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Commutative d-Torsion K-Theory and Its Applications

Cihan Okay

Abstract

Commutative $d$-torsion $K$-theory is a variant of topological $K$-theory constructed from commuting unitary matrices of order dividing $d$. Such matrices appear as solutions of linear constraint systems that play a role in the study of quantum contextuality and in applications to operator-theoretic problems motivated by quantum information theory. Using methods from stable homotopy theory we modify commutative $d$-torsion $K$-theory into a cohomology theory which can be used for studying operator solutions of linear constraint systems. This provides an interesting connection between stable homotopy theory and quantum information theory.

Commutative d-Torsion K-Theory and Its Applications

Abstract

Commutative -torsion -theory is a variant of topological -theory constructed from commuting unitary matrices of order dividing . Such matrices appear as solutions of linear constraint systems that play a role in the study of quantum contextuality and in applications to operator-theoretic problems motivated by quantum information theory. Using methods from stable homotopy theory we modify commutative -torsion -theory into a cohomology theory which can be used for studying operator solutions of linear constraint systems. This provides an interesting connection between stable homotopy theory and quantum information theory.

Paper Structure

This paper contains 19 sections, 15 theorems, 88 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Commutative $d$-torsion $K$-theory
  3. Classifying spaces
  4. Stabilization
  5. Spectra and $\Gamma$-spaces
  6. The spectrum
  7. Low dimensional homotopy groups
  8. $C(d,m)$-cohomology
  9. A quotient space
  10. $C(d,m)$ spectrum
  11. $C(d,m)$-cohomology
  12. Operator solutions of linear constraint systems
  13. Linear constraint systems
  14. Topological description
  15. Computing the homotopy classes
  16. ...and 4 more sections

Key Result

Theorem \ref{thm:Thm-stable}

Let $X$ be a connected CW complex. There is a commutative diagram \begin{tikzcd} & H^2(X,(\ZZ/d)_m) \arrow[d,hook]\arrow[r,"(i_m)_*"] & H^2(X,\ZZ/d) \arrow[d,equal] \\ k\mu_d(X) \arrow[d] \arrow{r}{\zeta} & C(d,m)(X) \arrow[d,two heads] \arrow{r}{\cl} & H^2(X,\ZZ/d) \\ H^1(X,\ZZ/d) \arrow[r,"(\pi_

Figures (3)

  • Figure 1: (Left figure) $\mathfrak{H}_{\text{sq}}$ consists of $9$ vertices and $6$ edges each consisting of $3$ vertices in each row and column. All the incidence weights are equal to $1$. The operator solution is given by tensor product of two Pauli matrices, where the notation is simplified by omitting $\otimes$. The function $\tau_\text{sq}$ takes the value $0$ for each hyperedge except the right-most column. (Right figure) A topological realization given by a torus together with a cell structure consisting of triangles. The operators are placed on the edges and each triangle corresponds to an hyperedge. The cocycle $\tau_{\text{sq}}$ assigns $0$ to each triangle except $\lbrace XX,YY,ZZ \rbrace$, which is assigned $1$.
  • Figure 2: (Left figure) $\mathfrak{H}_{\text{st}}$ consists of $10$ vertices and $5$ edges each consisting of $4$ vertices in each line, and all the incidence weights are equal to $1$. The function $\tau_\text{st}$ takes the value $0$ for each hyperedge except the horizontal line. (Right figure) On the torus $\tau_\text{st}$ specifies a $2$-cocycle that assigns $0$ to each cell except $\lbrace XXX,YYX,YXY,XYY \rbrace$ is assigned $1$.
  • Figure 3: Refined topological realization

Theorems & Definitions (43)

  • Theorem \ref{thm:Thm-stable}
  • Corollary \ref{cor:stable-LCS}
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6
  • ...and 33 more