Exact Entanglement Swapping through Single-Occupancy Measurements in Gaussian Fermion States
We determine the exact entanglement structure of the conditional state obtained by measuring $m$ corresponding rungs ($m\leq N/2$) in two identical copies of an arbitrary half-filled free fermion Gaussian state and post-selecting the same normalized single-fermion state ($|ψ\rangle = u|10\rangle+v|01\rangle$, where 0 and 1 denote the fermion occupancy on each sites) on each rung. For $m<N/2$, the conditional wavefunction generally depends on the initial state. Nevertheless, whenever the selected outcome has nonzero probability, the state on the unmeasured sites remains Gaussian and factorizes exactly into $N-m$ different orthogonal modes including $m$ inter-copy entangling modes and $N-2m$ spectator modes localized in one copy. Consequently, the entanglement entropy between the unmeasured parts of the two copies is $S=m h_2(|v|^2)$, independent of the initial state, where $h_2(x)=-x\ln x-(1-x)\ln(1-x)$. The success probability is given by $P_m=\det C_L\det(I_m-C_L)=\det(C_{LR}C_{RL})$, determined solely by the initial correlations and independent of ($u$,$v$). Equal-weight Bell post-selection serves as a special case that achieves the maximal entanglement swapping.