In quantum information, the Bell states are a set of four two-qubit states
which are simple and useful examples of quantum entanglement.
They are given by:
is the tensor product of qubit in state and in .
These states form an orthonormal basis for the two-qubit
More importantly, however,
is that the Bell states are maximally entangled,
which we prove here for .
Consider the following pure density operator:
The reduced density operator of qubit is then calculated as follows:
This result is maximally mixed, therefore is maximally entangled.
The same holds for the other three Bell states,
and is equally true for qubit .
This means that a measurement of qubit
has a 50-50 chance to yield or .
However, due to the entanglement,
measuring also has consequences for qubit :
As an example, if collapses into due to a measurement,
then instantly also collapses into , never ,
even if it was not measured.
This was a specific example for ,
but analogous results can be found for the other Bell states.
- J.B. Brask,
Quantum information: lecture notes,