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Quantum communication over large distances depends upon successful distribution of highly entangled quantum states. Due to unavoidable noise in quantum communication channels, the quality of entangled states generally decreases exponentially with channel length. Entanglement purification and distillation can overcome the degenerative influence of noisy quantum channels by transforming previously shared less entangled pairs into a smaller number of maximally entangled pairs by local operations and classical communication (LOCC). This is important because the precision of both local operations and classical communication are independent of the imperfection of the quantum channel. The objective is to share strongly correlated qubits between distant parties in order to allow reliable quantum teleportation or quantum cryptography. Besides its important application in quantum communication, entanglement purification also plays a crucial role in error correction for quantum computation, because it can significantly increase the quality of logic operations between different qubits.

History
The limits for entanglement dilution and distillation are due to Bennett, Bernstein, Popescu and Schumacher. Entanglement distillation protocols for mixed states were introduced by Bennett, Brassard, Popescu, Schumacher, Smolin and Wootters, and the connection to quantum error-correction developed in a ground-breaking paper by Bennett, DiVincenzo, Smolin and Wootters that has stimulated a lot of subsequent research.

Bell States
Main Article: Bell state

A two qubit system can be written as a superposition of possible computational basis qubit states: $$|00\rangle, |01\rangle, |10\rangle, |11\rangle$$, each with an associated complex coefficient $$\alpha\,\!$$:
 * $$|\psi\rangle = \alpha_{00}|00\rangle + \alpha_{01}|01\rangle + \alpha_{10}|10\rangle + \alpha_{11}|11\rangle$$

As in the case of a single qubit, the probability of measuring a particular computational basis state $$|x\rangle$$ is the amplitude of it’s associated coefficient $$|\alpha_{x}|^{2}\,\!$$, subject to the normalization condition $$\sum_{x \epsilon {0,1}}|\alpha_{x}|^{2}\,\!$$.

The Bell state is a particularly important example of a two qubit state:
 * $$\frac{|00\rangle+|11\rangle}{\sqrt{2}}$$

Bell states posses the property that measurement outcomes on it of a Bell state are correlated. As can be seen from the expression above, the two possible measurement outcomes are zero and one, both with probability of 50%. As a result, a measurement of the second qubit always gives the same result as the measurement of the first qubit.

The Quantum Channel
Main Article: Quantum channel

The quantum channel is inherently different from the classical communication channel due to its quantum mechanical nature. Either classical or quantum information can be transmitted over a quantum channel by encoding in the information in a quantum state. Suppose that two parties, Alice and Bob, would like to communicate classical information over a noisy quantum channel. Alice encodes the classical information that she she intends to send to Bob in a (quantum) product state, as a tensor product of reduced density matrices $$p_{1} \otimes p_{2} \otimes$$.... where each $$p\,\!$$ is diagonal and can only be used as a one time input for a particular channel $$\epsilon\,\!$$.

The capacity of a noisy quantum channel $$\epsilon\,\!$$ for quantum information is given by:

Quantifying Entanglement
Bell states can be used to quantify entanglement. Let m be the number of high-fidelity copies of a Bell state that can be produced using LOCC. Given a large number of Bell states the amount of entanglement present in a pure state $$|\psi\rangle$$ can then be defined as the ratio of $$\frac{n}{m}$$, called the distillable entanglement of a particular state $$|\phi\rangle$$, which gives a quantified measure of the amount of entanglement present in a given system. The process of entanglement distillation aims to saturate this limiting ratio.

Entanglement Distillation
Let an entangled state $$|\psi\rangle$$ have a Schmidt decomposition:
 * $$|\psi\rangle = \sum_{x}\sqrt{p(x)}|x_{A}\rangle|x_{B}\rangle

$$ Where coefficients p(x) form a probability distribution, and thus are positive valued and sum to unity. The tensor product of this state is then,


 * $$|\psi\rangle ^{\otimes m} = \sum_{x_{1},x_{2},...,x_{m}}\sqrt{p(x_{1})p(x_{2})...p(x_{m})}|x_{1A}x_{2A}...x_{mA}\rangle | x_{1B}x_{2B}...x_{mB}\rangle

$$

Now, omitting all terms $$x_{1},...,x_{m}\,\!$$ which are not part of the typical set :$$ A_{\epsilon}^{(n)} $$ the new state is


 * $$|\phi_{m}\rangle = \sum_{x \epsilon A_{\epsilon}^{(n)}}\sqrt{p(x_{1})p(x_{2})...p(x_{m})}|x_{1A}x_{2A}...x_{mA}\rangle| x_{1B}x_{2B}...x_{mB}\rangle

$$

And renormalizing,
 * $$|\phi_{m}^{'}\rangle = \frac{|\phi_{m}\rangle}{\sqrt{\langle\phi_{m}|\phi_{m}\rangle}}

$$

Then the fidelity
 * $$F(|\psi\rangle ^{\otimes m}, |\phi_{m}^{'}\rangle) \rightarrow 1$$ as $$m \rightarrow \infty$$.

Suppose that Alice and Bob are in possession of m copies of $$|\psi\rangle$$. Alice can perform a measurement onto the typical set $$A_{\epsilon}^{(n)}$$ subset of $$p_{\psi}\,\!$$, converting the state $$|\psi\rangle^{\otimes m} \rightarrow |\phi_{m}\rangle$$ with high fidelity. The theorem of typical sequences then shows us that $$1-\delta\,\!$$ is a lower bound on the probability that a sequence is in the set $$A_{\epsilon}^{(n)}$$, and may be made arbitrarily close to 1 for sufficiently large m, and therefore the Schmidt coefficients of the renormalized Bell state $$|\phi_{m}^{'}\rangle$$ will be at most a factor $$\frac{1}{\sqrt{1-\delta}}$$ larger. Alice and Bob can now obtain a smaller set of n Bell states by performing LOCC on the state $$|\phi_{m}^{'}\rangle$$with which they can overcome the noise of a quantum channel to communicate successfully.

Entanglement Dilution
The reverse process of entanglement distillation is entanglement dilution, where large copies of the Bell state $$|\phi\rangle$$ are converted into less entangled states $$|\psi\rangle$$ using LOCC with high fidelity. The aim of the entanglement dilution process, then, is to saturate the inverse ratio of n to m, defined as the distillable entanglement.