et al. [8].

Received December 18, 2018; Revised May 7, 2019; Accepted June 4, 2019

© The Author(s) 2019. Published by Oxford University Press. All rights reserved.
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SIC-POVMs and the Stark Conjectures

2 G. S. Kopp

A set of complex equiangular lines achieving this upper bound—d2 lines indimension d—is equivalent to a type of generalized quantum measurement known as a
SIC-POVM (symmetric informationally complete positive operator-valued measure). SICPOVMs were introduced in 1999 by Zauner [30, 31] and have applications to quantum
structing a quantum state from a set of measurements of copies of the state (see, e.g.,[19]). Their existence has implications for quantum foundations due to their presence inthe theory of quantum Bayesianism (or QBism) [10]. See the bibliography of Fuchs et al.
[9] for a complete list of references on the history and applications of SIC-POVMs.
SIC-POVMs are conjectured to exist in all dimensions. They have been proven toexist (by explicit construction) in dimensions 1–21, 24, 28, 30, 31, 35, 37, 39, 43, 48, 124,
and 323 [1–4, 11–13, 21, 22, 30], and approximate numerical solutions have been foundin every dimension up to 151 and several other dimensions up to 844 [9, 18, 20–22].
(Some further unpublished solutions have been found by Grassl and Scott [14]: an exact
solution in dimension 53 and numerical solutions in all dimensions up to 165, dimension
1155, and dimension 2208) All but one (the Hoggar lines [15]) of the known SIC-POVMs
are (up to unitary symmetry) so-called Heisenberg SIC-POVMs—the orbit of a single
vector under the action of a discrete Heisenberg group.
Appleby et al. [5, 6] recently discovered an empirical connection between SICPOVMs and Hilbert’s 12th problem for real quadratic fields. Hilbert’s 12th problem is alongstanding open problem in number theory. For a base field K, it asks for an explicit
construction of the abelian extensions of K, analogous to the Kronecker–Weber theorem
in the case when K = Q and the theory of complex multiplication in the case when K is
imaginary quadratic.
Appleby et al. [5, 6] observe that, for d > 3, the known examples of Heisenberg
SIC-POVMs are always defined over abelian extensions of the real quadratic field√
Q( (d − 3)(d + 1)). They predict the existence of a certain Galois orbit of SIC-POVMs
called a minimal multiplet, and they conjecturally identify the specific ray class fieldover which the SIC-POVMs in the minimal multiplet should be defined.
The applicability of SIC-POVMs to Hilbert’s 12th problem is currently aspirational. Although it is possible to use SIC-POVMs (found by other methods) to constructclass fields, the reverse direction (using class fields to construct SIC-POVMs) is moresensible from a computational perspective. Rather, the surprising connection between
class field theory and complex equiangular lines gives a telling hint about the algebraic
structure of class fields—and, in fact, the algebraic properties of derivative ray classzeta values at s = 0.

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information processing and especially quantum state tomography, the process of recon-SIC-POVMs and the Stark Conjectures 3

Specifically, in this paper, we formulate a pair of conjectures that imply astrong refinement of the rank 1 abelian Stark conjecture in an infinite family of classfields. In Conjecture 5.1, we predict the existence of a Galois orbit of real algebraic√
units in a certain class field over Q( (d − 3)(d + 1)) satisfying several strong algebraic
Conjecture 5.2 states that the Stark units are such an orbit.
This paper likewise provides a refinement of Zauner’s conjecture on SIC-POVM
existence. We give a conjectural construction of a Heisenberg SIC-POVM in infinitely
many dimensions d = 5, 11, 17, 23, 29, 41, . . . (i.e., those d that are odd primes congruent
to 2 modulo 3). Specifically, we prove in Theorem 5.4 that any set of units satisfying
the conditions of Conjecture 5.1 may be used to construct a SIC-POVM in dimension d.
Conjecture 5.2 gives an analytic formula for a set of real numbers that we expect tosatisfy the conditions of Conjecture 5.1, thus providing a construction of SIC-POVMs.
Of crucial importance to our construction are the conjectures of Stark on theleading terms of the Taylor expansions of Hecke and Artin L-functions at s = 1 ands = 0 [25–29]. Stark made his conjectures in the 1970s, and they remain open over everybase field except for Q (known to Dirichlet) and imaginary quadratic fields (proof due toStark [25]). Much work has been done toward attacking or refining the Stark conjectures;
one vital reference is the proceedings of the 2002 “International Conference on Stark’s
Conjectures and Related Topics” at Johns Hopkins University [7].
Specifically, we connect the SIC-POVM existance question to the rank 1 abelian
Stark conjecture in the real quadratic case [27, 28]. In the rank 1 case, the conjecture
predicts the existance of “Stark units” in ray class fields given by the analytic formula (0)) for a certain differenced ray class zeta function Z (s). The following
εA = exp(ZAA

observation was first made by the author in his PhD thesis [16] in the case d = 5, and wehave checked it numerically in a number of cases beyond those of odd prime dimensionsd ≡ 2 (mod 3) that are the focus of this paper.
Observation 1.1.

Squares of overlap phases of centred Heisenberg SIC-POVMs indimension d are Galois conjugate to integral powers of Stark units in abelian extensions√of Q( (d − 3)(d + 1).
Conjectures 5.1 and 5.2 make Observation 1.1 completely explicit in the casewhere d is an odd prime congruent to 2 modulo 3 and the SIC-POVM is in the “minimal
multiplet”. A congruence condition modulo a prime lying over (d) determines the signsof the overlap phases; see (5.2).

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4 G. S. Kopp

Our construction has been verified numerically in dimensions d = 5, 11, 17,
and 23. Moreover, our methods give exact solutions in these dimensions. Namely, we
can conjecturally determine Stark units as algebraic numbers using sufficiently higha SIC-POVM according to the recipe provided by our conjectures. Thus, we provide thefirst known exact expression for a SIC-POVM in dimension 23.
The paper is organized as follows. In Sections 2 and 3, we provide mathematicalbackground and an introduction to SIC-POVMs and the Stark conjectures, and we define
the notation to be used in the rest of the paper. In Section 4, we take a brief detour toprove a density result on primes ramifying in extensions conjecturally generated by SICPOVMs. In Section 5, we present our main results and conjectures. Finally, in Section 6,
we present data verifying our main conjecture numerically in dimensions d = 5, 11, 17,
and 23.
Work in progress (some of it joint with other authors) will give an explicit
formulation of Observation 1.1 in full generality, including the case of “non-minimal”
SIC-POVMs.
2 Complex Equiangular Lines
In this section, we introduce the definitions from quantum information theory anddesign theory that we need to state our main conjectures. Specifically, we define SICPOVMs and Heisenberg SIC-POVMs. We discuss extended unitary equivalence andGalois equivalence of SIC-POVMs and define the notion of a multiplet of Heisenberg
SIC-POVMs.

2.1 Definition of SIC-POVMs

The study of SIC-POVMs began with Zauner’s 1999 PhD thesis [30] (see English
translation [31]). The term “SIC-POVM” was attached to the concept in 2004 by Renes
et al. [18].
The maximal number of equiangular complex lines possible in Cd is d2 ; thiswas originally proved in 1975 by Delsarte, Goethals, and Seidel using orthogonal
polynomials [8].
Proposition 2.1 (Delsarte et al. [8]). Let α > 0. Consider a set V of unit vectors in Cd
spanning equiangular lines; that is, |v, w| 2 = α whenever v, w ∈ V and v = w. Then,|V| ≤ d2 .

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 (0)), and then verify directly that they produce
precision approximate values of exp(ZA

SIC-POVMs and the Stark Conjectures 5

The same authors also show that, for a set of d2 equiangular complex lines, thesize of the angle is determined.
Proposition 2.2 (Delsarte et al. [8]). For any set V of unit vectors in Cd spanning d2

and thus the common angle is arccos

α=

1
,
d+1




.

√1
d+1

((2.1))A SIC-POVM is defined as a set of rank 1 Hermitian operators on d-dimensional
Hilbert space satisfying certain properties, and it is equivalent to a set of equiangularcomplex lines achieving the upper bound from Proposition 2.1.
1



1
1
d 1 , d 2 , . . . , d d2 of cardinality
d2 consisting of rank 1 Hermitian d × d matrices d1 i such that each 2i = i , andDefinition 2.3 (SIC-POVM). A SIC-POVM is a setTr i j =

1

if i = j,1
d+1

if i = j.

((2.2))

Write i = vi vi† , where vi is a column vector, and vi† is its conjugate-transpose, a row
vector. Then, the vi define a set of d2 equiangular complex lines in Cd , and conversely
any set of d2 equiangular complex lines in Cd define a SIC-POVM. We will use the term
“SIC-POVM” interchangeably with “set of d2 equiangular complex lines in Cd .”There are two types of operators on Cd preserving the SIC-POVM property. A
SIC-POVM {Cv1 , . . . , Cvd2 } may be “rotated” by any unitary matrix U ∈ U(d) = {U ∈GL(Cd ) : UU † = 1} to obtain another SIC-POVM {CUv1 , . . . , CUvd2 }. Moreover, if Cd is thecomplex conjugation operator on Cd , so that Cd v := v, then Cd also preserves the SICPOVM property (and the same holds for any “antiunitary” operator of the form Cd U).
These operators may be collected together to form the extended unitary group.
Definition 2.4.
Lemma 2.5.

Define the extended unitary group EU(d) := U(d)The action of EU(d) takes SIC-POVMs to SIC-POVMs.

Cd U(d).

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equiangular lines with |v, w| 2 = α whenever v, w ∈ V and v = w,

6 G. S. Kopp

Proof.

The action of U(d) preserves the Hermitian inner product; Cd U(d) conjugates

the Hermitian inner product. Thus, both preserve its absolute value and thus the SICPOVM property.

the overlaps are the d4 complex numbers vi , vj . The overlap phases, for i = j, are the√complex numbers d + 1vi , vj , which lie on the unit circle.
2.2 Definition of Heisenberg SIC-POVMs

 
 
i
be
Heisenberg SIC-POVMs are a special class of SIC-POVMs. Let ζd = e d1 = exp 2πda dth root of unity.
Definition 2.7 (Heisenberg group). Let d = d if d is odd, d = 2d if d is even. Let I be
the d × d identity matrix. The Heisenberg group H(d) is the finite group of order d d2generated by the d × d scalar matrix ζd I and the d × d matrices
⎛

0 0

⎜
⎜ 1 0
⎜
⎜
X=⎜
⎜ 0 1
⎜ . .
⎜ .. ..
⎝
0 0

···
···
···
..
.
···

0 1

⎞

⎛

1

0

⎜
⎟
⎜ 0 ζ
0 0 ⎟
d
⎜
⎟
⎜
⎟
⎜ 0 0
0 0 ⎟
,
Z
=
⎜
⎟
⎜ . .
.. .. ⎟⎜ .. ..
⎟
. . ⎠
⎝
1 0
0 0

⎞

0

···

0

0

···

0

ζd2
..
.

···
..
.

0
..
.

0

···

ζdd−1

⎟
⎟
⎟
⎟
⎟.
⎟
⎟
⎟
⎠

((2.3))The Heisenberg group H(d) spans the vector space Md (C) of d × d complex
matrices, and a canonical basis is given as follows.
Definition 2.8 (Heisenberg basis). Let τd = exp

(d+1)π id= − expπid. The set of d2

matrices
Dm,n = τdmn X m Zn for 0 ≤ m, n ≤ d − 1((2.4))

forms a basis of Md (C) over C, and this basis is called the Heisenberg basis. (The Dm,nare known as displacement operators. Elsewhere in the literature, the displacement
operators have been defined both with and without the scalar phase factor of τdmn .)Empirically, all but one of the known SIC-POVMs are equivalent to orbits of the
Heisenberg group action, and that observation motivates the following definition.

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Definition 2.6 (Overlaps). If v1 , v2 , . . . , vd2 are unit vectors in Cd defining a SIC-POVM,

SIC-POVMs and the Stark Conjectures 7

Definition 2.9 (Heisenberg SIC-POVM). A Heisenberg SIC-POVM is a SIC-POVM ofthe form
CDm,n v : 0 ≤ m, n ≤ d − 1((2.5))The elements of EU(d) that preserve the property of being a Heisenberg SICPOVM are restricted to a finite group, the extended Clifford group EC(d), defined to be
the normalizer of H(d) inside EU(d).
Lemma 2.10.

If v is a fiducial vector for a Heisenberg SIC-POVM, and γ ∈ EC(d), thenγ v is also a fiducial vector for a Heisenberg SIC-POVM.
Proof.

The reader may work this out directly or see Scott and Grassl [21, Section 3]. Some authors consider the larger class of group covariant SIC-POVMs, thosewhich are orbits of some subgroup of U(d). The Hoggar lines are group convariant for
H(2) ⊗ H(2) ⊗ H(2), so all known SIC-POVMs are group convariant. In the case of prime
dimension, it was shown by Zhu that all group covariant SIC-POVMs are equivalent toHeisenberg SIC-POVMs [32].
2.3 SIC-POVM-generated fields and the Galois action on Heisenberg SIC-POVMs

In 2016, Appleby et al. [5, 6] empirically discovered a surprising connection between
SIC-POVMs and ray class fields of real quadratic fields. For all Heisenberg SIC-POVMs
they checked, they found that the ratios of the entries of the fiducial vector lie in an√
abelian extension of the real quadratic field Q(), where = d = (d − 3)(d + 1).
√
Every real quadratic field K = Q(
) arises this way—indeed, each K arises forinfinitely many d. While not every class field over K (nor a cofinal set of class fields)
is predicted to arise, a rich family of class fields are predicted, with ramification at a
positive density set of primes. We make this density claim explicit in Section 4.
The key property of the pair (d, ) is the existence of a unit(d − 1) +ε = εd =2√.

((2.6))

with the property that ε3 ≡ 1 (mod d). The presence of ε is related to the order 3 “Zaunersymmetry” enjoyed by many known SIC-POVMs. We call ε the Zauner unit. Note that

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for some vector v ∈ Cd . This v is called a fiducial vector.

8 G. S. Kopp

the Zauner unit is not always equal to the fundamental unit, but is sometimes a higher√
power of it; for example, ε15 = (2 + 3)2 .
Ignoring repeated fields, the sequence



4

,Q




5

,...,Q




d

,...

((2.7))

is essentially an ordering of real quadratic fields by the size of their regulators. Whilein the usual ordering by the size of the discriminant, a “random” quadratic field shouldhave a small class number (size O(o(1) )) and a large regulator (size O(

1/2+o(1) )); in

the ordering given in (2.7), class numbers are large (size O(d)) and regulators are verysmall (size O(log(d))). One might philosophize that ordering real quadratic fields by thesize of their regulators makes them behave more like imaginary quadratic fields, whose
explicit class field theory we understand.
Now, fix d, and let√= (d − 3)(d + 1) and K = Q(). Let v be a Heisenberg

fiducial vector, and let E be the field generated by the ratios of the entries of the v
along with the d th roots of unity, where d = d if d is odd, and d = 2d if d is even. If
σ ∈ Gal(E/K), then v σ is also a Heisenberg fiducial vector; however, v σ may or may notlie in the same EC(d) orbit as v. This Galois action respects orbits because (γ v)σ = γ σ v σ
and EC(d) is Galois-closed.
Definition 2.11 (Multiplet). The set of all those EC(d)-orbits of fiducial vectors, which
are Galois equivalent to a given EC(d)-orbit, is called a multiplet.
3 Explicit Class Field Theory and Zeta Functions
In this section, we give the number-theoretic background we need to state our conjectures. A much more complete exposition of class field theory may be found in Neukirch’s
book [17]. The real quadratic Stark conjectures may be found in [27].
3.1 Global class field theory

Let K be a number field, and let OK be its ring of algebraic integers, the maximal orderof K.
A modulus m is a pair m = (c, S), where c is an ideal of OK , and S is a subset ofthe (possibly empty) set of real embeddings K → R. Associate to the real embeddingsρ1 , . . . , ρr the “infinite primes” ∞1 , . . . , ∞r , and write m using the notation m = c ρj ∈S ∞j .
(An example of this notation is m = (7)∞1 ∞3 ).

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Q

SIC-POVMs and the Stark Conjectures 9

Definition 3.1.
Clm =If m = (c, S), define the ray class group modulo m to be{fractional ideals coprime to c}.
{principal frac. ideals (α) coprime to c with α ≡ 1 (mod c) and ρ(a) > 0 ∀ ρ ∈ S}((3.1))

imposing the stronger condition of coprimility to c .
Clm ∼={fractional ideals coprime to c }. ((3.2)){principal frac. ideals (α) coprime to c with α ≡ 1 (mod c) and ρ(a) > 0 ∀ ρ ∈ S}

Consider two moduli m = (c, S) and m = (c , S ). We say that m|m if c ⊃ c andS ⊂ S . The quotient maps πm ,m : Clm  Clm are defined by first imposing comprimality
to c in Clm , then modding out by the stronger congruence and positivity conditions
modulo m .
The abelian Galois extensions of K are associated to quotients of ray classgroups, by the following existence theorem of Takagi.
Theorem 3.2 (Existence theorem).

Let K be a number field, and let K ab be themaximal abelian extension of K (an infinite-degree extension). Then, there is a natural
isomorphism of the Galois group Gal(K ab /K) with the inverse limit lim← Clm withrespect to the quotient maps πm ,m . This isomorphism is called the Artin map:
Art : lim Clm → Gal(K ab /K).
←((3.3))

The field Lm /K corresponding to Clm by Galois theory—so that Gal(Lm /K) ∼= Clm underthe Artin map—is called the ray class field of K modulo m. If m = (c, S), then theextension Lm is ramified only at the primes dividing c and the real places in S.
Proof.

See [17, Chapter VI, especially Theorem 7.1].



3.2 Ray class zeta functions and Hecke L-functions

We now define two Dirichlet series, ζ (s, A) and ZA (s), attached to a ray ideal class A of
the ring of integers of a number field.
Definition 3.3 (Ray class zeta function).

Let K be any number field, and let c be

an ideal of the maximal order OK . Let S be a subset of the real places of K (i.e.,
the embeddings K → R). Let A be a ray ideal class modulo m = (c, S). Define theDownloaded from https://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rnz153/5610408 by guest on 14 November 2020

If c is any nonzero subideal of c, then the ray class group is unaffected by10 G. S. Kopp

zeta function of the ray class A by the sum over integral idealsζ (s, A) =

N(a)−s for Re(s) > 1.

((3.4))a∈A
a≤OK

Theorem 3.4.

This function ζ (s, A) has a meromorphic continuation to C. It has asimple pole at s = 1 with residue independent of A, and no other poles.
Proof.

See Neukirch [17, Chapter VII, Theorem 8.5] for the corresponding statement

about Hecke L-functions, from which this theorem follows.



The pole at s = 1 may be eliminated by considering the function ZA (s), defined
as follows. The function ZA (s) is holomorphic everywhere.
Definition 3.5 (Differenced ray class zeta function).

Let R be the element of Cm

defined by
R = aOK : a ≡ −1 (mod c) and a is positive at each place in S .

((3.5))

Define the differenced zeta function of the ray class A to be
ZA (s) = ζ (s, A) − ζ (s, RA).

((3.6))

Hecke L-functions (of finite-order Hecke characters) are linear combinations ofray class zeta functions and are ubiquitous in modern number theory, largely because
they have an Euler product. Conversely, ray class zeta functions may be expressed aslinear combinations of Hecke L-functions.
Definition 3.6 (L-function of a finite-order Hecke character). Let K be a number field
and m a modulus of K. Let χ : Clm → C× be a group homomorphism. The Hecke
L-function isL(s, χ ) =χ (A)ζ (s, A).

((3.7))A∈Clm

All our results and conjectures are stated using (differenced) ray class zetafunctions rather than Hecke L-functions, as they are cleaner that way. However, ourcomputer calculations (see Section 6) rely on Magma’s built-in algorithms for computing
Hecke L-functions.

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This function may be meromorphically continued to the whole complex plane.

SIC-POVMs and the Stark Conjectures 11
3.3 A Stark conjecture over real quadratic base field

The Existence Theorem 3.2 does not provide a procedure for actually building the rayclass field Lm . Explicit constructions are known when the base field K is Q or an
explicitly in general.
The Stark conjectures provide one approach to developing an explicit class field
theory. The following conjecture, due to Stark [27], gives a conjectural generator for a
ray class field over a real quadratic field, under certain conditions.
Conjecture 3.7 (Stark conjecture, rank 1 real quadratic case). Let K be a real quadratic
number field with real embeddings ρ1 and ρ2 . Let c be a nonzero ideal of the ring of×
integers of K with the property that, if ε ∈ OK
such that ε ≡ 1 (mod c), then ρ1 (ε) > 0.

Let A be a ray ideal class in Clc∞2 . Let Lc∞j be the ray class field of K modulo c∞j , andlet ρ̃j be a choice of embedding of Lc∞j extending ρj . (Here, ρ̃1 (Lc∞2 ) = ρ̃2 (Lc∞1 ) is a real
field, and ρ̃1 (Lc∞1 ) = ρ̃2 (Lc∞2 ) is a complex [non-real] field.) Then,(1) (0) = log(ρ̃ (α )) for some real algebraic unit α ∈ L
ZA
1 A
A
c∞2 , and furthermore

Lc∞2 = K(αA ).
(2)The units αA are compatible with the Artin map Art : Clc∞2 → Gal Lc∞2 /K .
Art(A)

Specifically, αA = αI

, where I ∈ Clc∞2 is the identity class.

4 Asymptotic Density of Primes Appearing in Moduli of SIC-POVM-Generated
Class Fields
This paper may be viewed as a contribution to explicit class field theory for a thin familyof class fields. In this section, we provide some evidence for the relevance of SIC-POVMs
to Hilbert’s 12th Problem by showing that the family of class fields predicted to arise
is, in a quantifiable sense, not very thin.
In their papers [5, 6], Appleby, Flammia, McConnell, and Yard make the following
conjecture.
Conjecture 4.1 (Class field hypothesis of Appleby, Flammia, McConnell, and Yard). Let√d ≥ 4, and let K = Q( (d − 3)(d + 1). There is a minimal multiplet M of Heisenberg
SIC-POVMs in dimension d containing a representative such that the field generated
by its overlaps is the ray class field Ld∞1 of modulus d∞1 over K, and such that theoverlaps of any other Heisenberg SIC-POVM in dimension d generate a field containing Ld∞1 .

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imaginary quadratic field; however, it is not known how to construct ray class fields

12 G. S. Kopp

We prove that, under the class field hypothesis, a positive density of primes
appear in the moduli of SIC-POVM-generated class fields over a fixed real quadratic
field. Moreover, we quantify that density explicitly.
Let√> 0 be a fundamental discriminant, and let K = Q(). Let P

be the set of primes

P = p prime : (∃d, f ) (d − 3)(d + 1) = f 2

and p|d .

((4.1))(Under the class field hypothesis, P is the set of primes dividing the moduli of minimal
SIC-POVM-generated class fields of K.) Then, P has asymptotic density3#{p ≤ N : p ∈ P }=
N→∞ #{p ≤ N : p is prime}
8μ(P ) := lim((4.2))

in the set of all primes.
Proof.

Let ε > 1 be the smallest totally positive unit of OK greater than 1. (So, ε iseither the fundamental unit or its square.)Consider some p ∈ P , along with some associated (d, f ). The equation (d−3)(d+1) = f 2

may be rewritten as the Pell equation(d − 1)2 − f 2

In other words, d−1+f
2

√

= 4.

((4.3))

is a totally positive unit of OK ; thus,
d−1+f
2

√
= εk for some k ∈ N.

((4.4))We see that εk + 1 + ε−k = d, so z = εk is a root of z2 + z + 1 ≡ 0 (mod p).
Conversely, suppose that for some k ∈ N, z = εk is a root of z2 +z+1 ≡ 0 (mod p).
Let d := εk + 1 + ε−k ∈ pN; we may then write
εk =

d−1+f
2

It follows that (d − 3)(d + 1) = f 2
We have now shown that P
0 (mod p)}.

√for some f ∈ N.

((4.5))as above. Thus, p ∈ P .
= {p prime : (∃k) z = εk is a root of z2 + z + 1 ≡Downloaded from https://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rnz153/5610408 by guest on 14 November 2020

Proposition 4.2.

SIC-POVMs and the Stark Conjectures 13

Write P as a union
P =R

S

I ,

((4.6))

where((4.7))


S = p ∈ P : p = 3 and p is split in K/Q ;((4.8))


I = p ∈ P : p = 3 and p is inert in K/Q .

((4.9))

is finite and thus irrelevant to the density calculation. If p ∈ S , then Fp

contains a cube root of unity, so p ≡ 1 (mod 3). We may rewrite S as
S = p prime : p ≡ 1 (mod 3) ,p
= 1, and 3|op (ε) ;((4.10))here op (ε) denotes the order of the reduction of ε in the multiplicative group

×
×

 2
∼
∼
OK /pOK, and the= F×
= F×
p . On the other hand, if p ∈ I , then OK /pOK
p2
Frobenius automorphism in Gal(Fp2 /Fp ) takes ε to ε−1 ; that is, εp ≡ ε−1 (mod p). Thus,
3|op (ε)|(p + 1), so
I = p prime : p ≡ 2 (mod 3) ,p
= −1, and 3|op (ε) .

((4.11))

We now write S and I as infinite disjoint unions

S =

∞


j

S ,

S

(j)



j−1
= p ∈ S : 3j |(p − 1) and ε ≡ u3
(mod p) for u a cubic nonresidue ,j=1((4.12))
I =

∞

j
I ,

I

(j)



j−1
= p ∈ I : 3j |(p + 1) and ε ≡ u3
(mod p) for u a cubic nonresidue .

j=1((4.13))jLet ζ3j be a primitive (3j )-th root of unity, and let αj be a solution to αj3 = ε. Consider
the Galois fields Kj = K(ζ3j , αj ). The Galois group Gj := Gal(Kj /Q) has a presentation

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The set R
R = p ∈ P : p = 3 or p is ramified in K/Q ;14 G. S. Kopp

Gj = σ1 , σ2 , σ3 , where
σ1 (

√

√
)=−

σ2 (

√

√

)=

σ3 (

√

)=

√

((4.14))σ2 (ζ3j ) = ζ32j

σ3 (ζ3j ) = ζ3j((4.15))σ1 (αj ) = αj−1

σ2 (αj ) = αj

σ3 (αj ) = ζ3j αj .

((4.16))(Note that the automorphism σ1 is central, and that σ2 generates the subgroup of
√
(j)
(j)
automorphisms fixing bothand αj .) We may rewrite the S and I as Chebotarev
sets:
(j)



 j−1
j−1
= rational primes p : Frobp (Kj ) = σ33 , σ2−1 σ33 σ2 ;((4.17))

(j)




j−1
j−1
= rational primes p : Frobp (Kj ) = σ1 σ33 , σ2−1 σ1 σ33 σ2 .

((4.18))S
I
Moreover,





(J)
⊆ rational primes p : Frobp (Kj ) = {id} =: S ;((4.19)) (j) (J)
I ⊆ rational primes p : Frobp (Kj ) = {σ1 } =: I .

((4.20))

S

(j)

j>J

j>J

Thus, we may bound S and I between finite unions of Chebotarev sets:


⎛
S

(j)

⊆S ⊆⎝

1≤j≤J



⎛
I

(j)

⊆I ⊆⎝

1≤j≤J



⎞
(j)
S ⎠

1≤j≤J



S

(J)

;

((4.21))

⎞
I

(j) ⎠

I

(J)

.

((4.22))

1≤j≤J

For an set of primes S, denote by μ(S) the asymptotic density of S in the set of all primes.
   (J) (j)(j)
By the Chebotarev density theorem, μ S= μ I= |G2 | = 2·312j−1 , and μ S=j (J) 1
μ I
= 4·32J−1. Taking densities in (4.21) and (4.22), we obtain the inequalities
J

j=1

1
2 · 32j−1

⎛
J

≤m≤⎝
j=1

⎞
1
1
⎠+
for both m = μ(S ) and m = μ(I ).
4 · 32J−1
2 · 32j−1((4.23))

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σ1 (ζ3j ) = ζ3−1j

SIC-POVMs and the Stark Conjectures 15

Sending J → ∞, we see thatμ(S ) = μ(I ) =

∞

j=1

1
3
=
.
2j−1
16
2·3

((4.24))

5 Toward an Infinite Family of Heisenberg SIC-POVMs
We now state our main conjectures and results. Our 1st conjecture predicts the existenceof a Galois orbit of real algebraic units in the ray class field L(d)∞2 satisfying certain
strong conditions, for d an odd prime such that d ≡ 2 (mod 3). We will show in
Theorem 5.4 that these conditions imply the existence of a Heisenberg SIC-POVM in
dimension d.
Conjecture 5.1. Let d be an odd prime such that d ≡ 2 (mod 3). Let = (d − 3)(d +√1) and K = Q(), and consider the class group Cl(d)∞2 . With indices m, n ∈ Z/dZ,(m, n) = (0, 0), let√Am,n = αOK : α ≡ m + n(mod d) and ρ2 (α) > 0 ∈ Cl(d)∞2 ,((5.1))


and let Art : Cl(d)∞2 → Gal L(d)∞2 /K denote the Artin map of class field theory. Then,
there is a real algebraic unit α such that the ray class field L(d)∞2 = K(α) and such thatits (real) Galois conjugates αm,n := α Art(Am,n ) over the Hilbert class field L(1) have thefollowing properties:
(1)

−1 .
α−m,−n = αm,n(2)The αm,n ≡ 1 (mod p) for all prime ideals p of OL(d)∞ dividing dOL(d)∞ .

(3)

The roots of (d + 1)x2 = αm,n are in L(d)∞2 .

(4)

Fix a choice of p dividing dOL(d)∞ , and let νm,n be the unique root of

2

2

2

(d + 1)x2 = αm,n((5.2))

satisfying νm,n ≡ 1 (mod p). Let ν0,0 = 1. Then, the matrix
1 νm,n D−m,−ndd−1 d−1

M=((5.3))

m=0 n=0

is idempotent (i.e., M 2 = M) and rank 1 (i.e., all the rank 2 minors of M
vanish). (See (2.4) for the definition of Dm,n .)Downloaded from https://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rnz153/5610408 by guest on 14 November 2020

3
3
Thus, μ(P ) = 16
+ 16
= 38 .

16 G. S. Kopp

The condition that d is prime and d ≡ 2 (mod 3) is necessary because (5.1) does
not make sense for all (m, n) = (0, 0) without it. For general d, the principal subgroup√of Cl(d)∞2 is represented by those Am,n such that (m + n) is relatively prime to (d).

Our 2nd conjecture provides an explicit analytic formula for a set of real
numbers that we expect to be Galois conjugate algebraic units with the desired (0) are known to be real numbers.)
properties. (The ZA

Conjecture 5.2.

Let d be an odd prime such that d ≡ 2 (mod 3). A unit α satisfying

Conjecture 5.1 and its Galois conjugates over K may be constructed as Stark units α Art(A) = exp ZA(0) ,((5.4))

for all A ∈ Cl(d)∞2 .
 Conjecture 3.7, due to Stark, predicts that exp ZA(0) = α Art(A) for an algebraic unit α generating L(d)∞2 over K. Point (1) of Conjecture 5.1 follows, because
ZRA (s) = −ZA (s). Points (2), (3), and (4) of Conjecture 5.1 provide, to our knowledge, newpredictions about Stark units.
 (0) appearThere are several analytic formulas for the derivative zeta values ZA

ing on the right-hand side of (5.4). Shintani’s Kronecker limit formula expresses 
exp ZA(0) as a product of special values of Barnes’s double gamma function [23, 24]. A
different formula appears in Chapter 4 of the author’s PhD thesis [16].
We state Conjectures 5.1 and 5.2 as two separate conjectures because it may bepossible to prove the 1st without proving the 2nd. One route by which this might be done
is through the use of p-adic zeta functions rather than Archimedian zeta functions. The
p-adic approach seems hopeful because p-adic analogues of the Stark conjectures havebeen proven in some cases. At least, the p-adic condition (2) for corresponding d-adic
special values looks potentially amenable to proof.
We now prove a lemma about the structure of the class fields over K, which we
require for our main result, Theorem 5.4.
Lemma 5.3.

√

Let K = Q(√), where= (d − 3)(d + 1) and d ≥ 4. Let εd = (d−1)+
2.

×If η ∈ OK
and η ≡ 1 (mod d), then η = εd3k for some k ∈ Z, and in particular η is totally

positive. It follows that the class fields L(d) , L(d)∞1 , L(d)∞2 , and L(d)∞1 ∞2 are all distinct.

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The condition that d is prime and d ≡ 2 (mod 3) is equivalent to (d) being prime in OK ,√that is, to (m + n) being relatively prime to (d) whenever (m, n) ≡ (0, 0) (mod d).

SIC-POVMs and the Stark Conjectures 17

Proof.

It suffices to prove that the minimal η > 1 in OK such that η ≡ 1 (mod d) is ε3 ;consider the minimal such η. We have ηn = εd3 for some n ∈ Z, n > 0.
Let η and ε denote the nontrivial Galois conjugates of η and ε, respectively. If
Thus, n ≤ 2. Suppose n = 2. Then εd has a square root is K; it is straightforward
to check that this happens exactly when d + 1 is a square, in which case εd = εd2 0 withd = d20 − 2d0 . Then
η = εd3=√(d0 − 1)(d − 2) + d2≡ −d0 (mod d) .

((5.5))((5.6))((5.7))

Thus, η ≡ 1 (mod d).
So we must have n = 1, which is what we wanted to prove.
It follows that the class groups Cl(d) , Cl(d)∞1 , Cl(d)∞2 , and Cl(d)∞1 ∞2 are all
distinct. Thus, by Theorem 3.2, the class fields L(d) , L(d)∞1 , L(d)∞2 , and L(d)∞1 ∞2 areall distinct.

We are now ready to prove that Conjecture 5.1 implies the existence of a
Heisenberg SIC-POVM.
Theorem 5.4.

Let d be an odd prime such that d ≡ 2 (mod 3). Assume Conjecture 5.1,

and let M be the matrix constructed therein. Let σ ∈ Gal(L(d)∞2 /Q) be any Galois√√
automorphism not fixing K; that is, σ () = −. Then σ (M) = vv † for a fiducial
vector v of a Heisenberg SIC-POVM.
Proof.

The αm,n are Galois conjugate real algebraic units that generate L(d)∞2 over K.

By Lemma 5.3, L(d) = L(d)∞2 . The αm,n must not be totally real, because if they were, they

−1 , so the αwould all lie in L(d) . Moreover, by (1), α−m,−n = αm,nm,n are Galois conjugate−1
to their inverses. Take τ ∈ Gal(L(d)∞2 /K) such that τ (α) = α −1 . Then, τ (αm,n ) = αm,nbecause Gal(L(d)∞2 /K) is abelian. Thus,(σ τ σ −1 )(σ (αm,n )) = σ (αm,n )−1 .

((5.8))We have τ = ArtR , where
R = βOK : β ≡ 1 (mod d) and ρ2 (β) > 0 ∈ Cl(d)∞2 .

((5.9))

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18 G. S. Kopp

Thus, by class field theory, the fixed field of τ is L(d) . It follows that the class fieldof σ τ σ −1 ∈ Gal(L(d)∞1 /K) is also L(d) . Thus, σ τ σ −1 must act by complex conjugation,
because L(d) is the real subfield of the complex field L(d)∞1 .

d+1

definition of M, we obtain Tr(MDm,n ) = νm,n . Thus, Tr σ (M)σ (D

m,n )

 2

 = Tr σ (M)σ (D

m,n )



 

(σ τ σ −1 ) Tr σ (M)σ (Dm,n )((5.10))

  

 
= σ Tr MDm,n τ Tr MDm,n((5.11))= σ νm,n ν−m,−n((5.12))

=

1
.
d+1

((5.13))Moreover, the action of σ on the Heisenberg group is determined by its action of Q(ζd ),and there is some λ ∈ (Z/dZ)× such that σ (Dm,n ) = σ (Dm,λn ). So, by changing the valueof n, we have Tr σ (M)D

m,n

 2
 =

1
,
d+1

((5.14))for all (m, n) = (0, 0). Now write σ (M) = vw † for some v, w ∈ Cd with w † v = 1, which ispossible because M, and thus σ (M), is rank 1 and idempotent. We will show that σ (M)is in fact a Hermitian projector; that is, w = v. Conjugation-transposition acts on σ (M)as follows:
σ (M)† =m

σ (νm,n )D†−m,−λn =

n


m

σ (ν−m,−n )D†m,λn = σ (M).

((5.15))nThus, w = v. So (5.14) may be rewritten as v, Dm,n v = √ 1

d+1

for (m, n) = (0, 0); also,

v, v = 1. In other words, v is a fiducial vector for a Heisenberg SIC-POVM in Cd .

Conjecture 5.1 has been verified numerically for d = 5, 11, 17, and 23. We discusseach case in turn in Section 6.
6 Examples
In this section, we use Conjectures 5.1 and 5.2 to compute exact SIC-POVMs in dimensions d = 5, 11, 17, and 23. In each case, our solutions represent the minimal multiplet(in the sense of Appleby et al. [6] and Conjecture 4.1), or (in the case of d = 23) we expect

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Thus, σ (αm,n ) lies on the unit circle. It follows that σ (νm,n ) = √ 1 . From theSIC-POVMs and the Stark Conjectures 19

them to. Originally, the case d = 5 is due to Zauner [30, 31], d = 11 is due to Scott andGrassl [21], and d = 17 is due to Appleby et al. [4]. The case d = 23 is new.
approxWe numerically compute the numbers αA

 (0)) for the ideal classes:= exp(ZA

approxThe αAare expected to be Galois conjugate algebraic numbers (see Conjecture 3.7);

in each case, we find that they indeed agree to high precision with Galois conjugate
exact . The α exact are found by using Mathematica’s built-in lattice
algebraic numbers αAAbasis-reduction algorithms to compute the coefficients of a minimal polynomial fd (x)over L(1) , the Hilbert class field of K.
Next, we verify conditions (2) and (3) of Conjecture 5.1 and compute the minimal
polynomial gd (x) of a choice of νA . The latter is done by factoring fd ((d + 1)x2 ) over L(1)in Magma. Finally, we verify that the νA satisfy condition (4) of Conjecture 5.1.
Let g̃d (x) denote a polynomial obtained from gd (x) by applying a Galois√ √
automorphism σ to the coefficients with the property that σ= −. The roots
νm,n of gd (x) are real numbers, and we have already computed them not only as a set,
but also as an orbit of the Galois group G = Gal(L(d)∞2 /L(1) ). The roots of g̃d (x) arecomplex numbers of absolute value √ 1 , and they are the overlaps of the SIC-POVM
d+1

we are looking for. In order to compute a fiducial vector, we need to compute the Galois
action on them as well.
In each case (d = 5, 11, 17, 23), the Galois group G = Gal(L(d)∞2 /L(1) ) ∼
=

×
×

d−1
OK /dOK /εd  is a cyclic group of order n = 3 . Let γ be a generator for OK /dOK ,so τ = Art(Aγ ) is a generator for G. Fix ν = ν1,0 . We compute τ (ν) in the monomial basis{1, ν, · · · ν n−1 } for L(d)∞2 over L(1) :τ (ν) = hd (ν) =n−1cj ν j ,((6.1))

j=0

where cj ∈ L(1) . This is done by solving the following linear system for the cj .
⎛

τ (ν)

⎞

⎛

1

ν

⎜ 2
⎟ ⎜
⎜ τ (ν) ⎟ ⎜ 1
τ (ν)
⎜
⎟ ⎜
⎜
⎟=⎜ .
..
.
⎜
⎟ ⎜ ..
..
.
⎝
⎠ ⎝
n−1
1 τ
ν
(ν)

···

ν n−1

···
..
.

τ (ν)n−1..
.

···

τ n−1 (ν)n−1

⎞⎛
⎟⎜
⎟⎜
⎟⎜
⎟⎜
⎟⎜
⎠⎝

c0
c1
..
.

⎞
⎟
⎟
⎟
⎟.
⎟
⎠

((6.2))

cn−1

The action on τ on the roots of gd (x) is given by applying the polynomial hd (x), so the
action of a generator τ̃ for Gal L(d)∞1 /L(1) on the roots of g̃d (x) is given by applying the

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A ∈ Cl(d)∞2 (K), using Magma’s built-in functions for evaluation of Hecke L-functions.

20 G. S. Kopp

polynomialh̃d (x) =n−1τ (cj )xj .

((6.3))

From the roots ν̃m,n of g̃d (x), a fiducial vector vd is computed by applying
σ to (5.3):
1 ν̃m,n D−m,−λn .
d
d−1 d−1

vd vd† = σ (M) =((6.4))

m=0 n=0

As we have not kept track of the action of σ on roots of unity, we must check different
values of λ ∈ (Z/dZ)× until we find one that works.
While this method produces an exact fiducial, we write down only a numericalfiducial in the examples that follow and in the accompanying text files. (We have alreadyspecified vd exactly up to a Galois action by writing down gd (x), and the minimal
polynomials of its entries have prohibitively large coefficients in some cases.)In the cases d = 5, 11, and 17, the coefficients of gd (x) live in K, so g̃d (x) is
determined by gd (x). In the case d = 23, the coefficients of gd (x) live in a degree 2
extension of K, so there are two choices for g̃d (x). The two choices lead to two different
PEC(23)-orbits, 23b and 23f.
The ancillary text files accompanying this paper contain expressions forfd (x), gd (x), hd (x), and vd ; in the case d = 23, expressions for related polynomialsg∗d (x) and h∗d (x) are given instead, as explained in Section 6.4. The ancillary files are
dim5.txt, dim11.txt, dim17.txt, dim13.txt, and hstar23.txt.

6.1 The example d = 5

In dimension d = 5, the corresponding quadratic discriminant is = (d−3)(d+1) = 12.
√√√
The real quadratic base field is K = Q(
) = Q( 3), with Zauner unit ε = d−1+=2√
2 + 3 equal to the fundamental unit.
There is exactly one PEC(5)-orbit of Heisenberg SIC-POVMs [21].
We construct “the” Heisenberg SIC-POVM in dimension 5 explicitly using∼ Z/8Z. The numbers
Conjecture 5.1 and Conjecture 5.2. The ray class group Cl(K) =approxαA

d∞2

 (0)) were computed to 50 digits of precision and were found to= exp(ZA

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j=0

SIC-POVMs and the Stark Conjectures 21

numerically satisfy the following polynomial f5 (x) over K.

The minimal polynomial of the corresponding νA is√√√
g5 (x) = 1296x8 − (648 + 1080 3)x7 + 648x6 + (864 + 360 3)x5 − (540 + 360 3)x4√√
+ (144 + 60 3)x3 + 18x2 − (3 + 5 3)x + 1.
((6.6))The overlaps of the Heisenberg SIC are the roots of the conjugate polynomial√√√g̃5 (x) = 1296x8 − (648 − 1080 3)x7 + 648x6 + (864 − 360 3)x5 − (540 − 360 3)x4√√
+ (144 − 60 3)x3 + 18x2 − (3 − 5 3)x + 1.
((6.7))

A numerical approximation to a fiducial vector is⎛0.24167903563278788347⎞⎟⎜⎜ −0.42393763943145804455 − 0.23674553208033493698i ⎟⎟⎜⎟⎜v5 ≈ ⎜ 0.67406464953559540185 + 0.19581007881800632630i ⎟ .
⎟⎜⎜ 0.04040992380093525849 − 0.19581007881800632630i ⎟⎠⎝0.34218699574986301405 + 0.23674553208033493698i((6.8))

6.2 The example d = 11

In dimension d = 11, the corresponding quadratic discriminant is = (d − 3)(d + 1) =√√√96. The real quadratic base field is K = Q(
) = Q( 6), with Zauner unit ε = d−1+=2√
5 + 2 6 equal to the fundamental unit.
There are three PEC(11)-orbits of SIC-POVMs [21], denoted 11a, 11b, and 11c.
They occur in two multiplets, {11a, 11b} and {11c}.
We construct the orbit 11c explicitly using Conjectures 5.1 and 5.2. The ray class∼ Z/40Z. The numbers α = exp(Z (0)) were computed to 50 digits ofgroup Cl
(K) =
d∞2

A

A

Downloaded from https://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rnz153/5610408 by guest on 14 November 2020√√√√
f5 (x) = x8 − (8 + 5 3)x7 + (53 + 30 3)x6 − (156 + 90 3)x5 + (225 + 130 3)x4√√√
− (156 + 90 3)x3 + (53 + 30 3)x2 − (8 + 5 3)x + 1.
((6.5))

22 G. S. Kopp

precision and were found to numerically satisfy the following polynomial f11 (x) over K.
√√
f11 (x) = (x40 + 1) + (−106 − 44 6)(x39 + x) + (10614 + 4334 6)(x38 + x2 )√+ (−877414856 − 358203120 6)(x35 + x5 )√+ (21232702036 + 8668214302 6)(x34 + x6 )√+ (−404105217077 − 164975264036 6)(x33 + x7 )√+ (6148885983306 + 2510272190954 6)(x32 + x8 )√+ (−75622522312964 − 30872765454828 6)(x9 + x31 )√+ (756937617777704 + 309018488445524 6)(x10 + x30 )√+ (−6189687857216636 − 2526929486213638 6)(x11 + x29 )√+ (41399992237530827 + 16901476056189164 6)(x12 + x28 )√+ (−226256732983154420 − 92368924446311502 6)(x13 + x27 )√+ (1007258543741292244 + 411211578537502754 6)(x14 + x26 )√+ (−3634879852543059214 − 1483933485842242424 6)(x15 + x25 )√+ (10563883893311542549 + 4312687540103174716 6)(x16 + x24 )√+ (−24534973105051444408 − 10016360826714109164 6)(x17 + x23 )√+ (45162757813812803926 + 18437618670122548666 6)(x18 + x22 )√+ (−65380387193394562674 − 26691431301568773124 6)(x19 + x21 )√+ (74013773227204686051 + 30215996390786346646 6)x20 .

((6.9))

Downloaded from https://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rnz153/5610408 by guest on 14 November 2020√√
+ (−652115 − 266222 6)(x37 + x3 ) + (27825305 + 11359634 6)(x36 + x4 )

SIC-POVMs and the Stark Conjectures 23

The minimal polynomial of the corresponding νA is√√+ (25848 + 10560 6)(1217 x37 + x3 ) + (−419472 − 171072 6)(1216 x36 + x4 )√+ (−18892224 − 7714080 6)(1215 x35 + x5 )√+ (−181457280 − 74074176 6)(1214 x34 + x6 )√+ (2141686656 + 874329984 6)(1213 x33 + x7 )√+ (62948109312 + 25698435840 6)(1212 x32 + x8 )√+ (337583904768 + 137818340352 6)(1211 x31 + x9 )√+ (−5182578339840 − 2115780175872 6)(1210 x30 + x10 )√+ (−83855167709184 − 34233724293120 6)(129 x29 + x11 )√+ (−202894373007360 − 82831288725504 6)(128 x28 + x12 )√+ (4929898807885824 + 2012622741651456 6)(127 x27 + x13 )√+ (47257471319703552 + 19292782106492928 6)(126 x26 + x14 )√+ (45726669189808128 + 18667833415925760 6)(125 x25 + x15 )√+ (−1783877902738587648 − 728265100678004736 6)(124 x24 + x16 )√+ (−11823467430652674048 − 4826910371186737152 6)(123 x23 + x17 )√+ (−11846897461773729792 − 4836475651119906816 6)(122 x22 + x18 )√+ (215144426763866603520 + 87832344582220677120 6)(12x21 + x19 )√+ (1246186807345680482304 + 508753633031010385920 6)x20 .

((6.10))Downloaded from https://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rnz153/5610408 by guest on 14 November 2020√√
g11 (x) = (1220 x40 + 1) + (48 + 22 6)(1219 x39 + x) + (1968 + 792 6)(1218 x38 + x2 )

24 G. S. Kopp

A numerical approximation to a fiducial vector is⎛⎞⎟−0.00727092982813886277 − 0.27361988848296183641i ⎟⎟⎟−0.02661472547965484998 − 0.37021984761997660380i ⎟⎟0.26103622782810404567 − 0.17519492308655237763i ⎟⎟⎟−0.17130782441847984905 − 0.00498132225998709619i ⎟⎟⎟0.115894663935244395576 + 0.023682353557509036148i ⎟ .
⎟−0.20515008071739008667 − 0.15408741485673275946i ⎟⎟⎟−0.12010487681658182011 + 0.12297653825975158437i ⎟⎟⎟0.24462906391022207471 − 0.16496679971724189093i ⎟⎟0.56644313698737356128 + 0.07879109944268139950i ⎟⎠0.16629395529532658791 − 0.07929717656822654842i((6.11))

6.3 The example d = 17

In dimension d = 17, the corresponding quadratic discriminant is= (d − 3)(d +√√1) = 252. The real quadratic base field is K = Q(
) = Q( 7), with Zauner unit
√
√
d−1+
ε=
= 8 + 3 7 equal to the fundamental unit.
2
There are three PEC(17)-orbits of SIC-POVMs [21], denoted 17a, 17b, and 17c.
They occur in two multiplets, {17a, 17b} and {17c}.
We construct the orbit 17c explicitly using Conjectures 5.1 and 5.2. The ray classgroup Cl(O ) ∼= Z/96Z. The numbers α = exp(Z (0)) were computed to 50 digits of
d∞2

28

A

A

precision and were found to numerically satisfy the a polynomial f17 (x) over K. Valuesof f17 (x), g17 (x), h17 (x), and a fiducial vector v17 may be found in the accompanying text
file dim17.txt.
6.4 The example d = 23

In dimension d = 23, the corresponding quadratic discriminant is= (d − 3)(d +√√1) = 480. The real quadratic base field is K = Q(
) = Q( 30), with Zauner unit
√
√
d−1+
ε=
= 11 + 2 30 equal to the fundamental unit. Unlike our previous examples,2
OK is not a principal ideal domain, but rather has class number 2.
According to previous numerical searches, there appear to be six PEC(23)-orbitsof SIC-POVMs [21], denoted 23a, 23b, 23c, 23d, 23e, and 23f. We have explicitly computed
one multiplet consisting of two PEC(23)-orbits using Conjecture 5.1; Markus Grassl has

Downloaded from https://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rnz153/5610408 by guest on 14 November 2020

⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
v11 ≈ ⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝

0.31885501173446151953

SIC-POVMs and the Stark Conjectures 25

checked that our solutions are unitary equivalent to {23b, 23f } [14]. We expect, but havenot shown, that the other four orbits form a multiplet {23a, 23c, 23d, 23e}.
approxThe ray class group Cld∞2 (K) ∼== Z/2Z × Z/176Z. The numbers αAapproxThe values αm,n
= exp(ZA(0)) (corresponding to the principal ideal classes) werem,nfound to numerically satisfy the a polynomial f23 (x) over the Hilbert class field L(1) =√ √Q( 5, 6).
√√In this example, the number d + 1 = 2 6 is in the Hilbert class field L(1) , so we √ 
simplify our calculation somewhat by computing g∗23 (x) = g23 2 6x instead of g23 (x),and by computing a polynomial h∗23 (x) for the action of a Galois group generator on aroot of g∗23 (x). Values of f23 (x), g∗23 (x), and fiducial vectors v23b and v23f for 23b and23f may be found in the accompanying text file dim23.txt; the value of h∗23 (x) may befound in h23star.txt.
Funding
This work was partially supported by the National Science Foundation [NSF grant DMS-1401224,
NSF grant DMS-1701576, and NSF RTG grant 0943832].
Acknowledgments
Thank you to Jeff Lagarias for many inspiring conversations about the SIC-POVM problem,
Hilbert’s 12th problem, and the Stark conjectures, and for helpful comments on drafts ofthis paper, especially regarding the formulation of the conjectures. This paper includes andextends work from Chapter 5 of the author’s PhD thesis, which was advised by Jeff Lagarias.
Proposition 4.2 has been obtained independently by Gary McConnell in unpublished work, andmy method of upper-bounding the density was inspired by a conversation with Andrew Booker.
I thank Marcus Appleby, Neil Gillespie, Markus Grassl, Gary McConnell, and Harold Stark forfruitful discussions. Finally, I thank the anonymous referees for helpful comments, corrections,and suggestions.

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