L-entropy: A new genuine multipartite entanglement measure

TL;DR

The paper introduces latent entropy ($L$-entropy) as a new genuine multipartite entanglement measure for pure states that extends to finite and infinite-dimensional systems. It derives $L$-entropy from an upper bound on reflected entropy and proves it is a pure-state entanglement monotone, with maximal values attained for GHZ states in the 3-party case and for $k\ge 2$-uniform states in 4- and 5-party configurations. The work explores the behavior of $L$-entropy in random states, exhibits a Page-curve–like structure in multiboundary wormhole setups, and extends the framework to finite temperatures via a multipartite thermal pure quantum (MTPQ) construction, including analysis in a multi-copy SYK model. Overall, the approach provides a versatile, information-theoretic toolkit linking quantum information, holography, and many-body physics for diagnosing multipartite entanglement across diverse physical contexts.

Abstract

We advance ``Latent entropy" (L-entropy) as a novel measure to characterize genuine multipartite entanglement in pure states, applicable to quantum systems with both finite and infinite degrees of freedom. This measure, derived from an upper bound on reflected entropy, attains its maximum for three-party GHZ states and $n=4,5$-party $2$-uniform states. We establish that it satisfies all essential properties of a genuine multipartite entanglement measure, including being a pure-state entanglement monotone. We further obtain an analogue of the Page curve by analyzing the behavior of L-entropy in multiboundary wormholes, emphasizing their connection to multipartite entanglement in random states. Specifically, for $n = 5$, we show that random states approximate $2$-uniform states, exhibiting maximal multipartite entanglement. Extending these ideas to finite temperatures, we introduce the Multipartite Thermal Pure Quantum (MTPQ) state, a generalization of the thermal pure quantum state to multipartite systems, and demonstrate that the entanglement structure in states of the multicopy SYK model exhibits finite-temperature $2$-uniform behavior.

Figures (3)

• Figure 1: $\ell_{A_1A_2\cdot\cdot\cdot A_n}$ for three-party (green), four-party (blue) and five-party (red) random states are plotted here with increasing dimension $d$. The dotted and the solid lines correspond to the numerical and the analytical results, respectively.
• Figure 2: (a) Three-boundary wormhole with black hole $B$ and radiation regions $R_1,R_2$. (b) Page curve of tripartite L-entropy $\ell_{R_1R_2B}/\ell_{\max}$ versus $q=\sqrt{2}\mathcal{A}_{R_1}/\mathcal{A}_{B_0}$. Phases I--III correspond to $\mathcal{A}_B>\mathcal{A}_{R_1}+\mathcal{A}_{R_2}$, $\mathcal{A}_{R_1}<\mathcal{A}_B<\mathcal{A}_{R_1}+\mathcal{A}_{R_2}$, and $\mathcal{A}_B<\mathcal{A}_{R_1}$, with $|R_1|=|R_2|$.
• Figure 3: $\ell_{A_iA_j}$ (blue) and its value for thermal 2-uniform state given by \ref{['Ltherm2uni']} (red) are plotted with increasing $\alpha$ for multi-copy SYK model each involving 5-parties.