Two-Qubit Basis Vectors are Orthogonal

Remember that in a two-dimensional (single qubit) state space, we wanted to have the basis state vectors perpendicular (orthogonal) to each other. That makes many of our calculations much easier to write out and to understand. It also reflects the fact that the basis states correspond to mutually exclusive measurement outcomes. We want to have that same orthogonality in our four-dimensional state space, though it is admittedly more difficult to picture that. We also need to remember that it is the state vectors that are perpendicular (in state space) and not the spins of Bob’s and Alice’s particles.