Universal and Maximal Entanglement Swapping in General Fermionic Gaussian States

TL;DR

The paper demonstrates a universal, maximal entanglement swapping mechanism in general fermionic Gaussian states by performing post-selected Bell measurements on half of a bilayer system composed of two identical copies. By recasting the doubled wavefunction with an SU(2) rotation and exploiting determinant/minor structure, it shows that, at half filling, the unmeasured region collapses to a product of Bell pairs, yielding maximal interlayer entanglement independent of microscopic details. Numerical results using correlation matrices corroborate the analytic findings, revealing universal entanglement scaling S_{A_R} = (L/2) log 2 for full half-system measurements and linear scaling with the number of measurements otherwise. The work also links the post-measurement probability to the initial entanglement spectrum and discusses extensions to deterministic state preparation via imaginary-time evolution and to general projective measurements beyond Bell states, highlighting a fermionic-origin universality grounded in antisymmetry and Gaussianity.

Abstract

Exploring universal entanglement structure in many-body systems is both fundamental and challenging, particularly when the system undergoes non-unitary operations. In this work, we uncover a universal mechanism for realizing maximal entanglement swapping in fermionic Gaussian states subjected to projective Bell measurements. We consider two initially decoupled, half-filled copies of a free-fermion system in arbitrary dimensions and perform post-selective Bell measurements on half of the corresponding sites across the two copies. Remarkably, the post-measurement state factorizes into a product of Bell pairs, establishing maximal interlayer entanglement entirely independent of the initial Gaussian state. We derive this post-measurement state exactly for general particle-number-conserving fermionic Gaussian states, establishing both the validity and universality of the mechanism, with numerical simulations serving as consistency checks. This phenomenon arises from a robust interplay between fermionic statistics and Gaussianity, revealing a distinct fermionic route to measurement-induced maximal entanglement.

Figures (5)

• Figure 1: Left: Two identical free-fermion layers $A$ (upper) and $B$(lower) in $d$ dimensions (illustrated for $d=2$), each with an even number of sites $L$. Each layer is prepared in a number-conserving Gaussian state at half filling. Middle: Projective Bell measurements are performed on the subsystem $S_P$. While $S_P$ can be chosen arbitrarily, here we take it to be the left half of the system, i.e., $S_P = A_L \cup B_L$, where we denote the left (right) half of $A$ as $A_L$ ($A_R$), and similarly for $B$. For every site $i \in \{1,\dots,L\}$ in layer $A$, there is a dual site $\overline{i}$ in layer $B$. On each rung $(i,\overline{i}) \subset S_P$ we apply the Bell projector defined in Eq. \ref{['Bell_rung']}. That is, each rung in $S_p$ is projected to the Bell state $\ket{+}_{i\overline i} = \tfrac{1}{\sqrt{2}}(c_i^\dagger + c_{\overline{i}}^\dagger)\ket{\mathrm{vac}}$. Right: After $L/2$ such rung measurements, the post-selected state factorizes across the bipartition as a tensor product of a state on $S_P$ and one on its complement $S_P^c=A_R\cup B_R$. On $S_P$ each rung $(i,\overline i)$ is in the Bell state $\ket{+}_{i\overline i}$ (orange lines), while on the unmeasured complement $S_P^c$ each rung $(i,\overline{i})$ is in the Bell state $\ket{-}_{i\overline i} = \tfrac{1}{\sqrt{2}}(c_i^\dagger - c_{\overline{i}}^\dagger)\ket{\mathrm{vac}}$ (purple lines).
• Figure 2: Numerical results of the EE between the unmeasured part on a single layer and the rest, after performing uniform Bell measurements $P_i^+$ ($i \in A_L, |A_L| = N_{\mathbf{m}}$). Left: The measurements are performed precisely on half system with $N_{\mathbf{m}} = L/2$, with varying the total system size. The EE displays a linear dependence $S = \frac{L}{2} \log 2$ with the system size of a single layer $L$. Right: The number of performed measurements $N_{\mathbf{m}} \le L/2$, with a fixed system size $2L=28$. The EE displays a linear dependence $S = N_{\mathbf{m}} \log 2$ with the total system size of a single layer $L$. Here we consider the ground state to the critical Hamiltonian $H_0 = -\frac{1}{2} \sum_{i} c_i^\dagger c_{i+1}^\dagger + h.c.$ with open boundary conditions, and we have tested that other states, including gapped ground states, leads to the same behavior. The size of the total bilayer system is $2L$.
• Figure 3: Numerical results for the entanglement entropy of $A_R$ (See Fig.\ref{['fig:sketch']}) when fewer than half of the rungs are measured in the imperfect Bell states $|+\rangle_{i\overline i}^\epsilon$. Solid points denote numerical results, and the red squares correspond to analytical results in \ref{['SAR_imperfect']} and \ref{['SAR_imperfect2']}. The system size is $L=10$.
• Figure 4: Numerical results for half-system Bell measurements in a bilayer with an imperfect copy. Left: EE between $A_R$ and $B_R$ (in units of $\log 2$) as a function of the single-layer length $L$. Right: overlap fidelity $F=|\langle \Phi_{\rm Ideal}|\Psi_{\rm post}\rangle|^2$ as a function of $L$.
• Figure 5: Numerical results of projection probability $P = |\Delta_{A_L} \Delta_{A_R}|^2$, where the initial states are chosen as the ground states of $H_0$ in Eq.\ref{['H0_appendixF']}. Left: $P$ plotted as a function of the single-chain system size $L$ for various values of the initial mass $m_0$. Middle: Replot of the left panel with the horizontal axis $L^2$ and the vertical axis $\log P$. Blue circles denote the numerical data, while solid green lines show linear fits of the form $\log P \simeq \alpha + \beta L^2$. Right: $P$ plotted as a function of the initial mass $m_0$ for several values of the system size $L$.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Lemma 2.1
• Proposition 1
• Lemma 2.2
• Lemma 2.3
• Proposition 2
• Lemma A.1
• Lemma A.2
• Proposition A.1