SiC のバナジウム中心に基づく通信波長帯量子光源に向けたナノビーム光共振器の設計

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.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.

arXiv:1707.09911Journal of Mathematical PhysicsNov 1, 201721 citations

通信波長帯量子中継器の実現に向けた光量子メモリおよびもつれ光子対源の研究

翔一郎安井

Mar 25, 20260 citations

Asymmetric SICs over finite fields

Joseph W. Iverson, Dustin G. Mixon

Zauner's conjecture concerns the existence of $d^2$ equiangular lines in $\mathbb{C}^d$ ; such a system of lines is known as a SIC. In this paper, we construct infinitely many new SICs over finite fields. While all previously known SICs exhibit Weyl--Heisenberg symmetry, some of our new SICs exhibit trivial automorphism groups. We conjecture that such \textit{totally asymmetric} SICs exist in infinitely many dimensions in the finite field setting.Zauner's conjecture concerns the existence of $d^2$ equiangular lines in $\mathbb{C}^d$; such a system of lines is known as a SIC. In this paper, we construct infinitely many new SICs over finite fields. While all previously known SICs exhibit Weyl--Heisenberg symmetry, some of our new SICs exhibit trivial automorphism groups. We conjecture that such \textit{totally asymmetric} SICs exist in infinitely many dimensions in the finite field setting.

arXiv:2506.20778Jun 25, 20250 citations

Exploring the geometry of qutrit state space using symmetric informationally complete probabilities

Gelo Noel M. Tabia, D. M. Appleby

We examine the geometric structure of qutrit state space by identifying the outcome probabilities of symmetric informationally complete (SIC) measurements with quantum states. We categorize the infinitely many qutrit SICs into eight SIC families corresponding to independent orbits of the extended Clifford group. Every SIC can be uniquely identified from a set of geometric invariants that we use to establish several properties of the convex body of qutrits, which include a simple formula describing its extreme points, an expression for the rotation between the probability vectors for distinct qutrit SICs, and a polar equation for its boundary states.We examine the geometric structure of qutrit state space by identifying the outcome probabilities of symmetric informationally complete (SIC) measurements with quantum states. We categorize the infinitely many qutrit SICs into eight SIC families corresponding to independent orbits of the extended Clifford group. Every SIC can be uniquely identified from a set of geometric invariants that we use to establish several properties of the convex body of qutrits, which include a simple formula describing its extreme points, an expression for the rotation between the probability vectors for distinct qutrit SICs, and a polar equation for its boundary states.

arXiv:1304.8075Physical Review AJul 31, 201325 citations

From Ghost SICs to SIC-POVMs: A Galois Reduction of Zauner's Conjecture

Hassan Khodaiemehr

Symmetric informationally complete positive operator-valued measures (SIC-POVMs, or SICs) are optimal minimal measurements for quantum state tomography. Problem~1 of Horodecki, Rudnicki and Życzkowski's open problems in quantum information asks for SICs in infinitely many dimensions. By work of Appleby--Flammia--Kopp and Radchenko--Wheeler, ghost SICs, which satisfy the SIC equations with Hermiticity up to parity, exist unconditionally for $d\ge4$ . We study the passage from ghosts to SICs. We compute all isolated points of the complexified SIC variety for $d=4,5,6$ ; exactly half are SICs. We prove that no symmetry of the SIC equations maps a ghost to a SIC, so the step is purely arithmetic, and that a SIC exists in dimension $d$ whenever the ghost overlap units satisfy a place law: each Galois conjugate is real or unimodular according to the infinite place of the real quadratic field $F$ below it. If the unit field is abelian over $F$ , the place law is the Stark relation $c_-=κ$ ; equivalently, a Galois conjugate of an associated near-group fusion category is unitary exactly when it is pseudo-unitary. For $d$ a power of a prime $p\equiv5\pmod 6$ or of $3$ , an infinite family, we prove that under the algebraic part of Stark's conjecture $c_-=κ$ is equivalent to the triviality of $κ$ on the Hilbert-class-field part of the unit field. That condition involves no complex conjugation, and it follows from a sign-free multiplet hypothesis. We state an explicit $GL_2$ reciprocity law for the squared ghost overlaps that implies $c_-=κ$ , and we prove it for the computed algebraic overlaps in $d=5,7,11,15$ by Artin reciprocity and a check in every ray class. Zauner's conjecture remains open: the algebraic part of Stark's conjecture and the multiplet hypothesis are not proved here.Symmetric informationally complete positive operator-valued measures (SIC-POVMs, or SICs) are optimal minimal measurements for quantum state tomography. Problem~1 of Horodecki, Rudnicki and Życzkowski's open problems in quantum information asks for SICs in infinitely many dimensions. By work of Appleby--Flammia--Kopp and Radchenko--Wheeler, ghost SICs, which satisfy the SIC equations with Hermiticity up to parity, exist unconditionally for $d\ge4$. We study the passage from ghosts to SICs. We compute all isolated points of the complexified SIC variety for $d=4,5,6$; exactly half are SICs. We prove that no symmetry of the SIC equations maps a ghost to a SIC, so the step is purely arithmetic, and that a SIC exists in dimension $d$ whenever the ghost overlap units satisfy a place law: each Galois conjugate is real or unimodular according to the infinite place of the real quadratic field $F$ below it. If the unit field is abelian over $F$, the place law is the Stark relation $c_-=κ$; equivalently, a Galois conjugate of an associated near-group fusion category is unitary exactly when it is pseudo-unitary. For $d$ a power of a prime $p\equiv5\pmod 6$ or of $3$, an infinite family, we prove that under the algebraic part of Stark's conjecture $c_-=κ$ is equivalent to the triviality of $κ$ on the Hilbert-class-field part of the unit field. That condition involves no complex conjugation, and it follows from a sign-free multiplet hypothesis. We state an explicit $GL_2$ reciprocity law for the squared ghost overlaps that implies $c_-=κ$, and we prove it for the computed algebraic overlaps in $d=5,7,11,15$ by Artin reciprocity and a check in every ray class. Zauner's conjecture remains open: the algebraic part of Stark's conjecture and the multiplet hypothesis are not proved here.

quant-pharXiv:2611.01524Nov 3, 2026

Zauner's conjecture concerns the existence of

d^2

equiangular lines in

\mathbb{C}^d

; such a system of lines is known as a SIC. In this paper, we construct infinitely many new SICs over finite fields. While all previously known SICs exhibit Weyl--Heisenberg symmetry, some of our new SICs exhibit trivial automorphism groups. We conjecture that such \textit{totally asymmetric} SICs exist in infinitely many dimensions in the finite field setting.

d\ge4

. We study the passage from ghosts to SICs. We compute all isolated points of the complexified SIC variety for

d=4,5,6

; exactly half are SICs. We prove that no symmetry of the SIC equations maps a ghost to a SIC, so the step is purely arithmetic, and that a SIC exists in dimension

d

whenever the ghost overlap units satisfy a place law: each Galois conjugate is real or unimodular according to the infinite place of the real quadratic field

F

below it. If the unit field is abelian over

F

, the place law is the Stark relation

c_-=κ

; equivalently, a Galois conjugate of an associated near-group fusion category is unitary exactly when it is pseudo-unitary. For

d

a power of a prime

p\equiv5\pmod 6

or of

3

, an infinite family, we prove that under the algebraic part of Stark's conjecture

c_-=κ

is equivalent to the triviality of

κ

on the Hilbert-class-field part of the unit field. That condition involves no complex conjugation, and it follows from a sign-free multiplet hypothesis. We state an explicit

GL_2

reciprocity law for the squared ghost overlaps that implies

c_-=κ

, and we prove it for the computed algebraic overlaps in

d=5,7,11,15

by Artin reciprocity and a check in every ray class. Zauner's conjecture remains open: the algebraic part of Stark's conjecture and the multiplet hypothesis are not proved here.