Entanglement Distillation - Quantifying Entanglement

Quantifying Entanglement

A two qubit system can be written as a superposition of possible computational basis qubit states:, each with an associated complex coefficient :

As in the case of a single qubit, the probability of measuring a particular computational basis state is the amplitude of its associated coefficient, subject to the normalization condition .

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

Bell states possess the amazing property that measurement outcomes 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.

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 can then be defined as the ratio of, called the distillable entanglement of a particular state, 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. The number of copies of a pure state that may be converted into a maximally entangled state is equal to the von Neumann entropy S(p) of the state, which is an extension of the concept of classical entropy for quantum systems. Mathematically, for a given density matrix p, the von Neumann entropy S(p) is . Entanglement can then be quantified as the entropy of entanglement, which is the von Neumann entropy of either or as:

Which ranges from 0 for a product state to 1 for a maximally entangled state.