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Section 8.4 Bell States

Not all entangled states are equally entangled. Some are very nearly separable: imagine tweaking the coefficients of 1/2 in (8.10) to values like \(0.49\) and \(0.51\text{.}\) Other states are far from separable.

There exist two-particle states that are, in some sense, maximally entangled. In these states, which are called Bell states, 1  the \(S_z\) value of the positron is completely determined by the \(S_z\) measurement on the electron. One example is the following:

\begin{equation} \ket{\psi} = \frac{1}{\sqrt{2}}\ket{\uparrow\uparrow} + \frac{1}{\sqrt{2}}\ket{\downarrow\downarrow}\text{.}\label{eq_bell_state_example}\tag{8.13} \end{equation}

For an electron-positron pair in this state, an electron spin measurement is equally likely to give spin up or spin down. But a subsequent positron measurement will always find the positron to have the same spin that you just measured for the electron.

By the way, the entanglement goes in both directions: you could measure the positron spin first and collapse the state of the electron to be the same spin as the positron. Also, it is possible to construct Bell states where the electron and positron spins are exactly opposite.

Named after John Bell, an Irish physicist. You'll be encountering more of his handiwork shortly.