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On the rational invariants of quantum systems of $n$-qubits

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

Abstract

For an $n$-qubit system, a rational function on the space of mixed states which is invariant with respect to the action of the group of local symmetries may be viewed as a detailed measure of entanglement. We show that the field of all such invariant rational functions is purely transcendental over the complex numbers and has transcendence degree $4^n - 2n-1$. An explicit transcendence basis is also exhibited.

On the rational invariants of quantum systems of $n$-qubits

Abstract

For an -qubit system, a rational function on the space of mixed states which is invariant with respect to the action of the group of local symmetries may be viewed as a detailed measure of entanglement. We show that the field of all such invariant rational functions is purely transcendental over the complex numbers and has transcendence degree . An explicit transcendence basis is also exhibited.
Paper Structure (15 sections, 9 theorems, 61 equations, 1 figure)

This paper contains 15 sections, 9 theorems, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The rationality problem
  3. The main result
  4. States and observables
  5. L-states
  6. Observables
  7. Correlation functions
  8. The Bloch model
  9. Quotients and rationality
  10. Invariant functions and quotients
  11. Rationality
  12. First proof of Theorem \ref{['bigthm:main']}
  13. An explicit transcendence basis
  14. Second proof of Theorem \ref{['bigthm:main']}
  15. Rational sections

Key Result

Theorem 1

The field $\mathbb C(\mathscr L_n)^G$ is purely rational and has transcendence degree $4^n - 3n -1$ over the complex numbers.

Figures (1)

  • Figure 3.1: Nowhere zero vector fields on 6 vertices and on 3 vertices.

Theorems & Definitions (34)

  • Theorem 1
  • Remark 1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Example 1.4: $n=2$
  • Definition 1.5
  • Example 1.6: $n = 2$
  • Remark 1.7
  • Remark 1.8
  • ...and 24 more