Global Coherence in Quantum Discord as a Resource

TL;DR

This work defines global coherence as the part of bipartite coherence not localized to either subsystem and formulates a resource theory with free operations given by local CPTP maps. It proves that global coherence is necessary for semi-device-independent (SDI) steering and introduces Mermin strength as a witness that exactly characterizes the resource enabling genuine remote state preparation (RSP) with two-qubit states. By connecting global coherence to the discord witness via correlation rank and to single-system coherence, the paper provides a unified operational framework linking coherence, discord, and steering tasks. The results clarify when discord reflects truly quantum correlations and offer concrete metrics (Schrödinger strength and Mermin strength) to quantify and certify resources for SDI steering and RSP in bipartite quantum systems.

Abstract

This work addresses which aspect of \textit{bipartite coherence} in \textit{quantum discord} is essential for \textit{genuinely quantum correlation}. To this end, \textit{global coherence} of bipartite states is defined as a form of bipartite coherence that is not local coherence in either of the subsystems or in both subsystems alone, and we identify it as being indicated by a witness of discord. With \textit{global coherence} as \textit{a resource}, \textit{any local operations} of the form $Φ_A \otimes Φ_B$, which may create coherence locally, are \textit{free operations}. This implies that with global coherence as a resource for operational tasks, any local operations can be freely used, but require \textit{classical randomness not to be freely} available. Using this identification, we demonstrate that \textit{global coherence} is a \textit{necessary resource} for the task of \textit{semi-device-independent steerability} of quantum discord. On the other hand, for the task of \textit{remote state preparation} using \textit{quantum discord} in two-qubit states, a \textit{necessary and sufficient quantum resource} is identified by invoking a \textit{witness of global coherence}.

Figures (3)

• Figure 1: One-sided semi-device-independent scenario JDS_PRA23 where the local Hilbert-space dimension $d_A$ of the shared bipartite state $\rho_{AB}$ in $\mathbb{C}^{d_A}\otimes\mathbb{C}^{d_B}$ producing the correlation $\{p(a,b|x,y)\}_{a,x,b,y}$ is also bounded. Here, the bound on $\lambda$ given by $d_{\lambda} \le d_A$ is used as a tool to witness superunsteerability; on the other hand, any amount of shared randomness $\lambda$ is not freely available if superunsteerability is used as a resource LBL+15JD23.
• Figure 2: Hierarchy of correlations in bipartite quantum states. The region $I$ represents correlations in incoherent states, and the regions $II$ and $III$ represent correlations in locally coherent states that include all one-way discordant and two-way discordant states that can be created locally; the regions $IV$-$VII$ represent correlations in globally coherent states. We have found that the regions $V$ and $VI$ (denoted as nonlocal discordant), which are subsets of all two-way discordant correlations that have global coherence, exhibit superunsteerability, implying SDI steerability. Nonlocal discordant correlations in region $V$ with $\Gamma = 0$ are not useful for the RSP.
• Figure 3: An abstract representation of the convex set of correlations in separable and unsteerable states in the two-setting scenario for two-qubit systems and the nonconvex set of correlations in the nonsuperunsteerable states. The straight line represents the Werner state given by Eq. (\ref{['Wers']}), which has the Schrödinger strength, $SS_{2}(\rho_W)=p>0$ for any $p>0$JDK+19. In the two-setting scenario, the Werner state has SDI steerability for any $p>0$, on the other hand, it has standard steerability for $p>1/\sqrt{2}$.

Theorems & Definitions (20)

• Definition 1
• Definition 2
• Definition 3
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Lemma 3