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arXiv:2609.04431

SIC dimension towers via cyclotomic polynomials

Gary McConnell

math.NTquant-phSubmitted Sep 3, 2026• Announced Sep 7, 2026
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Abstract

We prove a structure theorem for the dimension towers~$\{d_k(D)\}_{k\geq0}$ which arise in the number-theoretic formulation of Zauner's SIC-POVM conjecture over a real quadratic field~$K=\Q(\qD)$. If~$\eps$ denotes the first totally positive power of a fundamental unit of~$K$ and $t_k = \eps^k + \eps^{-k}$ the trace of its $k$-th power, then the SIC dimension tower~$\{d_k=1+t_k\}_{k\geq0}$ is the level~$m=3$ row of an infinite two-dimensional cyclotomic array~$\{Ψ_m(t_k)\}_{m\geq1,k\geq0}$ attached to~$K$, while the auxiliary factors~$(d_k+1)$ and $(d_k-3)$ are its ramified levels $m=2$ and $m=1$. Here~$Ψ_m$ denotes the minimal polynomial of~ $ζ_m+ζ_m^{-1} = 2\cos{2π/m}$. This construction arose initially from an attempt to formulate relations among SIC dimensions in $q$-algebraic terms. The central object is a single closed composite norm relation for the two-parameter family $c_{m,k}=1-ζ_m\eps^k$ over the cyclotomic field tower $\{K(μ_m)\}_{m\geq1}$. This framework sheds new light on the mod-$p$ analogue of Leopoldt's conjecture, by placing the central 3-symmetry of Zauner's conjecture within a broader arithmetic context. Away from the primes dividing~$2mD$, the valuations $v_p(Ψ_m(t_k))$ at every fixed level~$m$ are described exactly in terms of a single local unit valuation, which is then related, through the~$p$-adic class number formula, to the corresponding $p$-adic $L$-value.

Comments
15 pages; 1 table

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We prove a structure theorem for the dimension towers~{dk(D)}k≥0\{d_k(D)\}_{k\geq0}{dk​(D)}k≥0​ which arise in the number-theoretic formulation of Zauner's SIC-POVM conjecture over a real quadratic field~K=\Q(\qD)K=\Q(\qD)K=\Q(\qD). If~\eps\eps\eps denotes the first totally positive power of a fundamental unit of~KKK and tk=\epsk+\eps−kt_k = \eps^k + \eps^{-k}tk​=\epsk+\eps−k the trace of its kkk-th power, then the SIC dimension tower~{dk=1+tk}k≥0\{d_k=1+t_k\}_{k\geq0}{dk​=1+tk​}k≥0​ is the level~m=3m=3m=3 row of an infinite two-dimensional cyclotomic array~{Ψm(tk)}m≥1,k≥0\{Ψ_m(t_k)\}_{m\geq1,k\geq0}{Ψm​(tk​)}m≥1,k≥0​ attached to~KKK, while the auxiliary factors~(dk+1)(d_k+1)(dk​+1) and (dk−3)(d_k-3)(dk​−3) are its ramified levels m=2m=2m=2 and m=1m=1m=1. Here~ΨmΨ_mΨm​ denotes the minimal polynomial of~ ζm+ζm−1=2cos⁡2π/mζ_m+ζ_m^{-1} = 2\cos{2π/m}ζm​+ζm−1​=2cos2π/m. This construction arose initially from an attempt to formulate relations among SIC dimensions in qqq-algebraic terms.
The central object is a single closed composite norm relation for the two-parameter family cm,k=1−ζm\epskc_{m,k}=1-ζ_m\eps^kcm,k​=1−ζm​\epsk over the cyclotomic field tower {K(μm)}m≥1\{K(μ_m)\}_{m\geq1}{K(μm​)}m≥1​. This framework sheds new light on the mod-ppp analogue of Leopoldt's conjecture, by placing the central 3-symmetry of Zauner's conjecture within a broader arithmetic context. Away from the primes dividing~2mD2mD2mD, the valuations vp(Ψm(tk))v_p(Ψ_m(t_k))vp​(Ψm​(tk​)) at every fixed level~mmm are described exactly in terms of a single local unit valuation, which is then related, through the~ppp-adic class number formula, to the corresponding ppp-adic LLL-value.
We propose a recipe for constructing a fiducial vector for a symmetric informationally complete positive operator valued measure (SIC-POVM) in a complex Hilbert space of dimension of the form d = n2 + 3, focusing on prime dimensions d = p. Such structures are shown to exist in 13 prime dimensions of this kind, the highest being p = 19 603. The real quadratic base field K (in the standard SIC-POVM terminology) attached to such dimensions has fundamental units uK of norm −1. Let ZK denote the ring of integers of K; then, pZK splits into two ideals: p and p′. The initial entry of the fiducial is the square ξ2 of a geometric scaling factor ξ, which lies in one of the fields K(uK). Strikingly, each of the other p − 1 entries of the fiducial vector is a product of ξ and the square root of a Stark unit. These Stark units are obtained via the Stark conjectures from the value at s = 0 of the first derivatives of partial L-functions attached to the characters of the ray class group of ZK with modulus p∞1, where ∞1 is one of the real places of K.
Abstract The existence of d2d^2d2 pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in ddd-dimensional Hilbert space is known only for finitely many dimensions ddd. We prove that, if there exists a set of real units in a certain ray class field (depending on ddd) satisfying certain algebraic properties, a SIC-POVM exists, when ddd 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=0s=0s=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 first exact SIC-POVM in dimension 23.
Algebraic number theory relates SIC-POVMs in dimension d > 3 to those in dimension d(d − 2). We define a SIC in dimension d(d − 2) to be aligned to a SIC in dimension d if and only if the squares of the overlap phases in dimension d appear as a subset of the overlap phases in dimension d(d − 2) in a specified way. We give 19 (mostly numerical) examples of aligned SICs. We conjecture that given any SIC in dimension d, there exists an aligned SIC in dimension d(d − 2). In all our examples, the aligned SIC has lower dimensional equiangular tight frames embedded in it. If d is odd so that a natural tensor product structure exists, we prove that the individual vectors in the aligned SIC have a very special entanglement structure, and the existence of the embedded tight frames follows as a theorem. If d − 2 is an odd prime number, we prove that a complete set of mutually unbiased bases can be obtained by reducing an aligned SIC to this dimension.
The existence of pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in -dimensional Hilbert space is known only for finitely many dimensions ⁠. We prove that, if there exists a set of real units in a certain ray class field (depending on ⁠) satisfying certain algebraic properties, a SIC-POVM exists, when 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 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 first exact SIC-POVM in dimension 23.
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It has been observed that the mutual scalar products of the vectors in a SIC-POVM are given by algebraic units, and at least in some cases by square roots of Stark units. The full picture is somewhat more complicated, especially if non-minimal SIC-POVMs are considered. We present a mixture of exact and numerical evidence suggesting that the overlap units are always products of integral powers of square roots of Stark units from ray class fields all of which are attached to the maximal ring of integers in the base field. In the non-minimal case a lattice of such ray class fields is involved. In every second dimension (counted in a certain way) some of the overlap units equal ±1\pm 1±1, and we show that this follows from a special property of the ray class fields. Our observations are complementary to but consistent with the claim that the overlap units can be calculated directly from the Shintani--Faddeev modular cocycle.