# Entanglement distillation by local operations and post-selection using one entanglement pair

1. Consider the state $|X\rangle = \sqrt{0.9} |00\rangle + \sqrt{0.1} |11\rangle$, shared between Alice and Bob, who are located far apart.

2. Alice brings in an ancilla qubit at her location (left-most qubit in the kets): $|X\rangle = \sqrt{0.9} |000\rangle + \sqrt{0.1} |011\rangle$.

3. Now Alice performs a CNOT gate with the control being her entangled qubit, and the target being the ancilla: $|X\rangle = \sqrt{0.9} |000\rangle + \sqrt{0.1} |111\rangle$.

4. Then Alice measures the ancilla in the basis $\{\sqrt{0.1} |0\rangle + \sqrt{0.9} |1\rangle , \sqrt{0.9} |0\rangle - \sqrt{0.1} |1\rangle\}$. Supposing the measurement outcome is $+1$, i.e., the ancilla collapsed to the state $\sqrt{0.1} |0\rangle + \sqrt{0.9} |1\rangle$ , the remaining state of the initial $2$ qubits will be $|X\rangle = \sqrt{0.1 \times 0.9} |00\rangle + \sqrt{0.9 \times 0.1} |11\rangle$, which is the maximally entangled state up to a normalization factor.

5. We started from a state that was not maximally entangled, and we were able to boost the entanglement by doing a local measurement and post-selecting on the outcome.

Is entanglement distillation using post-selection as I have described above feasible?

• What are your criteria of feasibility? – DaftWullie May 13 '18 at 16:11
• Can it be made better (equally as good, or nearly as good) as currently existing entanglement distillation protocols, whatever the criteria they use are? – user120404 May 13 '18 at 16:22
• This basically is a standard entanglement distillation protocol, just people usually describe the measurement using a POVM instead of having to introduce an extra qubit. – DaftWullie May 13 '18 at 16:25
• But in standard distillation protocols they use many copies of entangled pairs and combine them into a (nearly) maximally entangled pair -- as far as I understand. In the protocol I described above, only one pair is needed. – user120404 May 13 '18 at 16:27