Reduction-induced Variation of Partial Von Neumann Entropy

TL;DR

The work addresses the challenge of quantifying entanglement in bipartite mixed states, where existing measures are often computationally intensive or limited in scope. It introduces Reduction-induced Variation of Partial Von Neumann Entropy (RIVPVNE), a unifying entanglement metric that reduces to the familiar PVNE for pure states and remains computationally lightweight. RIVPVNE is defined as $\mathrm{RIVPVNE}(\rho)=S(\rho_i)-S_{\max}(\rho(i)_{\mathrm{eff}})$, and is shown to be an entanglement monotone under stochastic LOCC, with an effective density matrix $\rho_{\mathrm{eff}}$ capturing entanglement features. Through Bell-like and mixed-state examples, the authors compare RIVPVNE to EOF, REE, and Concurrence, demonstrating favorable behavior and lower computational burden, arguing for its practical relevance and potential extension to multipartite entanglement.

Abstract

TThe organization and structure of bipartite mixed-state quantum entanglement (QE) are more complex and less well understood compared to bipartite pure-state QE. Bipartite mixed-state QEs and their measures play a crucial role in both theory and practical applications. Some existing measures involve quantifying the minimum QE and reflect the inherently complex nature of their computation, while others are only applicable to highly limited-dimensional quantum systems. In this context, we propose a method termed Reduction-induced Variation of Partial Von Neumann Entropy to quantify QE in any bipartite states, particularly focusing on bipartite mixed states. Partial Von Neumann Entropy is merely a special case of this method,Its intuitive and clear physical representation, along with easy computation and wide applicability, facilitates exploring its potential applications. Furthermore, we present examples to demonstrate the superiorities of this method in identifying bipartite QE by comparing with other existing bipartite mixed-state QE measures through both their physical implications and mathematical structures.

Figures (2)

• Figure 1: Reduction-induced Variation of Partial Von Neumann Entropy as well as other QE measures such as the Partial Von Neumann Entropy, Entanglement of Formation, Relative Entropy of Entanglement and Concurrence varying with $\alpha$. When $\alpha=\frac{\pi}{4}$, $\mathop{\mathrm{RIVPVNE}}\nolimits(\rho)=\mathop{\mathrm{PVNE}}\nolimits(\rho)=S(\rho_i)=\mathop{\mathrm{EOF}}\nolimits(\rho)=\mathop{\mathrm{REE}}\nolimits(\rho)=\mathop{\mathrm{C}}\nolimits(\rho)=1$; whereas for the values $\alpha=0, \frac{\pi}{2}$, $\mathop{\mathrm{RIVPVNE}}\nolimits(\rho)=\mathop{\mathrm{PVNE}}\nolimits(\rho)=S(\rho_i)=\mathop{\mathrm{EOF}}\nolimits(\rho)=\mathop{\mathrm{REE}}\nolimits(\rho)=\mathop{\mathrm{C}}\nolimits(\rho)=0$; and in the ranges $0<\alpha<\frac{\pi}{4}$ and $\frac{\pi}{4}<\alpha<\frac{\pi}{2}$, $\mathop{\mathrm{RIVPVNE}}\nolimits(\rho)=\mathop{\mathrm{PVNE}}\nolimits(\rho)=S(\rho_i)=\mathop{\mathrm{EOF}}\nolimits(\rho)=\mathop{\mathrm{REE}}\nolimits(\rho)<\mathop{\mathrm{C}}\nolimits(\rho)$.
• Figure 2: Reduction-induced Variation of Partial Von Neumann Entropy as well as other QE measures such as Entanglement of Formation, Relative Entropy of Entanglement and Concurrence varying with $\alpha$. When $\alpha=\frac{\pi}{2}$, $\mathop{\mathrm{RIVPVNE}}\nolimits(\rho)=\mathop{\mathrm{EOF}}\nolimits(\rho)=\mathop{\mathrm{REE}}\nolimits(\rho)=\mathop{\mathrm{C}}\nolimits(\rho)=1$; whereas for $\alpha=0, \pi$, $\mathop{\mathrm{RIVPVNE}}\nolimits(\rho)=\mathop{\mathrm{EOF}}\nolimits(\rho)=\mathop{\mathrm{REE}}\nolimits(\rho)=\mathop{\mathrm{C}}\nolimits(\rho)=0$. While within the ranges $0<\alpha<\frac{\pi}{2}$ and $\frac{\pi}{2}<\alpha<\pi$, $\mathop{\mathrm{RIVPVNE}}\nolimits(\rho)<\mathop{\mathrm{EOF}}\nolimits(\rho)<\mathop{\mathrm{REE}}\nolimits(\rho)=\mathop{\mathrm{C}}\nolimits(\rho)$.