User:FiberOptix/Sandbox

Entanglement Distillation Aka Entanglement Purification

Overview

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.

Bell States A two qubit system can be written as a superposition of possible computational basis qubit states: \ket{00}, \ket{01}, \ket{10}, \ket{11}, each with an associated complex coefficient \alpha:

$\ket{\psi} = \alpha_{00}\ket{00} + \alpha_{01}\ket{01} + \alpha_{10}\ket{10} + \alpha_{11}\ket{11}$

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

The Bell state is a particularly important example of a two qubit state:

/frac{\ket{00}+\ket{11}}{/sqrt{2}}

What makes these Bell states special lies in the property that measurement outcomes of a Bell state are correlated. As can be seen from the expression above, the two possible measurement outcomes are 0 (outcome state \ket{00}) and 1 (outcome state \ket{11}), both with probability 1/2. As a result, a measurement of the second qubit always gives the same result as the measurement of the first qubit.

---Measurement correlations in the Bell state are stronger than could ever exist between classical systems.---

Quantifying Entanglement

Bell states can be used to quantify entanglement. We define m as the number of high-fidelity copies of a Bell state that can be produced using LOCC. Given a large number of Bell states n we can then define the amount of entanglement present in a pure state \ket{\psi} to be the ratio of n/m, called the distillable entanglement of a particular state \ket{\phi}, which gives a measurement of entanglement.

The Quantum Channel

The quantum channel is inherently different from the classical communication channel due to its quantum mechanical nature.

Entanglement Distillation

Let an entangled state \ket{\psi} have a Schmidt decomposition:

\ket{\psi} = \Sigma_{x}\sqrt{p(x)}\ket{x_{A}}\ket{x_{B}}

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

\ket{\psi}^{\tensor m} = \Sigma_{x_{1},x_{2},...,x_{m}}\sqrt{p(x_{1})p(x_{2})...p(x_{m})}|\ket{x_{1A}x_{2A}...x_{mA}}\ket{ x_{1B}x_{2B}...x_{mB}}

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

\ket{\psi}^{\tensor m} = \Sigma_{x \epsilon A_{\epsilon}^{(n)}}\sqrt{p(x_{1})p(x_{2})...p(x_{m})}|\ket{x_{1A}x_{2A}...x_{mA}}\ket{ x_{1B}x_{2B}...x_{mB}}

And renormalizing,

\ket{\psi ’} = \frac{\ket{\psi}}{\sqrt{\braket{\psi}{psi}}}

Then the fidelity F(\ket{\psi}^{\tensor m}, \ket{\psi’}) \rarrow 1 as m \rarrow \infty.

Suppose that Alice and Bob are in posession of m copies of \ket{\psi}. Alice can perform a measurement onto the typical set A_{\epsilon}^{(n)} subset of x, converting the state \ket{\psi}^{\tensor m} \rarrow \ket{\psi_{m}’} with high fidelity. The schmidt coefficients of the renormalized state \ket{\psi_{m}’} will be at most a factor \frac{1}{\sqrt{1-\delta}} larger since the theorem of typical sequences tells us that 1 — /delta is a lower bound on the probability that a sequence is in the set set A_{\epsilon}^{(n)}, and may be made arbitrarily close to 1 for sufficiently large m.

Distillation Optimization

Entanglement Distillation and Quantum Error-Correction <\math>