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On the variety of X-states

Luca Candelori, Vladimir Y. Chernyak, John R. Klein

Abstract

We introduce the notion of an X-state on $n$-qubits. After taking the Zariski closure of the set of X-states in the space of all mixed states, we obtain a complex algebraic variety $\scr X$ that is equipped with the action of the Lie group of local symmetries $G$. We show that the field of $G$-invariant rational functions on $\scr X$ is purely transcendental over the complex numbers of degree $2^{2n-1}-n-1$.

On the variety of X-states

Abstract

We introduce the notion of an X-state on -qubits. After taking the Zariski closure of the set of X-states in the space of all mixed states, we obtain a complex algebraic variety that is equipped with the action of the Lie group of local symmetries . We show that the field of -invariant rational functions on is purely transcendental over the complex numbers of degree .
Paper Structure (30 sections, 34 theorems, 164 equations)

This paper contains 30 sections, 34 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Background from physics
  3. Preliminaries
  4. Varieties
  5. Invariant functions and algebraic quotients
  6. Rational sections
  7. Rationality
  8. L-states
  9. Qubits
  10. L-states on $n$-qubits
  11. The Bloch model for L-states
  12. X-states on $n$-qubits
  13. The Bloch model for X-states
  14. Longitudinals and transversals
  15. A vector bundle model for X-states
  16. ...and 15 more sections

Key Result

Theorem 1

The field $\mathbb C(\mathscr X)^G$ is purely transcendental over $\mathbb C$ of degree $2^{2n-1} - n-1$.

Theorems & Definitions (84)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 1.1
  • Definition 1.2
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • ...and 74 more