Constructing exact symmetric informationally complete measurements from numerical solutions
Marcus Appleby, Tuan-Yow Chien, Steven Flammia, Shayne, Waldron
TL;DR
This paper introduces a novel method to convert high-precision numerical solutions of SIC POVMs into exact algebraic solutions using Galois symmetries, significantly expanding the set of known solutions and supporting related number-theoretic conjectures.
Contribution
The authors develop a new approach combining integer relation algorithms and Galois symmetries to find exact SIC solutions from numerical data, surpassing previous methods in scope.
Findings
Calculated 69 new exact SIC solutions
Discovered solutions in high-degree number fields up to 12,288
Confirmed SIC solutions obey number-theoretic conjectures
Abstract
Recently, several intriguing conjectures have been proposed connecting symmetric informationally complete quantum measurements (SIC POVMs, or SICs) and algebraic number theory. These conjectures relate the SICs and their minimal defining algebraic number field. Testing or sharpening these conjectures requires that the SICs are expressed exactly, rather than as numerical approximations. While many exact solutions of SICs have been constructed previously using Gr\"obner bases, this method has probably been taken as far as is possible with current computer technology (except in special cases where there are additional symmetries). Here we describe a method for converting high-precision numerical solutions into exact ones using an integer relation algorithm in conjunction with the Galois symmetries of a SIC. Using this method we have calculated 69 new exact solutions, including 9 new…
| minimal multiplet | maximal multiplet | ||
| minimal multiplet | intermediate multiplet(s) | maximal multiplet | ||
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| generator | in terms of radicals | minimal polynomial | |
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Constructing exact symmetric informationally complete measurements from numerical solutions
∗Marcus Appleby
,
†Tuan-Yow Chien
,
*∗,‡*Steven Flammia
and
†Shayne Waldron
∗Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, Australia
†Department of Mathematics, University of Auckland, Auckland, New Zealand
‡Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, USA
Abstract.
Recently, several intriguing conjectures have been proposed connecting symmetric informationally complete quantum measurements (SIC POVMs, or SICs) and algebraic number theory. These conjectures relate the SICs and their minimal defining algebraic number field. Testing or sharpening these conjectures requires that the SICs are expressed exactly, rather than as numerical approximations. While many exact solutions of SICs have been constructed previously using Gröbner bases, this method has probably been taken as far as is possible with current computer technology (except in special cases, where there are additional symmetries). Here we describe a method for converting high-precision numerical solutions into exact ones using an integer relation algorithm in conjunction with the Galois symmetries of a SIC. Using this method we have calculated 69 new exact solutions, including 9 new dimensions where previously only numerical solutions were known, which more than triples the number of known exact solutions. In some cases the solutions require number fields with degrees as high as 12,288. We use these solutions to confirm that they obey the number-theoretic conjectures and we address two questions suggested by the previous work.
1. Introduction
Symmetric informationally complete quantum measurements (SIC POVMs as they are often called, or SICs as we will call them in this paper) are collections of equiangular lines in in a -dimensional complex vector space. They were originally introduced by Zauner[1] and Renes et al [2], and they have many applications to quantum information [3, 4, 5, 6, 7, 8, 9], as well as playing a central role in the QBist approach to the interpretation of quantum mechanics [10]. They also have applications to classical signal processing [11, 12]. SICs have been calculated numerically[2, 13, 14, 15, 16, 17] in every dimension up to 151, and for a handful of other dimensions up to . Exact solutions[1, 18, 13, 19, 20, 16, 17] have been calculated in dimensions –, , , , , , , , . This encourages the conjecture that SICs exist in every finite dimension, but a proof continues to elude us. The SICs in dimensions 2 and 3, together with the Hoggar lines [21] in dimension 8, are sometimes called sporadic SICs, on the grounds that they have a number of special properties [22, 23]. In this paper we exclude them from consideration. In the following the term “SIC” will therefore always mean “non-sporadic SIC”. In particular, we will always assume without comment that the dimension is greater than 3.
In physics a numerical solution to a system of equations is usually perfectly adequate, especially when, as here, it is known to 35 digits of precision or more. However, as we discuss below, SICs have important features (important for the applications as well as from a pure mathematical point of view) which are only accessible if one has an exact solution in hand. Here we face the difficulty that, aside from a few low dimensional cases where a hand calculation is possible, all previously known exact solutions were obtained using a Gröbner basis. Moreover every solution in dimensions higher than 16 relied on the existence of additional symmetries which are only present in special cases. The great majority of SICs do not have such symmetries, and for these the Gröbner basis method has probably been pushed about as far as it is possible to go without massive increases in computer speed and memory. We therefore need a new method for calculating exact SICs. In this paper we describe such a method. The idea is to start with a high-precision numerical solution (much easier to calculate) and then convert it into an exact solution.
The method relies on, and is motivated by some recently discovered connections between SICs and algebraic number theory [24, 25]. Dimension 3 excepted, the standard-basis matrix elements of every known exact fiducial projector are algebraic numbers. More than that, they are expressible in radicals (i.e. the components can be built up from the integers using the standard arithmetical operations together with the operation of taking roots), meaning that the associated Galois group is solvable. It turns out that the number fields they generate have some remarkable properties [26, 27, 28]. We outline the most salient points of this number-theoretic connection here assuming some familiarity with the basic concepts underlying SICs; however, in section 2 below we provide all of the necessary definitions and background.
Let be a SIC fiducial projector in dimension , and let be the field generated over the rationals by the standard-basis components of together with . We refer to as the SIC field. It is easily seen that only depends on the orbit to which belongs (where is the extended Clifford group in dimension ). Let where is the square-free part of . Then one finds, in every case that has been calculated,
- •
.
- •
.
- •
The Galois group is Abelian.
(where the notation means that is a normal extension of ). In each dimension for which the full set of exact SICs has been calculated there is exactly one SIC field of minimal degree; every other SIC field being a low-degree extension of that. In ref. [27] it was shown that, in every known case, the minimal field is a very special kind of field extension: Namely, the ray class field over with conductor and ramification at both infinite places (where if is odd, and if is even).
If SICs existed in every finite dimension, and if the above statements were generally true, it would be a fact of some interest to algebraic number theorists. Ray class fields are used to classify fields having an Abelian Galois group over some given base-field. The Kronecker-Weber theorem states that the ray class fields over are precisely the fields , where is a complex root of unity. Kronecker further conjectured, and it was subsequently proved, that the ray class fields over , for a positive integer, are generated by the coordinates of certain distinguished points on an elliptic curve. Hilbert’s problem, still unsolved, asks for the generalization of these results; the obvious place to start being ray class fields over , for a positive integer—i.e., fields of precisely the kind that SICs generate (in the handful of cases we have been able to calculate). This does not mean that solving the SIC problem is equivalent to solving Hilbert’s problem. In the first place SICs only give us some of the ray class fields over a given . In the second place a solution to the SIC problem would not necessarily give us the analogues of the exponential and elliptic functions that Hilbert was asking for. Against that, it was shown in ref. [27] that, for each square-free positive integer , there is an infinite sequence of dimensions for which . So although a solution to the SIC problem would not give us the full set of ray class fields over a given , it might give us an infinite subset. Moreover, the SIC problem may conceivably reduce to proving a set of special-function identities. If so that would give us the functions Hilbert wanted. In short, it seems fair to say that a constructive solution to the SIC problem might be a significant step in the direction of solving Hilbert’s problem for real quadratic fields.
We have stressed the potential relevance of SICs to a major unsolved problem in algebraic number theory. The reverse is also true: the number-theoretic aspects give important insight into SIC geometry, as appears from ref. [29] which describes a series of significant geometrical results, none of which would have been suspected without the clue from number theory, and which cannot be properly understood in isolation from the number theory. Two of the authors are actively engaged in pursuing these connections between number theory and geometry further.
For many purposes a high-precision approximate SIC is completely satisfactory. However, if one wants to investigate the connections with number theory exact solutions are essential. As we discussed above, the Gröbner basis method, used to obtain most of the previously known exact solutions, is extremely demanding computationally and is probably not capable of being taken much further on existing computer hardware. Fortunately, number theory, besides creating the demand, also supplies the means of satisfying it. That is, one can exploit the conjectured number-theoretic properties of a SIC to bump the high precision approximate SICs in refs [13, 14] up to infinite precision. Success in this enterprise incidentally provides additional evidence in support of the conjectures on which the method is based.
The essential idea is as follows. Suppose one is given a real number specified to some finite degree of precision, and suppose one knows that is an approximation to an exact real number in a specified algebraic number field . Let be a basis for over the rationals. Then for some set of rational numbers . Multiplying through by the LCM of the we deduce
[TABLE]
for some set of integers . If we can calculate the we will have managed to find an exact number which approximates. With high enough precision and computational power, these particular integers will satisfy , and we will have recovered , the specific exact number we’re after. This procedure can be done using an integer-relation algorithm [30], such as the PSLQ algorithm [31].
A few remarks are in order. In the first place, there are infinitely many sequences satisfying Eq. (1) to the specified degree of accuracy, and there is no guarantee that the algorithm will return the one we want. Basically, the algorithm is a systematic guessing procedure. However, for our purposes that is good enough, since we can verify the guess (by checking that the final result really is an exact SIC fiducial).
In the second place, one does not expect to get more information out than one initially puts in. Suppose is given to digits of precision. Then if one does not want to generate spurious results one needs to be larger than a number , where is the maximum number of digits in the integers . In practice [30] one needs to be 10–15% larger than . The problem is, of course, that one does not initially know . As a practical rule of thumb we therefore proceed by repeatedly running the algorithm at successively higher levels of precision until the result is stable. Again, the justification for this procedure is the fact that the end result demonstrably is an exact SIC fiducial.
In the third place, there is the problem that we do not always know the field . As discussed above, we have a conjecture regarding the minimal SIC field in each dimension. However, many fiducials lie in an extension of this field for which we currently have no conjecture. A further problem is that in the cases of interest the SIC field is often of degree or more. For the reasons discussed above this means that we need to work to very high precision, and the calculations are correspondingly slow. For the calculations in this paper we therefore exploited a conjecture in ref. [26], which implies that the coefficients of suitably chosen polynomials lie in a much smaller field, which is easily inferred from the numerical data, and for which the calculations are much faster.
Scott-Grassl [13] calculated 179 numerical fiducials in dimensions –, and argued that with high probability their list is complete, in the sense that it includes exactly one representative of every orbit of unitarily equivalent SICs in these dimensions. Scott [14] calculated a further 323 numerical fiducials in dimensions –, restricting the search to fiducials having an or symmetry. He states that it is “likely” that this list is complete (by contrast, he is “confident” that the list in ref. [13] is complete). Another 76 numerical fiducials have been calculated [14, 15, 16, 17] in dimensions – and for a handful of additional dimensions up to , but with no claim as to completeness. This gives us a total of unitarily inequivalent numerical fiducials for . By contrast, there are only orbits for which an exact solution has been calculated [1, 18, 13, 19, 20, 16, 17] with . We have calculated exact representatives for a further orbits, and of these orbits we provide a complete analysis for of them. This means that we now have exact representatives for total orbits, more than half of the orbits in dimensions between and . We would have liked to calculate exact representatives and provide a complete analysis for every orbit for which a numerical fiducial is known, but unfortunately our method is not completely automated and is still too time consuming for us to justify this additional effort. In choosing which fiducials to calculate we were guided by two considerations. In the first place the conjectures in ref. [26], as was there noted, cannot be true of the type- orbits , , , (where we employ the Scott-Grassl [13] labeling scheme). We have calculated exact representatives of these orbits in order to find modified conjectures which stand a chance of holding for every type- orbit. In the second place, we noted above that the minimal SIC field in each dimension seems always to be the ray class field over with conductor . This raises the question, what can be said of the non-minimal fields in each dimension. We therefore set out to find a complete set of SICs in each dimension. Thus, Scott and Grassl only give an exact representative for one of the orbits in dimensions , , , , and . We have found exact representatives for all the other orbits in these dimensions. In particular we have found full sets of exact solutions for dimensions and —the two dimensions less than for which the number of distinct orbits is greatest. This information may help us to find a conjecture similar to the ray-class field conjecture applying to the non-minimal SIC fields.
The complete list of dimensions for which we provide exact solutions, and where previously only a numerical solution was known, is . From this list, we have neglected to provide a complete analysis for , although our methods could also be applied there by a sufficiently motivated individual.
The plan of the paper is as follows. Section 2 is devoted to necessary preliminaries. Mostly this material is in refs. [18, 13, 26, 27, 28]. Some of it, however, is new. Section 3 is devoted to the type- orbits , , , mentioned in the last paragraph. We show how the conjectures in ref. [26] naturally generalize to these orbits also. Section 4 concerns the relationship between the SIC fields in a given dimension. Scott and Grassl [13] find at most three exact solutions in a given dimension, and no more than two distinct SIC fields. With the new solutions calculated here we now have up to ten exact solutions for a given dimension, and up to seven distinct SIC fields. In section 4 we summarize this information. In section 5 we describe the method used to calculate the new solutions. Finally, in Appendix A we present our results. Many of the fields we calculate are much higher degree111Calculating with fields this enormous leads to numerous technical difficulties. For instance, we estimated that it would take many months to directly check the irreducibility of the successive polynomials used to define some of the field towers (by trying to factor them in Magma), and we therefore had recourse to an indirect argument based on the subfield structure. Again, when calculating with such fields it is essential to devise ways of minimizing the number of distinct arithmetical operations (especially the number of divisions) if one does not want to devote months or even years of CPU time to the task. One also needs to give careful thought to the amount of RAM taken up by some of the intermediate results. than the fields calculated in ref. [13], and this creates some presentational difficulties. For instance the print-out for exact fiducial occupies almost a thousand A4 pages (font size 9 and narrow margins). There can therefore be no question of presenting the fiducials themselves in the appendix. Instead we have made the fiducials available online [32], and confined ourselves here to a description of the fields and Galois groups.
2. Preliminaries
The purpose of this section is to fix notation, and to summarize some relevant results from refs. [18, 13, 14, 26, 27, 28]. Some of the statements in this section are proven facts while others are observations only known to be true for the SICs which have actually been calculated. The latter will always be clearly identified as such. As explained in the Introduction we assume without comment that the SICs with which we deal are non-sporadic in the sense of ref. [23]. In particular it will always be assumed that the dimension , and that the SICs with which we deal are covariant with respect to the Weyl-Heisenberg (WH) group.
We first define the WH group. Let be a -dimensional Hilbert space, and let be the standard basis. Let , by the operators which act according to
[TABLE]
where and addition of indices is mod . For each define the displacement operator
[TABLE]
where . The WH group then consists of all operators of the form , for an integer.
The displacement operators constitute a nice unitary error basis [33, 34]. In particular, an arbitrary operator has the expansion
[TABLE]
We next define the Clifford and extended Clifford groups. Let (respectively ) if is odd (respectively even), and define the extended symplectic group to consist of all matrices with entries in and determinant . The symplectic group is the subgroup consisting of matrices with determinant . For each there exists an operator on , unique up to an overall phase, such that
[TABLE]
The operator is a unitary if and an anti-unitary if . We refer to it as a symplectic unitary in the first case and an anti-symplectic anti-unitary in the second. For all
[TABLE]
where means “equal up to a phase” (so the map is a projective representation of ). The Clifford group (respectively extended Clifford group ) consists of all operators of the form with (respectively ) and an arbitrary phase.
There are [18, 35] explicit formulae for the operators . If the symplectic matrix
[TABLE]
is such that the associated unitary is given by
[TABLE]
where is an arbitrary phase, and is the multiplicative inverse of considered as an element of . Matrices in not satisfying the condition can always be written as a product of two matrices which do satisfy it. An antisymplectic matrix can be written as where and
[TABLE]
The anti-unitary acts by complex conjugation in the standard basis. Using these facts we can calculate for arbitrary .
By definition a WH-covariant SIC consists of the operators where is a rank-1 projector with the property
[TABLE]
for some set of phases . We refer to as the fiducial, to the quantities as the overlaps, and to the numbers as the overlap phases. It can be seen from Eq. (4) that the overlap phases completely determine the fiducial.
Observation 1. In every case which has been calculated, the overlap phases are units in the field generating a subgroup of the full unit group whose properties are described in ref. [27].
If is a SIC fiducial projector then so is for every . Consequently the fiducial projectors split into a collection of orbits. Scott and Grassl [13] have, with high probability, identified every orbit for and Scott [14] has extended the calculations to higher dimensions.
Observation 2. At the time of writing it appears that there are only finitely many orbits in every dimension .
We define the stability group of a SIC fiducial to be the set of all such that .
Observation 3. In every known case the stability group contains a unitary of the form , where .
Such a unitary is necessarily order 3, up to a phase. We refer to it as canonical order 3.
It is convenient to choose standard canonical order 3 unitaries. If or every canonical order 3 unitary is conjugate modulo a phase to , where
[TABLE]
We refer to as the Zauner matrix222We define the Zauner matrix this way to secure consistency with ref. [13]. The Zauner matrix is often defined to be which is order 3, unlike which is order 6 when is even. This makes no difference at the level of the unitaries since and are equal modulo a phase..
Observation 4. In the cases where it is likely that we know the full set of orbits it is found that if the additional conjugacy class never gives rise to SICs, while if it always does.
When we choose [13], as representative of the additional conjugacy class , where
[TABLE]
We say that an orbit is type- (respectively type-) if it contains fiducials stabilized by (respectively ).
We say that a subgroup of is displacement-free if it consists entirely of operators of the form , with . Following refs. [27, 28] we shall say that a fiducial is centred if its stability group (a) contains a canonical order 3 unitary, and (b) is displacement free (in ref. [26] such fiducials were called simple).
Observation 5. Every known orbit of SIC fiducials contains at least one centred fiducial.
Given a centred fiducial we define to consist of all such that is in the stability group of .
We next consider the Galois symmetries of the Clifford unitaries and SIC projectors. We say that an operator is in (respectively generates) a field as shorthand for the statement that its standard basis matrix elements are in (respectively generate) . Similarly we write to mean . It is immediate that the displacement operators generate the cyclotomic field . The same is true of the symplectic unitaries, provided the phase in Eq. (8) is chosen appropriately. This was shown in ref. [36] for the case of prime dimensions using an argument based on Gauss sums. The argument is easily generalized to show, for arbitrary , that if one chooses
[TABLE]
then , and consequently is in for all . In the sequel we shall always assume that this choice of phase has been made.
Now let be any SIC fiducial projector and define .
Observation 6. In every known case the extension is finite degree and normal. In particular contains complex conjugation implying .
Observation means that in every known case is the same for every fiducial projector on the same orbit. We refer to it as the SIC-field for that orbit333Note that in ref. [26] the extension was also introduced, to ensure that the Clifford unitaries would be in the field. The considerations in the previous paragraph show that this was unnecessary..
Observation 7. Let be a centred fiducial, and let be complex conjugation. In every known case breaks into the tower
[TABLE]
where
- (1)
The notation means “ is a normal extension of ”. 2. (2)
is the fixed field of the centralizer of . 3. (3)
, where is the square-free part of . 4. (4)
is Abelian, 5. (5)
For all
- (a)
fixes if and only if is a SIC fiducial projector , 2. (b)
fixes if and only if is a fiducial on the same orbit as . 6. (6)
. In particular is a degree 2 extension of . 7. (7)
Let be the non-trivial element of and let be any element of which does not fix . Then
[TABLE] 8. (8)
For all and some
[TABLE] 9. (9)
, , are normal over , but is not.
Note that it follows from item (7) that .
We say that a fiducial to which Observation 7 applies is strongly centred if it is centred and , so that the overlaps are all in . If centred fiducials are automatically strongly centred[26]. If the concepts are not equivalent. However
Observation 8. For every orbit for which an exact solution is known there exist strongly centred fiducials whose stability group includes (for a type- orbit) or (for a type- orbit)444This fact was not appreciated at the time ref. [26] was written. Consequently, the representatives of orbits , , , in that paper are not strongly centred. They become strongly centred if is replaced with , where . This transformation leaves the stability group unchanged. Furthermore, if is strongly-centred then in every known case the set of overlaps generates over . .
We refer to a set of orbits sharing the same SIC field as a multiplet.
Observation 9. In every known case the Galois group acts transitively on the elements of a multiplet.
Observation 10. In those cases where Scott and Grassl [13] give exact fiducials for what is probably the full set of orbits there are at most two distinct multiplets. We refer to the field of lowest (respectively highest) degree as the minimal (respectively maximal) field, and to the corresponding multiplet as the minimal (respectively maximal) multiplet. It is found that when there are two multiplets, so that the minimal and maximal fields are distinct, the minimal field is a subfield of the maximal field.
Observation 11. Let be the minimal field in dimension , and let be the associated tower. In every known case[27, 28]
- (1)
, are ray class fields over for which the finite part of the conductor is , 2. (2)
is the ray class field with ramification allowed at both infinite places, 3. (3)
is the ray class field with ramification only allowed at the infinite place taking to a positive real number, 4. (4)
is the Hilbert class field over .
We next describe the action of on the Clifford unitaries. For any operator in define
[TABLE]
where is the standard basis. Let be the unique integer in the interval which is co-prime to and such that , and let
[TABLE]
Then acts on the elements of according to
[TABLE]
Observation 12. Let be the orbits of any of the known multiplets in dimension . Let be the associated tower, and let be a strongly centered fiducial on orbit . Then one finds that for each and each index there exists and an index such that
[TABLE]
Also
[TABLE]
for all , where
[TABLE]
If then in Eqs. (20) and (21). The matrices , are not unique: Indeed, they can be replaced with arbitrary elements of the cosets , respectively, where
[TABLE]
For the automorphism (as defined in Observation 7) we can choose[26]
[TABLE]
for all .
Let be a tower to which the previous observations apply. With notation as in Observation 12, choose define where is a lift of to . It is easily seen that this definition does not depend on which of the two lifts is taken. Then it can be shown [26] that the are all in , the normalizer of in , and that for each fixed the map is an injective homomorphism of into . It is natural to ask, what is the range of this homomorphism. The answer depends on whether the orbit is type- or type-.
Observation 13. Suppose the orbit is type-. Then, in every known case, one finds that the range of the homomorphism is , where is the centralizer of in . So we have an isomorphism
[TABLE]
The conjecture, that this isomorphism is generally true of a type- orbit, plays an important role in the calculations in the next section. Its restriction to the case of minimal SIC fields would, if true, also be of number-theoretical interest: For it would mean that the Galois group of a ray class field over the Hilbert class field of a real quadratic field is, in many cases, isomorphic to the quotient of two matrix groups.
Turning to type- orbits one has:
Observation 14. For the two type- orbits for which exact solutions are given in ref. [13] one finds [26]
[TABLE]
just as is the case for the type- orbits.
However this observation is of limited significance. The problem [26] is that one can see from the numerical data that the group is non-Abelian555By contrast, it can be shown [26] that for a type- orbit is necessarily Abelian. for the type- orbits , , , . So for these orbits one of two things must be true: Either is not an Abelian extension of , or else is not isomorphic to . We discuss this problem further in the next section.
3. Type- fiducials
An alternative to the isomorphism of Eq. (25) holding for type- fiducials was conjectured in ref. [26]. In this section we propose a stronger conjecture, which holds for the previously known fiducials , , and also for fiducial calculated using the methods described in Section 5.
Suppose . Then where and
[TABLE]
The fact that is coprime to 3 means (see Appendix B of ref. [19]) that there is a natural isomorphism
[TABLE]
It is straightforward to show that extends to an isomorphism of onto , and that, for arbitrary ,
[TABLE]
where (respectively ) if is odd (respectively even). Applying this to we find that
[TABLE]
where
[TABLE]
Observe that , implying that is conjugate to the Zauner matrix in dimension .
Now consider the isomorphism of Eq. (25). Suppose, to begin with, that is a strongly centred type- fiducial. Without loss of generality we may assume that is an eigenvector of . Assume also that is Abelian (as is the case for every known fiducial with ). It is shown in ref. [26] that , the centralizer of in , is Abelian, from which it follows that so that the isomorphism becomes
[TABLE]
On the assumption that the left hand side is Abelian (as is the case for every known exact fiducial) this relation cannot hold for an arbitrary type- orbit. Indeed, it can be seen from Eq. (30) that
[TABLE]
The fact that is conjugate to the Zauner matrix in dimension means that is Abelian; however is non-Abelian, implying that is non-Abelian. This means that, if is generated by (as it is in, for example, orbits , , , ), then is non-Abelian.
As a generalization of Eq. (32) holding for every orbit, irrespective of whether it is type- or type-, we propose:
Conjecture 1**.**
Let be a strongly-centred fiducial, and let be such that the unitary is canonical order 3. Then
[TABLE]
where is a maximal Abelian subgroup of containing .
Note that in the case of type- fiducials this conjecture reduces to Eq. (32). It holds for every known exact fiducial. In particular, it holds for the type- fiducials and (calculated in ref. [13]) and (calculated here).
In the case of a type- fiducial there is exactly one maximal Abelian subgroup containing ; namely, the centralizer . But for a type- fiducial there are several. Indeed, one sees from Eq. (30) that a subgroup containing is maximal Abelian if and only if
[TABLE]
where is an arbitrary maximal Abelian subgroup of . One finds that must be conjugate to one of the three groups
[TABLE]
which are order , and respectively. Consequently, must be conjugate to one of the three groups
[TABLE]
We say that an orbit is type- if is isomorphic to . Of the known cases orbit is type- while orbits , are type-. It is an open question, whether there exist any type- orbits.
In ref. [19] the following, weaker conjecture was proposed for strongly-centred type- fiducials:
Conjecture 2**.**
Let be a strongly-centred type- fiducial and let be such that is a canonical order 3 unitary. Then
[TABLE]
where
[TABLE]
for some matrix such that .
To see that this conjecture is indeed weaker, let be a strongly centred type- fiducial satisfying Conjecture 1. Then for some . The fact that is conjugate to the Zauner matrix in dimension means, in view of Lemma 12 in ref. [26],
[TABLE]
Define
[TABLE]
(note that the fact that means is an integer). Then we also have , implying
[TABLE]
It is straightforward to show that
[TABLE]
where
[TABLE]
Using the Chinese Remainder Theorem [37] one finds
[TABLE]
where
[TABLE]
The fact that
[TABLE]
means . So satisfies Conjecture 2. To see that Conjecture 2 is strictly weaker than Conjecture 1 consider the order subgroup of .
[TABLE]
where . Let \mathcal{H}_{2}=\chi^{-1}\bigl{(}(\bar{G},\bar{\mathcal{H}}_{2})\bigr{)}. Then
[TABLE]
where
[TABLE]
Moreover . So a fiducial for which would satisfy Conjecture 2. However, it would not satisfy Conjecture 1 because is not maximal Abelian, being properly contained in , , .
The group plays an important role in the reconstruction of exact type- fiducials from numerical ones, as we will discuss in the next section. This is because is the centre of , which means that is contained in every maximal Abelian subgroup of containing . In view of its importance it may be worth noting that, in addition to Eq. (51), one also has
[TABLE]
Indeed, it is easily verified that
[TABLE]
(note that the fact that means and are both divisible by ). Together with the relation this means
[TABLE]
4. Fields and Multiplets
It is shown in ref. [27] that in every known case the minimal SIC field in each dimension is the ray-class field over with conductor and ramification at both infinite places. We will refer to the conjecture, that this is always the case, as the ray-class conjecture. However in many dimensions (not all) there are other SIC fields, having higher degree. This raises the question, whether one can generalize the ray-class conjecture to these additional fields.
For the dimensions where Scott and Grassl give a full set of exact solutions there are either one or two SIC fields. We refer to the field of lowest (respectively highest) degree as the minimal field, denoted (respectively maximal field, denoted ). Associated to the two fields are two multiplets, which we refer to as the minimal and maximal multiplets. This information is presented in Table 1.
It raises several questions:
- (1)
In the four dimensions listed for which there are two distinct SIC fields, the maximal field is an extension of the minimal one. One would like to know if that is merely a low dimensional accident, or whether it is generally true. 2. (2)
There are at most two distinct fields in the dimensions listed. One would like to know if in higher dimensions there are sometimes more than two. 3. (3)
In the dimensions listed it never happens that there is more than one multiplet associated to a given field. One would like to know if this correspondence between fields and multiplets persists in higher dimensions. 4. (4)
In the dimensions listed is the ray class field over with conductor and ramification at both infinite place. One would like to formulate a conjecture applying to the non-minimal fields in each dimension. For this purpose it would be useful to have more examples than the four cases , , , .
One of our aims in the calculations reported here was to address these questions. Our results reveal that the number of fields in a given dimension can be larger than . However, in every case calculated, one continues to find that there is a unique field of minimal degree over , and a unique field of maximal degree over . The field is always the ray class field over with conductor and ramification at both infinite places. The field always contains , and additional fields , when they exist, always lie between these two:
[TABLE]
We accordingly refer to them as intermediate fields, and to the associated multiplets as intermediate multiplets. Finally, one continues to find that there is a correspondence between fields and multiplets. This information is summarized in Table 2 and Fig. 1
5. Method for Calculating Exact Fiducials
We first describe the obvious, or “brute-force” approach to the problem. We then describe the refinements actually used to perform the calculations in this paper. Suppose that is a known high precision numerical fiducial corresponding to an unknown exact fiducial in dimension . Suppose, in addition, that it is somehow known that is the minimal SIC-field for dimension . If the conjectures in ref. [27, 28] are correct is then the ray class field over with conductor and ramification at both infinite places. Using Magma we can easily calculate a basis for the field over . Let be such a basis. Then for all , there exist integers such that
[TABLE]
For the numerical fiducial one then has
[TABLE]
to a high degree of precision. If the precision is sufficiently great we can use an integer relation algorithm to find the integers . Substituting these integers in
[TABLE]
gives us the exact matrix element for each , . There are three problems with this “brute-force” approach:
- •
It requires us to guess which numerical fiducials correspond to an exact fiducial generating the ray-class field.
- •
We do not currently have a conjecture for when it is not the minimal field.
- •
In the cases of interest to us the degree of the minimal field is large, which means that the required precision is large, and the calculation correspondingly slow.
To deal with these problems we therefore modified the procedure. We actually employed two different methods, which we refer to as methods 1 and 2. Method 1 was the original method used to calculate the exact fiducials in dimensions –. Method 2 is a much improved method used to calculate the exact fiducials for (and also for which, although the dimension is smallest, was actually calculated last).
Method 1
We describe the method as it applies for a type- fiducial, and then indicate the modifications needed to deal with a type- fiducial at the end. Define
[TABLE]
Let be the centralizer of the symmetry group , and let be the orbits of \bigl{(}\mathbb{Z}/d^{\prime}\mathbb{Z}\bigr{)}^{2} under the action of . If (respectively666We need to treat the case when differently because one cannot tell if a numerical fiducial is strongly centred or not. If the fiducial is not strongly centred then only equals up to a cube root of unity [26] (where is any element of ), and the coefficients of are consequently guaranteed to be in the field . Cubing the overlaps obviates this difficulty. ) let be the set of distinct numbers (respectively ) obtained as runs over , and define
[TABLE]
Of course, we do not know the overlaps , so the polynomial is also unknown. However, , and the orbits can all be calculated just from a knowledge of . Let be the set of distinct numbers ( if ) obtained as runs over , and define
[TABLE]
The known polynomials will then be high precision approximations to the unknown polynomials . If the conjecture that is isomorphic to , is correct (c.f. Eq. (25)) then the coefficients of all belong to the field . Since the degree of is much less than that of (no greater than for the fiducials calculated in this paper), the degree of , and consequently the degree of its numerical approximant , is also comparatively small. It is consequently possible to use an integer relation algorithm to calculate the coefficients of from those of without knowing the field in advance, and without an impractically high degree of precision and CPU time (using, for instance, the Magma function MinimalPolynomial or the Mathematica function RootApproximant). In practice it is most efficient to use this method to find the next-to-leading coefficient of the lowest degree polynomial (because this is the coefficient which involves the smallest integers, and is therefore easiest to find). Once has been ascertained one can then use a function such as IntegerRelation (in Magma) or FindIntegerNullVector (in Mathematica) which assumes a knowledge of the field to find every coefficient of every polynomial. The total CPU time needed to obtain exact expressions for the full set of polynomials was second or less in every case. The precision needed was decimal digits or less. We then used Magma to construct the the ray class field over with conductor and ramification at both infinite places. For a minimal SIC fiducial exact values of the overlaps were found directly, by factoring the over this field. If the fiducial was not minimal we factored the as far as possible, and then used Magma to find simplified generators for as an extension of the ray class field. The overlaps can then be expressed in terms of the generators. This part of the calculation took day of CPU time. Finally, the exact fiducial was calculated from the overlaps using Eq. (4).
For a type- fiducial the above procedure needs to be modified slightly. This is because, if Conjecture 1 is correct, is isomorphic to for some maximal Abelian subgroup which, unlike for a type- fiducial, cannot be determined from the numerical data. To get round this problem we use the fact that, as shown in Section 3, the group necessarily contains the group , which can be calculated in advance of knowing the exact fiducial from Eq. (53). The calculation then mirrors the calculation for a type- fiducial, but using instead of . Specifically, let be the orbits of \bigl{(}\mathbb{Z}/d^{\prime}\mathbb{Z}\bigr{)}^{2} under the action of , and let be the set of distinct numbers obtained as runs over ; then define . Let
[TABLE]
Then the coefficients of are in , the fixed field of . It follows from the results in Section 3 that . Consequently, the degree of is no more than times greater than the degree of , which means one can apply the technique described in the last paragraph for a type- fiducial to determine the coefficients of the from their approximants , and hence to determine the exact fiducial.
There are two main steps to the method just described. The first step is to find the coefficients of the polynomials using an integer relation algorithm; the second is to factor the polynomials. As we noted above the first step is fast, taking second of CPU time in every case tried. The second step is much slower. Moreover, the computation time grows rapidly with the degree of the number field. To deal with this problem we developed a new method, which does not rely so heavily on factoring, and is therefore much more efficient. While developing this method we also discovered the fact noted in Section 2, that every known centred fiducial becomes strongly centred when multiplied by the appropriate displacement operator. This obviated the need when to first calculate the cubed overlaps and then to take the cube root at the end of the calculation.
Method 2
As before we first describe the method as it applies to type- fiducials, the modifications needed to handle type- fiducials being described afterwards.
As with method 1, we begin by calculating high precision numerical approximations to the polynomials ; with, however, the difference that even when the numbers in Eq. (62) are the overlaps themselves, as opposed to their cubes. We then use the Magma function MinimalPolynomial to obtain exact versions of the coefficients. If we take to be the field generated by the coefficients. If we repeat the calculation for each of the nine fiducial vectors for which , and then take the desired strongly centred fiducial to be the one for which the polynomials obtained by applying MinimalPolynomial to the coefficients have lowest degree. If the SIC is minimal can be taken to be the appropriate ray-class field. Otherwise is an extension of this field. We find a set of generators for the extension by factoring some of the polynomials . It is crucial to the success of this method that in practice one does not need to factor all the . More than that: it turns out that one can get the additional field generators by factoring a polynomial whose degree is order or less (which particular polynomial being ascertained by trial and error). The fact that the degree is so small means that the factoring can be done in minutes at most. If instead one had to factor the of highest degree this method would have few, if any advantages over Method 1 described above.
Once the fields and have been determined we use the conjectured isomorphism of Eq. (25) to reduce the rest of the calculation to a problem in linear algebra. Let be a basis for the extension , let and be explicit listings of the elements of and respectively, and let be the set of permutations with the property that
[TABLE]
is an isomorphism of onto . Given any overlap
[TABLE]
for some . The conjecture is that for some permutation
[TABLE]
for all . If correct this means
[TABLE]
and , are the vectors
[TABLE]
Of course, we do not know the permutation . To find it we consider in turn each of the permutations in , and use it in conjunction with Eq. (67) to calculate a high-precision candidate for the vector (in practice digits of precision was sufficient). Let be the component of this candidate vector, and let be a basis for . We use the Magma function IntegerRelation to find integers solving the equation . We find that for one choice of permutation the norms are many orders of magnitude smaller than they are for any of the others. We take this permutation to correspond to the true isomorphism between and , and
[TABLE]
to be the exact value of the vector . We then invert Eq. (67) to obtain the exact value of the overlaps . By repeating this procedure we obtain an exact value for every overlap and then use Eq. (4) to obtain the exact fiducial. Finally, we check that the vector so obtained really is an exact fiducial.
In this description we have assumed that the fiducial to be calculated is type-. To calculate a type- fiducial the method can be modified along the lines indicated at the end of our description of method 1.
Appendix A Results
We used the method described in Section exact fiducials for orbits , , , , . The fiducials themselves are available online [32]. The purpose of this appendix is to describe the associated fields, Galois groups and the isomorphism of Eq. (34).
When is not a multiple of we always calculate an exact version of the corresponding Scott-Grassl [13] numerical fiducial. When is a multiple of we transform the Scott-Grassl fiducial so as to make it strongly-centred. Let be an exact version of the Scott-Grassl numerical fiducial in such a dimension, and let be the strongly-centred fiducial we have calculated. Then
[TABLE]
where is the displacement operator specified in Table 3.
The multiplet structure for these dimensions is specified in Table 2. The maximal field for each dimension is specified in Tables 4–8, using the same notational conventions as in ref. [26] to denote the field generators:
- •
denotes , the square-free part of .
- •
denote square roots of integers.
- •
denotes or .
- •
denote numbers constructed recursively from by taking sums, products and roots.
- •
denotes the square root of .
The , and generators are tabulated in Table 4; the generators in Tables 5, 6, 7. The minimal polynomials of the cubic generators are tabulated in Table 8. The fields in the tower of Eq. (14) are tabulated in Table 9, for both the maximal and the minimal and intermediate multiplets in each dimension.
Information regarding the Galois group is tabulated dimension by dimension, in a series of boxes. Let be the tower for the maximal multiplet in dimension . The first box specifies the action of the generators of . The generators are denoted , , and are chosen so that
- (1)
is an order extension of the non-trivial element of . 2. (2)
fix . In particular, they are mutually commuting. Moreover
[TABLE] 3. (3)
is the non-trivial element of . 4. (4)
Complex conjugation is given by (c.f. Eq. (15)).
For each generator , Box also tabulates the order of , and . In view of Eq. (72) it therefore completely specifies . Box specifies and its various subgroups for every multiplet in dimension . It also specifies how the Galois group switches between the different orbits in a multiplet. Finally, we tabulate the matrices defined by Eq. (21). In dimension let , be two orbits such that generator takes to , and let , be the exact fiducials on these orbits given in ref. [32]. Then
[TABLE]
where are the matrices tabulated in box 3. We only tabulate the matrices for cases where the action is non-trivial, so that . The matrices in Eq. (20) can then be obtained by inverting Eq. (22).
Dimension 15
order
b^{\prime}_{2}=2\operatorname{Re}\Bigl{(}e^{\frac{2\pi i}{3}}(10+30i)^{\frac{1}{3}}\Bigr{)}
orbit
interchanges , and restricts to the identity on , ; restricts to the identity on
Dimension 17
order
b^{\prime}_{1}=32\Bigl{(}1938+765a-462r_{2}-181ar_{2}-2(-171-72a+67r_{2}+32ar_{2})t2(-5844-2303a+1404r_{2}+551ar_{2})t^{2}+4(-173-64a+229r_{2}+112ar_{2})t^{3}-8(-2592-1021a+624r_{2}+245ar_{2})t^{4}-16(39+24a+113r_{2}+56ar_{2})t^{5}+32(-348-137a+84r_{2}+33ar_{2})t^{6}+512(2+a)(1+r_{2})t^{7}\Bigr{)}^{\frac{1}{2}}
b^{\prime\prime}_{1}=32i\Bigl{(}\frac{1}{2}(-204+119a+60r_{2}-33ar_{2})-8(-39+20a-3r_{2})t+16(-81+25a-39r_{2}+17ar_{2})t^{2}+4(-491+240a-51r_{2})t^{3}-8(-600+211a-216r_{2}+91ar_{2})t^{4}-16(-243+112a-27r_{2})t^{5}+96(-36+13a-12r_{2}+5ar_{2})t^{6}+256(-9+4a-r_{2})t^{7}\Bigr{)}^{\frac{1}{2}}
b^{\prime}_{2}=2\operatorname{Re}\Bigl{(}e^{\frac{4\pi i}{3}}\bigl{(}17(44+3i\sqrt{21})\bigr{)}^{\frac{1}{3}}\Bigr{)}
orbit
interchanges orbits , and restricts to the identity on .
Dimension 18
order
b^{\prime}_{1}=-\bigl{(}(a-15)(5+r_{1})\bigr{)}^{\frac{1}{2}}
b^{\prime\prime}_{1}=i\bigl{(}a+15)(5-r_{1})\bigr{)}^{\frac{1}{2}}
b^{\prime}_{2}=2\operatorname{Re}\Bigl{(}e^{\frac{4\pi i}{3}}\bigl{(}-11+i\sqrt{95}\bigl{)}^{\frac{1}{3}}\Bigr{)}
b^{\prime}_{3}=2\operatorname{Re}\Bigl{(}e^{\frac{4\pi i}{3}}\bigl{(}\frac{3}{2}(1+i\sqrt{95})^{\frac{1}{3}}\bigr{)}\bigr{)}
orbit
interchanges orbits , .
,
Dimension 19
order
b^{\prime}_{2}=2\operatorname{Re}\Bigl{(}e^{\frac{4\pi i}{3}}\bigl{(}38(13+3i\sqrt{5})\bigr{)}^{\frac{1}{3}}\Bigr{)}
orbit
interchanges orbits , and restricts to the identity on , , . restricts to the identity on ; restricts to the identity on , .
, , , , , ,
Dimension 20
order
2
4
24
2
2
2
b^{\prime}_{1}=\bigl{(}(a-17)(17+r_{1})\bigr{)}^{\frac{1}{2}}
b^{\prime\prime}_{1}=i\bigl{(}(a+17)(17-r_{1})\bigr{)}^{\frac{1}{2}}
b^{\prime}_{2}=2\operatorname{Re}\bigl{(}10e^{\frac{2\pi i}{3}}(19+i\sqrt{119})^{\frac{1}{3}}\bigr{)}
b^{\prime}_{3}=\Bigl{(}\frac{1}{3}\Bigl{(}\bigl{(}(-a+39)r_{3}r_{4}+(-7a+21)r_{3}\bigr{)}t-\bigl{(}(-2a+15)r_{2}+3a\bigr{)}r_{4}
\phantom{b^{\prime}_{3}=\Bigl{(}}-3(-a+18)r_{2}+195\Bigr{)}\Bigr{)}^{\frac{1}{2}}
b^{\prime\prime}_{3}=-\Bigl{(}\frac{1}{3}\Bigl{(}\bigl{(}-(a+39)r_{3}r_{4}+(7a+21)r_{3}\bigr{)}t^{\prime}+\bigl{(}(2a+15)r_{2}-3a\bigr{)}r_{4}
\phantom{b^{\prime\prime}_{3}=\Bigl{(}}-3(a+18)r_{2}+195\Bigr{)}\Bigr{)}^{\frac{1}{2}}
orbit
interchanges orbits , .
, , ,
Dimension 21
order
b^{\prime}_{2}=-\bigl{(}33+66a+22r_{1}+33ar_{1}+(11-8a-5ar_{1})b_{1}\bigr{)}^{\frac{1}{2}}
b^{\prime\prime}_{2}=i\bigl{(}-33+66a+22r_{1}-33ar_{1}-(77+4a-44r_{1}-3ar_{1})b_{1}\bigr{)}^{\frac{1}{2}}
b^{\prime}_{3}=\bigl{(}363+66a+66r_{1}+33ar_{1}+(55+24a-7ar_{1})b_{1}\bigr{)}^{\frac{1}{2}}
b^{\prime\prime}_{3}=-\bigl{(}363-66a-66r_{1}+33ar_{1}-(33-20a-44r_{1}+15ar_{1})b_{1}\bigr{)}^{\frac{1}{2}}
b^{\prime}_{4}=\operatorname{Re}\Bigl{(}e^{\frac{4\pi i}{3}}\bigl{(}4(-4+9a+i\sqrt{465+72a})\bigr{)}^{\frac{1}{3}}\Bigr{)}
b^{\prime\prime}_{4}=\operatorname{Re}\Bigl{(}e^{\frac{2\pi i}{3}}\bigl{(}4(-4-9a+i\sqrt{465-72a})\bigr{)}^{\frac{1}{3}}\Bigr{)}
orbit
cycles orbits in the order ; restricts to the identity on .
, , , , , ,
Dimension 24
order
b^{\prime}_{3}=4\operatorname{Re}\Bigl{(}e^{\frac{2\pi i}{3}}(1+i\sqrt{7})^{\frac{1}{3}}\Bigr{)}
orbit
interchanges , and restricts to the identity on .
Dimension 28
order
b^{\prime}_{3}=2\operatorname{Re}\Bigl{(}e^{\frac{2\pi i}{3}}(189+21i\sqrt{87})^{\frac{1}{3}}\Bigr{)}
orbit
interchanges , and restricts to the identity on .
Dimension 30
order
b^{\prime}_{2}=2\operatorname{Re}\Bigl{(}e^{\frac{2\pi i}{3}}(1+2i\sqrt{31})^{\frac{1}{3}}\Bigr{)}
b^{\prime}_{3}=2\operatorname{Re}\Bigl{(}e^{\frac{2\pi i}{3}}\Bigl{(}4+b^{\prime}_{2}+i\sqrt{48-8b^{\prime}_{2}-b^{\prime 2}_{2}}\Bigr{)}^{\frac{1}{3}}\Bigr{)}
b^{\prime}_{4}=2\operatorname{Re}\Bigl{(}e^{\frac{2\pi i}{3}}(70+10i\sqrt{31})^{\frac{1}{3}}\Bigr{)}
orbit
cycles orbits in the order , and restricts to an order automorphism on .
Dimension 35
order
b^{\prime}_{2}=2\operatorname{Re}\Bigr{(}e^{\frac{2\pi i}{3}}\bigl{(}280+210i\sqrt{6}\bigr{)}^{\frac{1}{3}}\Bigr{)}
b^{\prime}_{4}=\operatorname{Re}\Bigl{(}e^{\frac{2\pi i}{3}}\bigl{(}28+84i\sqrt{3}\bigr{)}^{\frac{1}{3}}\Bigr{)}
orbit
cycles orbits in the order and interchanges orbits ; it satisfies the relations on , on ; it restricts to the identity on , , . satisfies the relations on (where is the identity) and on .
Dimension 39
order
-
b^{\prime}_{5}=\operatorname{Re}\Bigl{(}e^{\frac{2\pi i}{3}}\bigl{(}-676+10140i\sqrt{3}\bigr{)}^{\frac{1}{3}}\Bigr{)}
b^{\prime}_{6}=\operatorname{Re}\Bigl{(}\bigl{(}180+4a+4i\sqrt{6753-90a}\bigr{)}^{\frac{1}{3}}\Bigr{)}
b^{\prime\prime}_{6}=\operatorname{Re}\Bigl{(}e^{\frac{2\pi i}{3}}\bigl{(}180-4a+4i\sqrt{6753+90a}\bigr{)}^{\frac{1}{3}}\Bigr{)}
orbit
interchanges the pairs , , , and . interchanges the pairs and and restricts to the identity on and . restricts to the identity on and .
Dimension 48
order
b^{\prime}_{6}=2\operatorname{Re}\Bigr{(}e^{\frac{2\pi i}{3}}\bigl{(}7+i\sqrt{15}\bigr{)}^{\frac{1}{3}}\Bigr{)}
orbit
cycles in the order and restricts to the identity on , . restricts to the identity on , .
Acknowledgements
This research was supported in part by the Australian Research Council via EQuS project number CE11001013. SF acknowledges support from an Australian Research Council Future Fellowship FT130101744.
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