More on Entangled States

There is a nice way of expressing the conditions for entangled states that allows us to extend the results to more than two qubits. These more general results will be useful when we discuss quantum error correction in the Error Correction lesson. This is also a nice example of where a careful choice of mathematical notation makes the results easier to remember. Building on $\ket S = c\ket{\uparrow_A\uparrow_B} + d\ket{\uparrow_A\downarrow_B} + e\ket{\downarrow_A\uparrow_B} + f\ket{\downarrow_A\downarrow_B}$, let’s make two generalizations:

1. We will use computational basis states instead of spin states

2. We change the coefficients from $c$, $d$, $e$, and $f$ to symbols that are easier to associate with the basis states:

   $$\ket\Psi = a_{00}\ket{00} + a_{01}\ket{01} + a_{10}\ket{10} + a_{11}\ket{11}.$$