bibkey: toth2026wassersteinfidelities
authors: Géza Tóth; József Pitrik
year: 2026
title: "Quantum Wasserstein distance and its relation to several types of fidelities"
doi: null
url: https://arxiv.org/abs/2506.14523v6
claim: "For Hermitian observables H_1,...,H_N, the paper compares the De Palma-Trevisan quantum Wasserstein distance with its variant optimized over separable couplings, and conjectures (Conjecture 1) that the modified separable distance never exceeds the modified De Palma-Trevisan distance for arbitrary mixed states."
strata_touched:
  - D5/S3/Quantum/Information/SeparableWassersteinComparisonRefutation
license: citation-only
triage: anchor

Quantum Wasserstein distance and its relation to several types of fidelities

Géza Tóth and József Pitrik, arXiv:2506.14523v6 (quant-ph), 2026-02-12. Quotations are from the v6 TeX source.

The De Palma–Trevisan distance (Section II):

One of the definitions of the quantum Wasserstein distance has been given by De Palma and Trevisan as \cite{DePalma2021Quantum} \begin{align} \begin{split} D_{\rm DPT}(\varrho,\sigma)^2\quad\=\frac 1 2 \min_{ \varrho_{12}}\sum_{n=1}^N;& \trace[(H_n^T\otimes \openone-\openone\otimes H_n)^2 \varrho_{12} ],\ \textrm{s.~t. }& \varrho_{12}\in\mathcal D,\label{eq:DPT} \ & {\rm Tr}1(\varrho{12})=\varrho^T, \ & {\rm Tr}2(\varrho{12})=\sigma, \end{split}

The modified distance:

\begin{split} \tilde{D}{\rm DPT}(\varrho,\sigma)^2&=D{\rm DPT}(\varrho,\sigma)^2\ &-[D_{\rm DPT}(\varrho,\varrho)^2+D_{\rm DPT}(\sigma,\sigma)^2]/2.\label{eq:Dmod} \end{split} \end{align}

The separable distance (Section III):

\begin{split} D_{\rm DPT, sep}(\varrho,\sigma)^2\quad\=\frac 1 2 \min_{ \varrho_{12}\in\mathcal D}\sum_{n=1}^N;& \trace[(H_n\otimes \openone-\openone\otimes H_n)^2 \varrho_{12} ],\ \textrm{s.~t. }& \varrho_{12}\in\mathrm{Sep},\label{eq:GMPC_distance_sep} \ & {\rm Tr}1(\varrho{12})=\varrho, \ & {\rm Tr}2(\varrho{12})=\sigma, \end{split} \end{align}

Its modified form:

Following the idea for the definition of the modified quantum Wasserstein distance in \EQ{eq:Dmod}, let us now define an analogous quantity for the quantum Wasserstein distance $D_{\mathrm{DPT},\mathrm{sep}}(\varrho ,\sigma )$ as \begin{align} \begin{split} &\tilde D_{\rm DPT, sep}(\varrho,\sigma)^2=D_{\rm DPT, sep}(\varrho,\sigma)^2\ &\quad-[D_{\rm DPT,sep}(\varrho,\varrho)^2+D_{\rm DPT, sep}(\sigma,\sigma)^2]/2.\label{eq:tildeDPTsep} \end{split}

Conjecture 1:

{\bf Conjecture 1.} The following inequality seems to hold \begin{align} \tilde D_{\rm DPT, sep}(\varrho,\sigma)^2 \le \tilde D_{\rm DPT}(\varrho,\sigma)^2\label{eq:DPTsepDPTConj} \end{align} if $\varrho$ and $\sigma$ are arbitrary mixed states.

Verified locator

• URL: https://arxiv.org/abs/2506.14523v6 (v6, 2026-02-12; source swap_fidelity_and_separability61.tex, md5 cf228e87bcfd90de27e7425f05127d95): the De Palma–Trevisan distance (l. 151–161), its modification (l. 228–232), the separable distance (l. 246–255) and its modification (l. 797–803), and Conjecture 1 (l. 866–870).