Witnessing Entanglement in Mixed-Particle Quantum Systems

Abstract

We introduce an entanglement witness that identifies off-diagonal long-range order (ODLRO) -- a distinctive form of entanglement -- in systems containing both fermionic and bosonic particles. By analyzing the particle-hole reduced density matrices of each subsystem, the approach detects ODLRO independently in both fermionic and bosonic sectors and identifies when long-range order develops across the entire mixed-particle system. The witness also quantifies the magnitude of ODLRO within each particle type, revealing how fermionic and bosonic correlations combine to form the total entanglement of the system, including a bosonic condensation of particle-hole pairs driven by many-body correlations rather than particle statistics. Using the Lipkin-Meshkov-Glick spin model, we show how the transition from ODLRO localized to one particle type to ODLRO shared by both particle types captures the onset of collective entanglement in a mixed-particle environment, providing new insight into systems where fermionic and bosonic correlations coexist.

Figures (4)

• Figure 1: Illustration of entanglement witnessing in a mixed fermion-boson system, where the fermionic degrees of freedom are initially strongly correlated. The largest eigenvalue of the fermion and boson blocks of the mixed-particle particle–hole reduced density matrix, $\lambda_G$, captures entanglement arising in the fermionic and bosonic modes, respectively.
• Figure 2: Heatmaps of the entanglement witness (a) $\lambda_G^{(f)}$ and (b) $\lambda_G^{(b)}$ for the fermionic and bosonic particle sectors (f) and (b), respectively, shown as functions of the fermionic correlation parameter $V_f$ and fermion-boson interaction parameter $\mu$, with $V_b = -2$. The system consists of 6 fermions and 6 bosons in 24 orbitals with $\varepsilon_f = \varepsilon_b= 5$. Darker regions indicate a stronger entanglement witness signal, reflecting a greater extent of ODLRO within the corresponding particle degrees of freedom.
• Figure 3: Entanglement witnesses $\lambda_G^{(f)}$ (solid lines) and $\lambda_G^{(b)}$ (dashed lines) for fermions and bosons, respectively, in systems with 8 particles---4 fermions and 4 bosons (blue)---and 12 particles---6 fermions and 6 bosons (pink)---as functions of the interaction parameter $\mu$. For the 8-particle system $V_f = -0.4$, $V_b = -2.0$, and $\varepsilon_f = \varepsilon_b = 1$; for the 12-particle system $V_f = -0.4$, $V_b = -2.0$, and $\varepsilon_f = \varepsilon_b = 5$. In both systems, the fermions, though initially non-interacting, develop increasing ODLRO with stronger interactions $\mu$.
• Figure 4: Entanglement witness values $\lambda_G^{(e)}$ and $\lambda_G^{(p)}$ for the electron–phonon model system for 12 particles---6 electrons and 6 phonons---in 24 orbitals as functions of interaction parameter $\mu$. Dashed lines show both particle type environments initially uncorrelated ($V_p = 0.0$, $\varepsilon_p = 0.3$; $V_e = -0.08$, $\varepsilon_e = 3.0$), reaching maximal $\lambda_G = 3$ only in the strong-coupling regime ($\mu > 0.5$). Solid lines show both particle type environments initially strongly correlated ($V_p = -0.08$, $\varepsilon_p = 0.3$; $V_e = -0.8$, $\varepsilon_e = 3.0$), saturating to maximal $\lambda_G$ at modest coupling ($\mu > 0.2$).