Matching works

3 works

quant-ph — Quantum Physics

Pauli problem in dimension 4

Contributed by Dmitry Grinko

The finite-dimensional Pauli problem asks how many measurements in orthonormal bases determine every pure state of a $d$ -level quantum system up to a global phase. Four bases always suffice, and the answer is known to be three for $d=2$ and four for $d=3$ and $d\geq5$ . Dimension four remained open because the embedding argument used in other dimensions fails there: the pure-state space $\mathbb{CP}^3$ embeds in $\R^9$ , the space of the nine independent probabilities of three bases. We show that three bases do not suffice in $\C^4$, so exactly four are needed. The same argument shows that no ten vectors in $\C^4$ do phase retrieval. Since eleven vectors are known to suffice, the smallest phase-retrieval frame and the smallest rank-one POVM distinguishing all pure states in $\C^4$ both have eleven elements. All three lower bounds follow from one statement: every six-dimensional real space of traceless Hermitian $4\times4$ matrices with a common isotropic vector, that is, a nonzero $e$ with $e^*Te=0$ for all $T$ in the space, contains a nonzero matrix of rank at most two. We prove it with complex $K$ -theory: otherwise, an odd unitary map on the five-sphere would have a $K^1$ -class that is nonzero by antipodal symmetry but vanishes because of the isotropic vector.

Primarily AI-generated textHuman understanding: some parts

Sep 29, 2026PDF ↗View work →

hexagon:2609.00112v1

math.AT — Algebraic Topology

Swan induction for the finite-local sphere

Contributed by Akhil Mathew

We establish rational Swan induction for ordinary perfect modules over the finite-local sphere $L_n^{p,f}\mathbb S$ . For a finite abelian ambient group, subgroups of $p$ -rank at most $n+1$ suffice, and this bound is sharp. The proof combines cyclic homotopy fixed points in telescopic spectra with the isotropy filtration of a chromatic quotient of finite genuine spectra. The result proves Conjecture~7.22 of Clausen--Mathew--Naumann--Noel for Morava $E$ -theory and gives a new proof of their chromatic upper bound for the algebraic $K$ -theory of $L_n^{p,f}\mathbb S$ -linear categories. The quotient argument also produces finite complexes realizing the induction relations. We formulate the problem of explicit realizations and give two geometric models at the prime two.

Primarily AI-generated textHuman understanding: some partsBurnside ringsSwan inductionchromatic homotopy theory

Sep 27, 2026PDF ↗View work →

hexagon:2609.00111v1

math.AG — Algebraic Geometry

Torsion in $K_1$ and $p$ -adic vanishing cycles

Contributed by Akhil Mathew

For every odd prime $p$ , we construct a regular strictly henselian local ring $A$ whose generic fibre has \'etale cohomology classes in $H^2_{\et}(A[1/p],\mu_p^{\otimes2})$ which are not sums of cup products of degree-one classes. The ring is the strict henselization of a local ring on a finite-type arithmetic scheme. A unit on a principal divisor gives an element of order $p$ in $SK_1(A[1/p])$ , detected by its localization boundary on the special fibre. Adams--Riemann--Roch and the connectivity of the motivic filtration show that this element has nonzero image in $H^3_{\mot}(A[1/p],\Z(2))$. The integral coefficient sequence then gives the cohomological obstruction. Only the eigenvalues $4$ and $8$ of $\psi^2$ are needed.

Primarily AI-generated textHuman understanding: all partsAdams operationsMilnor K-theoryp-adic vanishing cycles

Sep 27, 2026PDF ↗View work →

math.AC — Commutative Algebra
math.AG — Algebraic Geometry
math.AP — Analysis of PDEs
math.AT — Algebraic Topology
math.CA — Classical Analysis and ODEs
math.CO — Combinatorics
math.CT — Category Theory
math.CV — Complex Variables
math.DG — Differential Geometry
math.DS — Dynamical Systems
math.FA — Functional Analysis
math.GM — General Mathematics
math.GN — General Topology
math.GR — Group Theory
math.GT — Geometric Topology
math.HO — History and Overview
math.IT — Information Theory
math.KT — K-Theory and Homology
math.LO — Logic
math.MG — Metric Geometry
math.NA — Numerical Analysis
math.NT — Number Theory
math.OA — Operator Algebras
math.OC — Optimization and Control
math.PR — Probability
math.QA — Quantum Algebra
math.RA — Rings and Algebras
math.RT — Representation Theory
math.SG — Symplectic Geometry
math.SP — Spectral Theory
math.ST — Statistics Theory
cs.CC — Computational Complexity
cs.CG — Computational Geometry
cs.DM — Discrete Mathematics
cs.DS — Data Structures and Algorithms
cs.FL — Formal Languages and Automata Theory
cs.IT — Information Theory
cs.LO — Logic in Computer Science
cs.MS — Mathematical Software
cs.NA — Numerical Analysis
cs.SC — Symbolic Computation
math-ph — Mathematical Physics
quant-ph — Quantum Physics
stat.TH — Statistics Theory

Matching works

Pauli problem in dimension 4

Swan induction for the finite-local sphere

Torsion in K1K_1K1​ and ppp-adic vanishing cycles

Torsion in $K_1$ and $p$ -adic vanishing cycles