Entanglement swapping

Two initial maximally entangled pairs $A{B}_1$ and $B_{2}C$ can be turned into an entangled pair $AC$ by a generalized Bell basis measurement on $B_1{B}_2$, followed by a local correction at an endpoint using its classical label. This is qudit teleportation of $B_1$, including its correlations with $A$, to $C$. The initial two pairs are consumed. Without the classical label, averaging the conditional endpoint states need not leave entanglement.

Start from three normalized Bell states ${\Phi }_{12}^{+}{\Phi }_{34}^{+}{\Phi }_{56}^{+}$. A polarizing beam splitter parity filter ${\Pi }_Z^{25}$ followed by $X$-basis outcome $++$, and a second filter ${\Pi }_X^{46}$ followed by $Z$-basis outcome $00$, have effective bras $⟨{\Phi }^{+}{|}_{25}/\sqrt{2}$ and $⟨{\Phi }^{+}{|}_{46}/\sqrt{2}$. Their contraction leaves ${\Phi }_{13}^{+}/8$. The herald probability is therefore $1/64$ per ideal three-pair input, or $1/16$ conditional on both parity-routing filters having succeeded. This distinction prevents renormalizing away the optical postselection cost.

Suppose independent resource pairs have density matrices $v_j|{\Phi }^{+}⟩⟨{\Phi }^{+}|+(1-v_j)I_4/4$. An ideal complete Bell-basis measurement, followed by outcome-dependent Pauli gates, produces a chosen Bell state with white-noise visibility $v_1{v}_2$: either white-noise input makes the remote pair maximally mixed. If the reported label is reliable with probability $q$ and otherwise uniformly random, averaging the erroneous corrections makes that branch maximally mixed too, so $v_{\mathrm{out}}=v_1{v}_{2}q$. A second independent detector of the same quality during quantum teleportation adds another factor $q$ to the teleported Bloch vector.