Exact quantification of bipartite entanglement in unresolvable spin ensembles

TL;DR

This work tackles the challenge of quantifying mixed-state entanglement in unresolvable spin ensembles by exploiting permutational invariance. The authors prove that for logically pure PI states, the entanglement of formation $E_F$ is exactly the average entanglement entropy over the standard decomposition, yielding a closed-form expression involving sums over angular-momentum couplings and CG coefficients. They apply the framework to angular-momentum eigenstates and metrologically relevant superpositions such as GHZ-like and spin-squeezed states, uncovering intricate dependencies on $N$, $J$, $M$, and $n$, including parity-induced zigzag patterns. The results provide a practical, exact entanglement quantifier for ensemble-based quantum technologies and offer benchmarks for entanglement witnesses in unresolvable spin systems.

Abstract

Quantifying mixed-state entanglement in many-body systems has been a formidable task. In this work, we quantify the entanglement of states in unresolvable spin ensembles, which are inherently mixed. By exploiting their permutationally invariant properties, we show that the bipartite entanglement of a wide range of unresolvable ensemble states can be calculated exactly. Our formalism is versatile; it can be used to evaluate the entanglement in an ensemble with an arbitrary number of particles, effective angular momentum, and bipartition. We apply our method to explore the characteristics of entanglement in different physically motivated scenarios, including states with definite magnetization and metrologically useful superpositions such as Greenberger-Horne-Zeilinger (GHZ) states and spin-squeezed states. Our method can help understand the role of entanglement in spin-ensemble-based quantum technologies.

Figures (10)

• Figure 1: (a) An ensemble of $N$ spin-1/2 particles is divided into two subensembles: one with $n$ particles (red) and the other with $N-n$ particles (green). Without the ability to resolve individual spins, the ensemble state is permutationally invariant and generally mixed. Our goal is to exactly quantify the entanglement of formation $E_F$ between the subensembles, provided that all collective properties of the ensemble are known, i.e., logically pure. (b) A logically pure PI state $\rho_{\text{PI}}^J$ is generally an equal mixture of all degenerate states that share the same collective properties, cf. Eq. (\ref{['Superposition PI state']}). Each degenerate component (denoted as cloud) is composed of subensemble states that belong to a different combination of subensemble angular momentum ($j_1$,$j_2$) and degeneracy indices ($i^{j_1}$,$i^{j_2}$). Since the subensemble bases are different, the superposition amplitudes and hence entanglement are generally different in each component. We show that the entanglement of formation of $\rho_{\text{PI}}^J$ is exactly given by the average entanglement entropy of all components in the standard decomposition, cf. Eq. (\ref{['Basis state being split with i']}).
• Figure 2: $E_F$ vs. $M$ for (a) $N=50$ (even) and (b) $N=51$ (odd), shown in both splitting scenarios. We only show the data for non-negative $M$, because the states with $-M$ have exactly the same amount of entanglement as those with $+M$.
• Figure 3: $E_F$ vs. $J$ for (a) $M=J$ and (b) $M=0,1/2$ states, shown in both splitting scenarios.
• Figure 4: Probability distribution components with qu$d$it levels $d=2\min(j_1,j_2)+1$, $\text{Prob}(d)=\sum_{\{j_1,j_2|d=2\min{(j_1,j_2)+1}\}}d^{j_1}_n d^{j_2}_{N-n}/d^J_N$ for the six lowest $J$ states with $N=50$ and $M=0$. The distribution is plotted up to $d=14$ since higher values have negligible probabilities. Components with larger $d$ permit more levels for superposition and hence entanglement. The population of low $d$ states $(d=2,4)$ peaks when $J=2,3$.
• Figure 5: $E_F$ vs. total spin number $N$ with minimum $|M|$ for (a) even splitting and (b) single-spin splitting. We compare the lowest $J$ states ($J=0$ for even $N$ and 1/2 for odd $N$) with Dicke states ($J=N/2$).