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Ingemar Bengtsson

Publications and source records attributed to Ingemar Bengtsson.

At least 19 recordsLinked to original sources
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.

quant-ph↗
arXiv2025

A Conjecture on Almost Flat SIC-POVMs

A well supported conjecture states that SIC-POVMs -- maximal sets of complex equiangular lines -- with anti-unitary symmetry give rise to an identity expressing some of its overlaps as squares of the (rescaled) components of a suitably chosen fiducial vector. In number theoretical terms the identity essentially expresses Stark units as sums of products of pairs of square roots of Stark units. We investigate whether the identity is enough to determine these Stark units. The answer is no, but the failure might be quite mild.

quant-ph↗
arXiv2025

Ultra-massive spacetimes in 2+1 dimensions with positive Λ

We show that ultra-massive spacetimes exist also in 2 + 1 dimensions with a positive cosmological constant Λ > 0. They can be created through the collapse of a spherical null dust shell. The exterior of the shell is then a Mess spacetime, that is to say a locally de Sitter spacetime that cannot be obtained as a quotient of de Sitter space.

gr-qc↗
arXiv2025

Evolutionary constraints: Gluing in a toy model

It is possible to solve the Einstein constraint equations as an evolutionary rather than an elliptic system. Here we consider the Gauss constraint in electrodynamics as a toy model for thist. We use a combination of the evolutionary method with the gluing construction to produce initial data for an electromagnetic pulse surrounded by vacuum. It turns out that solving the evolutionary form of the constraint is straightforward, and explicitly yields the desired type of initial data. In contrast, proving the existence of a solution to the same problem within the elliptic setting requires sophisticated arguments based on functional analysis.

gr-qc↗
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.

quant-ph↗
arXiv2023

Lars Brink and SIC_POVMs

The notion of SIC-POVMs comes from quantum information theory, and they were not on the horizon when I was Lars Brink's student in the early 80s. In the summer of 2022 I told Lars that I know how to use number theoretical insights to construct SIC-POVMs in any Hilbert space of dimension $n^2+3$, and that the construction provides a geometric setting for some deep number theoretical conjectures. I will give a sketch of this development, of what it was like to be Lars' student, and of what his reaction to our construction was.

quant-ph↗
arXiv2022

Energy in Newtonian gravity

In Newtonian gravity it is a moot question whether energy should be localized in the field or inside matter. An argument from relativity suggests a compromise in which the contribution from the field in vacuum is positive definite. We show that the same compromise is implied by Noether's theorem applied to a variational principle for perfect fluids, if we assume Dirichlet boundary conditions on the potential. We then analyse a thought experiment due to Bondi and McCrea that gives a clean example of inductive energy transfer by gravity. Some history of the problem is included

gr-qc↗
arXiv2022

Bogdan Mielnik, geometry and quanta

We review selected achievements of the late Bogdan Mielnik in the field of theoretical physics, with an emphasis on his attempts to go beyond quantum mechanics. Some of his original views on the problems of contemporary society and organization of science are also recalled.

physics.hist-ph↗
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 $K$.

quant-ph↗
arXiv2022

Dimension towers of SICs. II. Some constructions

A SIC is a maximal equiangular tight frame in a finite dimensional Hilbert space. Given a SIC in dimension $d$, there is good evidence that there always exists an aligned SIC in dimension $d(d-2)$, having predictable symmetries and smaller equiangular tight frames embedded in them. We provide a recipe for how to calculate sets of vectors in dimension $d(d-2)$ that share these properties. They consist of maximally entangled vectors in certain subspaces defined by the numbers entering the $d$ dimensional SIC. However, the construction contains free parameters and we have not proven that they can always be chosen so that one of these sets of vectors is a SIC. We give some worked examples that, we hope, may suggest to the reader how our construction can be improved. For simplicity we restrict ourselves to the case of odd dimensions.

quant-ph↗
arXiv2020

Compounds of symmetric informationally complete measurements and their application in quantum key distribution

Symmetric informationally complete measurements (SICs) are elegant, celebrated and broadly useful discrete structures in Hilbert space. We introduce a more sophisticated discrete structure compounded by several SICs. A SIC-compound is defined to be a collection of $d^3$ vectors in $d$-dimensional Hilbert space that can be partitioned in two different ways: into $d$ SICs and into $d^2$ orthonormal bases. While a priori their existence may appear unlikely when $d>2$, we surprisingly answer it in the positive through an explicit construction for $d=4$. Remarkably this SIC-compound admits a close relation to mutually unbiased bases, as is revealed through quantum state discrimination. Going beyond fundamental considerations, we leverage these exotic properties to construct a protocol for quantum key distribution and analyze its security under general eavesdropping attacks. We show that SIC-compounds enable secure key generation in the presence of errors that are large enough to prevent the success of the generalisation of the six-state protocol.

quant-ph↗
arXiv2020

Algebraic units, anti-unitary symmetries, and a small catalogue of SICs

In complex vector spaces maximal sets of equiangular lines, known as SICs, are related to real quadratic number fields in a dimension dependent way. If the dimension is of the form $n^2+3$ the base field has a fundamental unit of negative norm, and there exists a SIC with anti-unitary symmetry. We give eight examples of exact solutions of this kind, for which we have endeavoured to make them as simple as we can---as a belated reply to the referee of an earlier publication, who claimed that our exact solution in dimension 28 was too complicated to be fit to print. An interesting feature of the simplified solutions is that the components of the fiducial vectors largely consist of algebraic units.

quant-ph↗
arXiv2020

The Hawking energy on photon surfaces

The Hawking energy has a monotonicity property under the inverse mean curvature flow on totally umbilic hypersurfaces with constant scalar curvature in Einstein spaces. It grows if the hypersurface is spacelike, and decreases if it is timelike. Timelike examples include Minkowski and de Sitter hyperboloids, and photon surfaces in Schwarzschild.

gr-qc↗
arXiv2020

SICs: Some explanations

The problem of constructing maximal equiangular tight frames or SICs was raised by Zauner in 1998. Four years ago it was realized that the problem is closely connected to a major open problem in number theory. We discuss why such a connection was perhaps to be expected, and give a simplified sketch of some developments that have taken place in the past four years. The aim, so far unfulfilled, is to prove existence of SICs in an infinite sequence of dimensions.

quant-ph↗
arXiv2019

Tight Frames, Hadamard Matrices and Zauner's Conjecture

We show that naturally associated to a SIC (symmetric informationally complete positive operator valued measure or SIC-POVM) in dimension d there are a number of higher dimensional structures: specifically a projector and a complex Hadamard matrix in dimension d squared and a pair of ETFs (equiangular tight frames) in dimensions d(d-1)/2, d(d+1)/2. We also show that a WH (Weyl Heisenberg covariant) SIC in odd dimension d is naturally associated to a pair of symmetric tight fusion frames in dimension d. We deduce two relaxations of the WH SIC existence problem. We also find a reformulation of the problem in which the number of equations is fewer than the number of variables. Finally, we show that in at least four cases the structures associated to a SIC lie on continuous manifolds of such structures. In two of these cases the manifolds are non-linear. Restricted defect calculations are consistent with this being true for the structures associated to every known SIC with d between 3 and 16, suggesting it may be true for all d greater than 2.

quant-ph↗
arXiv2019

Simplified exact SICs

In the standard basis exact expressions for the components of SIC vectors (belonging to a symmetric informationally complete POVM) are typically very complicated. We show that a simple transformation to a basis adapted to the symmetries of a fiducial SIC vector can result in a massive reduction in complexity. We rely on a conjectural number theoretic connection between SICs in dimension $d_j$ and SICs in dimension $d_{j+1} = d_j(d_j-2)$. We focus on the sequence 5, 15, 195, ... . We rewrite Zauner's exact solution for the SIC in dimension 5 to make its simplicity manifest, and use our adapted basis to convert numerical solutions in dimensions 15 and 195 to exact solutions. Comparing to the known exact solutions in dimension 15 we find that the simplification achieved is dramatic. The proof that the exact vectors are indeed SIC fiducial vectors, also in dimension 195, is a long calculation guided by the standard ray class hypothesis about the algebraic number fields generated by the SICs. We conjecture that our result generalizes to every dimension in the particular sequence we consider.

quant-ph↗
arXiv2018

Black Hole Lattices Under the Microscope

It is known how to choose initial data for Einstein's equations describing an arbitrary number of black holes at a moment of time symmetry. This idea has been used to give insight into the cosmological averaging problem. We study the local curvature of the initial data space, for configurations of 8, 120, or 600 black holes obtained by choosing points either regularly or randomly on the 3-sphere. We conclude that the asymptotic regions are remarkably close to that of Schwarzschild, while the region in between shows interesting behaviour. The cosmological back reaction as defined in the recent literature is actually a bit smaller for the random configurations.

gr-qc↗