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Isometries of the qubit state space with respect to quantum Wasserstein distances

Richárd Simon, Dániel Virosztek

TL;DR

The paper investigates isometries of the quantum Wasserstein geometry on the qubit state space, focusing on $d_{\text{sym}}$ and $D_z$. It proves that $d_{\text{sym}}$-isometries are exactly the unitary and anti-unitary conjugations (Wigner symmetries), while $D_z$-isometries are precisely the Bloch-length-preserving maps that either preserve or negate the $z$-component of the Bloch vector, linking symmetry groups to the geometry of the Bloch ball. The proofs combine Bloch-sphere representations, cost operators $C_{\text{sym}}$ and $C_z$, and self-distance/diameter arguments to establish necessity and sufficiency. These results clarify symmetry-preserving transformations in quantum OT for qubits and set the stage for understanding isometries of related divergences and higher-dimensional generalizations.

Abstract

In this paper we study isometries of quantum Wasserstein distances and divergences on the quantum bit state space. We describe isometries with respect to the symmetric quantum Wasserstein divergence $d_{sym}$, the divergence induced by all of the Pauli matrices. We also give a complete characterization of isometries with respect to $D_z$, the quantum Wasserstein distance corresponding to the single Pauli matrix $σ_z$.

Isometries of the qubit state space with respect to quantum Wasserstein distances

TL;DR

The paper investigates isometries of the quantum Wasserstein geometry on the qubit state space, focusing on and . It proves that -isometries are exactly the unitary and anti-unitary conjugations (Wigner symmetries), while -isometries are precisely the Bloch-length-preserving maps that either preserve or negate the -component of the Bloch vector, linking symmetry groups to the geometry of the Bloch ball. The proofs combine Bloch-sphere representations, cost operators and , and self-distance/diameter arguments to establish necessity and sufficiency. These results clarify symmetry-preserving transformations in quantum OT for qubits and set the stage for understanding isometries of related divergences and higher-dimensional generalizations.

Abstract

In this paper we study isometries of quantum Wasserstein distances and divergences on the quantum bit state space. We describe isometries with respect to the symmetric quantum Wasserstein divergence , the divergence induced by all of the Pauli matrices. We also give a complete characterization of isometries with respect to , the quantum Wasserstein distance corresponding to the single Pauli matrix .
Paper Structure (5 sections, 2 theorems, 51 equations)

This paper contains 5 sections, 2 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Motivation and main results
  3. Basic notions and notations
  4. Isometries of the $d_{sym}$ quantum Wasserstein divergence
  5. Isometries of the $D_z$ quantum Wasserstein distance

Key Result

Theorem 1

Let $\Phi: S(\mathbb{C}^2) \to S(\mathbb{C}^2)$ be an isometry of the quantum Wasserstein divergence corresponding to the symmetric cost operator $C_{\text{sym}}$, that is, such that $\Phi$ maps pure states to pure states, that is $\Phi(\mathcal{P}_1(\mathbb{C}^2)) \subset \mathcal{P}_1(\mathbb{C}^2)$. Then $\Phi$ is necessarily of the form of $\Phi(\rho)=U\rho U^*$, where $U$ is either a unitary

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof