Purification of single-photon entanglement with linear optics

We show that single-photon entangled states of the form $\mid 0⟩\mid 1⟩+\mid 1⟩\mid 0⟩$ can be purified with a simple linear-optics-based protocol, which is eminently feasible with current technology. Besides its conceptual interest, this result is relevant for attractive quantum repeater protocols.

Figure 1

(Color online) Scheme for entanglement purification of single-photon entanglement. Alice and Bob share two entangled single-photon states ${\rho }_{a_1{b}_1}$ and ${\rho }_{a_2{b}_2}$ with fidelity $F$. Both parties apply local unitary linear-optical transformations $U$ and $V$ to their respective modes. For appropriately chosen $U$ and $V$, the detection of one photon in either $d_a$ or $d_b$ projects modes $\tilde{a}$ and $\tilde{b}$ into a single-photon entangled state with higher fidelity.

Figure 2

(Color online) Output fidelity as a function of input fidelity for the optimized protocol, where the parameter $\theta$ is adapted to the input fidelity (${\tilde{F}}_{\mathit{opt}}$, solid line), and for a simplified protocol where $\theta =\frac{\pi }{8}$ for all input fidelities (${\tilde{F}}_1$, dashed line). As a reference, the straight line $\tilde{F}=F$ is also shown (dotted line).

Figure 3

(Color online) Fidelity improvement ${\tilde{F}}_1-F$ as a function of $F$ for Hong-Ou-Mandel dip visibilities $V=1$, 0.99, and 0.98 (top to bottom).