Quantum Steering and Entanglement in a Tritter: Hierarchy under Loss

TL;DR

This work analyzes a three-mode Gaussian state produced by a tritter fed with a two-mode squeezed vacuum and a coherent state, focusing on the hierarchy between entanglement and EPR steering under loss. Using exact covariance-matrix methods, the authors show that correlation strength depends solely on the squeezing parameter via $\lambda=\tanh r$ and is independent of the displacement, yielding genuine tripartite entanglement but no pairwise steering in the ideal case. When losses are introduced across five asymmetric configurations, steering decays more rapidly than entanglement, with finite transmissivity thresholds and direction-dependent behavior; monogamy of steering remains valid. Overall, the paper demonstrates a strict inclusion: steering is a subset of entanglement, and the tritter setup offers a practical platform for asymmetric quantum protocols, including one-sided device-independent tasks, in noisy environments.

Abstract

Multipartite entangled states of continuous variables are fundamental resources for scalable quantum information processing. We study the correlation hierarchy in a tripartite state engineered by mixing a two-mode squeezed vacuum with a coherent state on a tritter, a key linear optical element for multimode state generation. Using the covariance matrix formalism, we comprehensively analyze the entanglement and Einstein-Podolsky-Rosen (EPR) steering among the output modes. The strength of both correlations is governed solely by the squeezing parameter and is independent of the coherent amplitude. We further examine the impact of inevitable optical losses in various channel configurations. The results show that while losses degrade correlations, EPR steering remains monogamous and exhibits stricter resilience thresholds than entanglement. Our analysis, supported by parameter extension techniques, confirms that the steering condition is more stringent than the inseparability criterion, clearly demonstrating that steering forms a strict subset of entanglement. These results elucidate the correlation structure in a readily generated multimode state and offer practical insights for developing asymmetric quantum protocols, such as one-sided device-independent tasks, where EPR steering serves as a critical resource.

Figures (13)

• Figure 1: Schematic diagram of a three-mode optical splitter (Tritter). The blue components represent beam splitters, the green components represent phase shifters, and the black components represent mirrors.
• Figure 2: Evolution of entanglement with squeezing parameter $\lambda$. Blue dashed line: tripartite entanglement $E^{k|ij}$; red dotted line: bipartite entanglement $E^{i|j}$.
• Figure 3: The model describing how the input state undergoes Tritter processing to produce the output state under noise influence.
• Figure 4: Bipartite entanglement $E^{i|j}$ versus reflectivity $(1-T)$ for different squeezing parameters $\lambda$.
• Figure 5: Curves of tripartite entanglement versus reflectivity under different noise loss conditions. (a) Two cases with no loss in mode k, (b) Three cases with loss in mode k.