Skip to main content

Section 8.3 Separable States

Not all two particle states are entangled. An example of a state that is not entangled would be the following:

\begin{equation} \ket{\psi} = \frac{1}{2}\ket{\uparrow\uparrow} + \frac{1}{2}\ket{\uparrow\downarrow} + \frac{1}{2}\ket{\downarrow\uparrow} + \frac{1}{2}\ket{\downarrow\downarrow}\text{.}\label{eq_separable_state}\tag{8.10} \end{equation}

Following the procedure of the previous section leads to

\begin{align*} \ket{\psi} \amp = \ket{\uparrow} \left(\frac{1}{2}\ket{\uparrow} +\frac{1}{2}\ket{\downarrow}\right) + \ket{\downarrow} \left(\frac{1}{2}\ket{\uparrow} +\frac{1}{2}\ket{\downarrow}\right)\\ \amp =\frac{1}{\sqrt{2}}\ket{\uparrow} \left(\frac{1}{\sqrt{2}}\ket{\uparrow} +\frac{1}{\sqrt{2}}\ket{\downarrow} \right) +\frac{1}{\sqrt{2}}\ket{\downarrow} \left(\frac{1}{\sqrt{2}}\ket{\uparrow} +\frac{1}{\sqrt{2}}\ket{\downarrow} \right) \end{align*}

where in the second line we have made normalized positron states inside the parentheses.

And now something interesting has happened: the two positron states inside parentheses are identical. This means that if we measure the electron's spin, the resulting state collapse leads to the same positron state, regardless of what we obtain for the electron spin. The positron is no longer entangled with the electron!

Mathematically, we note that having the same positron state in both parentheses above means we can factor our state even further:

\begin{equation} \ket{\psi} = \left(\frac{1}{\sqrt{2}}\ket{\uparrow} +\frac{1}{\sqrt{2}}\ket{\downarrow} \right) \left(\frac{1}{\sqrt{2}}\ket{\uparrow} +\frac{1}{\sqrt{2}}\ket{\downarrow} \right)\tag{8.11} \end{equation}

In this form we can see directly that the two-particle state is really just a product of a single-particle electron state and a single-particle positron state. In general, whenever this happens that we can fully factor the two-particle state into

\begin{equation} \ket{\psi} = \ket{\phi_\text{ electron } }\ket{\phi_\text{ positron } }\text{.}\tag{8.12} \end{equation}

then the particles are not entangled. We call these separable states.

In summary, two-particle states are separable if they can be fully factored into a product of single-particle states. And if they cannot be factored, then they are entangled.