General Two-Qubit States

To explore the properties of multi-qubit states, we will start with a system of just two qubits. The generalization to more than two qubits is conceptually straightforward but is very messy to write out in detail.

To be specific, we will use two spin-$½$ qubits (though they might be different kinds of spin-$½$ particles, for instance, one electron and one neutron) and write the system basis states in a form like $\ket S = \ket A \ket B$. We will label my qubits $A$, and use $B$ for Bob’s. Later we will translate the formalism into the computational basis states $\ket0$ and $\ket1$.

For two spin-$½$ qubits there are four possible combinations of spin-up and spin-down for observations along a chosen direction. That means, as we mentioned in the previous section, we have a four-dimensional state space and four basis states. There are several different ways of writing the basis states; we shall use all of them as we proceed. The following table shows three of the four basis states in four different formats: