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

Question

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?

Answer 1

There are two different contexts where the term "entanglement distillation" is used, and are largely incomparable, even if they are conceptually extremely close (and I'm sure you'll be able to find papers that blur these boundaries).

In the first, Alice and Bob share a known quantum state which is (usually) a pure state. They use this to make a maximally entangled state with some probability. You can see this, for example, in section 12.5.1 of Nielsen & Chuang ("Transforming bi-partite pure state entanglement"). The protocol described in the question is the standard protocol in this context, except that the measurements are often expressed as POVMs instead of projective measurements in a larger Hilbert space.

In the second, one has many copies of a mixed state that one wishes to to make more entangled. This situation may be referred to as "purification" because one of the aims is, essentially, to make the state as pure as possible. The use of the mixed state, as usual, may describe part of an entangled system, or may describe some lack of knowledge/stochastic preparation procedure etc. In essence, there is some lack of knowledge about what you have, and variation between the copies, and it is this extra complication that necessitates multiple copies. See, for example, section 12.5.3 of Nielsen & Chuang ("Entanglement distillation and quantum error-correction").