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Quantum Wasserstein isometries of the $n$-qubit state space: a Wigner-type result

Gergely Bunth, Eszter Szabó, Dániel Virosztek

Abstract

We determine the isometry group of the $n$-qubit state space with respect to the quantum Wasserstein distance induced by the so-called symmetric transport cost for all $n \in \mathbb{N}.$ It turns out that the isometries are precisely the Wigner symmetries, that is, the unitary or anti-unitary conjugations.

Quantum Wasserstein isometries of the $n$-qubit state space: a Wigner-type result

Abstract

We determine the isometry group of the -qubit state space with respect to the quantum Wasserstein distance induced by the so-called symmetric transport cost for all It turns out that the isometries are precisely the Wigner symmetries, that is, the unitary or anti-unitary conjugations.
Paper Structure (5 sections, 7 theorems, 52 equations)

This paper contains 5 sections, 7 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Motivation and main result
  3. Basic notions, notation
  4. The spectral resolution of the $n$-qubit symmetric transport cost operator
  5. Quantum Wasserstein isometries of the $n$-qubit state space with respect to the symmetric transport cost --- the proof of Theorem \ref{['thm:n-qubit-symmetric']}

Key Result

Theorem 1

A map $\Phi: \mathcal{S}\left( \mathbb{C}^{2^n} \right) \to \mathcal{S}\left( \mathbb{C}^{2^n} \right)$ is a quantum Wasserstein isometry with respect to the symmetric transport cost, that is, if and only if $\Phi$ is a Wigner symmetry, that is, there exists a unitary or anti-unitary operator $U: \mathbb{C}^{2^n} \to \mathbb{C}^{2^n}$ such that $\Phi(\rho)=U\rho U^*$ for all $\rho \in \mathcal{S}

Theorems & Definitions (13)

  • Theorem 1
  • Definition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • ...and 3 more