Understanding Two-Qubit States: The Bell States

[aaaa202]

An example of a two-qubit state is one of the Bell states, for example:

lB> = 1/√2 (l00> + l11>)

In my book it is stated that the Bell states form an orthonormal basis for the set of two qubit states. But what exactly is the general form of a two-qubit state? Is it any vector of the form:

lq> = 1√2 (la>lb> + lc>ld>)

, where la>, lb>, lc> and ld> is any normalized linear combination of l0> and l1>.
If so how can I see that the bell states form an orthonormal basis?

 

[jfizzix]

A general two-qubit state can be expressed as:

[itex]|\Psi\rangle = a |00\rangle + b|01\rangle + c|10\rangle + d|11\rangle[/itex]

You can show that the four Bell states form an orthonormal basis by showing that they are all orthogonal to each other and have unit magnitude.
The inner product between any two different bell states is zero,
and the inner product of a bell state with itself is unity.

Alternatively, you can with a bit of algebra, show that
[itex]|\Psi\rangle = a' |\psi^{+}\rangle + b'|\psi^{-}\rangle + c'|\phi^{+}\rangle + d'|\phi^{-}\rangle[/itex]
where
[itex]|\psi^{+}\rangle, |\psi^{-}\rangle, |\phi^{+}\rangle,[/itex] and [itex]|\phi^{-}\rangle[/itex] are the 4 bell states.