arXiv subjects

Gary McConnell

Publications and source records attributed to Gary McConnell.

12 recordsLinked to original sources

arXiv2026

SIC dimension towers via cyclotomic polynomials

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~$\{\Psi_m(t_k)\}_{m\geq1,k\geq0}$ attached to~$K$, while the auxiliary factors~$(d_k+1)$ and $(d_k-3)$ are its ramified levels $m=2$ and $m=1$. Here~$\Psi_m$ denotes the minimal polynomial of~ $\zeta_m+\zeta_m^{-1} = 2\cos{2\pi/m}$. 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 $c_{m,k}=1-\zeta_m\eps^k$ over the cyclotomic field tower $\{K(\mu_m)\}_{m\geq1}$. 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 $v_p(\Psi_m(t_k))$ 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.

math.NT↗

arXiv2026

How Stark units enter SIC overlaps

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 $\pm 1$, 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.

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arXiv2024

Some new infinite families of non-$p$-rational real quadratic fields

Fix a finite collection of primes $\{ p_j \}$, not containing $2$ or $3$. Using some observations which arose from attempts to solve the SIC-POVMs problem in quantum information, we give a simple methodology for constructing an infinite family of simultaneously non-$p_j$-rational real quadratic fields, unramified above any of the $p_j$. Alternatively these may be described as infinite sequences of instances of $\mathbb{Q}(\sqrt{D})$, for varying $D$, where every $p_j$ is a $k$-Wall-Sun-Sun prime, or equivalently a generalised Fibonacci-Wieferich prime. One feature of these techniques is that they may be used to yield fields $K=\mathbb{Q}(\sqrt{D})$ for which a $p$-power cyclic component of the torsion group of the Galois groups of the maximal abelian pro-$p$-extension of $K$ unramified outside primes above $p$, is of size $p^a$ for $a\geq1$ arbitrarily large.

math.NT↗

arXiv2024

SIC-POVMs from Stark Units: Dimensions n^2+3=4p, p prime

The existence problem for maximal sets of equiangular lines (or SICs) in complex Hilbert space of dimension $d$ remains largely open. In a previous publication (arXiv:2112.05552) we gave a conjectural algorithm for how to construct a SIC if $d = n^2+3 = p$, a prime number. Perhaps the most surprising number-theoretical aspect of that algorithm is the appearance of Stark units in a key role: a single Stark unit from a ray class field extension of a real quadratic field serves as a seed from which the SIC is constructed. The algorithm can be modified to apply to all dimensions $d = n^2+3$. Here we focus on the case when $d= n^2+3 = 4p$, $p$ prime, for two reasons. First, special measures have to be taken on the Hilbert space side of the problem when the dimension is even. Second, the degrees of the relevant ray class fields are `smooth' in a sense that facilitates exact calculations. As a result the algorithm becomes easier to explain. We give solutions for seventeen different dimensions of this form, reaching $d = 39604$. Several improvements relative to our previous publication are reported, but we cannot offer a proof that the algorithm works for any dimensions where it has not been tested.

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arXiv2022

SIC-POVMs from Stark units: Prime dimensions n^2+3

We propose a recipe for constructing a SIC fiducial vector in complex Hilbert space of dimension of the form $d=n^2+3$, focussing on prime dimensions $d=p$. Such structures are shown to exist in thirteen prime dimensions of this kind, the highest being $p=19603$. The real quadratic base field $K$ (in the standard SIC terminology) attached to such dimensions has fundamental units $u_K$ of norm $-1$. Let $\mathbb{Z}_K$ denote the ring of integers of $K$, then $p\mathbb{Z}_K$ splits into two ideals $\mathfrak{p}$ and $\mathfrak{p}'$. The initial entry of the fiducial is the square $ξ^2$ of a geometric scaling factor $ξ$, which lies in one of the fields $K(\sqrt{u_K})$. Strikingly, the other $p-1$ entries of the fiducial vector are each the 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 $\mathbb{Z}_K$ with modulus $\mathfrak{p}\infty_1$, where $\infty_1$ is one of the real places of $K$.

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arXiv2021

Evidence for and against Zauner's MUB Conjecture in $\mathbb{C}^6$

The problem of finding provably maximal sets of mutually unbiased bases in $\mathbb{C}^d$, for composite dimensions $d$ which are not prime powers, remains completely open. In the first interesting case, $d=6$, Zauner predicted that there can exist no more than three MUBs. We explore possible algebraic solutions in $d=6$ by looking at their `shadows' in vector spaces over finite fields. The main result is that if a counter-example to Zauner's conjecture were to exist, then it would leave no such shadow upon reduction modulo several different primes, forcing its algebraic complexity level to be much higher than that of current well-known examples. In the case of prime powers $q \equiv 5 \bmod 12$, however, we are able to show some curious evidence which -- at least formally -- points in the opposite direction. In $\mathbb{C}^6$, not even a single vector has ever been found which is mutually unbiased to a set of three MUBs. Yet in these finite fields we find sets of three `generalised MUBs' together with an orthonormal set of four vectors of a putative fourth MUB, all of which lifts naturally to a number field.

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arXiv2019

Generating Ray Class Fields of Real Quadratic Fields via Complex Equiangular Lines

For certain real quadratic fields $K$ with sufficiently small discriminant we produce explicit unit generators for specific ray class fields of $K$ using a numerical method that arose in the study of complete sets of equiangular lines in $\mathbb{C}^d$ (known in quantum information as symmetric informationally complete measurements or SICs). The construction in low dimensions suggests a general recipe for producing unit generators in infinite towers of ray class fields above arbitrary real quadratic $K$, and we summarise this in a conjecture. There are indications [19,20] that the logarithms of these canonical units are related to the values of $L$-functions associated to the extensions, following the programme laid out in the Stark Conjectures.

math.NT↗

arXiv2017

SICs and Algebraic Number Theory

We give an overview of some remarkable connections between symmetric informationally complete measurements (SIC-POVMs, or SICs) and algebraic number theory, in particular, a connection with Hilbert's 12th problem. The paper is meant to be intelligible to a physicist who has no prior knowledge of either Galois theory or algebraic number theory.

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arXiv2014

Some non-standard ways to generate SIC-POVMs in dimensions 2 and 3

The notion of Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) arose in physics as a kind of optimal measurement basis for quantum systems. However the question of their existence is equivalent to that of the existence of a maximal set of \emph{complex equiangular lines}. That is to say, given a complex Hilbert space of dimension $d$, what is the maximal number of (complex) lines one can find which all make a common (real) angle with one another, in the sense that the inner products between unit vectors spanning those lines all have a common absolute value? A maximal set would consist of $d^2$ lines all with a common angle of $\arccos{\frac{1}{\sqrt{d+1}}}$. The same question has been posed in the real case and some partial answers are known. But at the time of writing no unifying theoretical result has been found in the real or the complex case: some sporadic low-dimensional numerical constructions have been converted into algebraic solutions but beyond this very little is known. It is conjectured that such structures always arise as orbits of certain fiducial vectors under the action of the Weyl (or generalised Pauli) group. In this paper we point out some new construction methods in the lowest dimensions ($d=2$ and $d=3$). We should mention that the SIC-POVMs so constructed are all unitarily equivalent to previously known SIC-POVMs.

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arXiv2012

An entropic partial order on a parabolic quotient of S6

Let m and n be any integers with n>m>=2. Using just the entropy function it is possible to define a partial order on S_mn (the symmetric group on mn letters) modulo a subgroup isomorphic to S_m x S_n. We explore this partial order in the case m=2, n=3, where thanks to the outer automorphism the quotient space is actually isomorphic to a parabolic quotient of S_6. Furthermore we show that in this case it has a fairly simple algebraic description in terms of elements of the group ring.

math.CO↗

arXiv2012

On the spectral dependence of separable and classical correlations in small quantum systems

We study the correlation structure of separable and classical states in 2x2- and 2x3-dimensional quantum systems with fixed spectra. Even for such simple systems the maximal correlation - as measured by mutual information - over the set of unitarily accessible separable states is highly non-trivial to compute; however for the 2x2 case a particular class of spectra admits full analysis and allows us to contrast classical states with more general separable states. We analyse a particular entropic partial order on the set of spectra and prove for the qubit-qutrit case that this partial order alone picks out a unique classical maximum state for mutual information. Moreover the 2x3 case is the largest system with such a property.

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arXiv2007

Efficient 2-designs from bases exist

We show that in a complex d-dimensional vector space, one can find O(d) bases whose elements form a 2-design. Such vector sets generalize the notion of a maximal collection of mutually unbiased bases (MUBs). MUBs have manifold applications in quantum information theory (e.g. in state tomography, cloning, or cryptography) -- however it is suspected that maximal sets exist only in prime-power dimensions. Our construction offers an efficient alternative for general dimensions. The findings are based on a framework recently established in [A. Roy and A. Scott, J. Math. Phys. 48, 072110 (2007)], which reduces the construction of such bases to the combinatorial problem of finding certain highly nonlinear functions between abelian groups.

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