Entangled states. The singlet spin configuration (Equation 12.1) is the classic example of an entangled state—a two-particle state that cannot be expressed as the product of two one-particle states, and for which, therefore, one cannot really speak of “the state” of either particle separately. You might wonder
whether this is somehow an artifact of bad notation—maybe some linear combination of the one-particle states would disentangle the system. Prove the following theorem:
\frac{1}{\sqrt{2}}(|\uparrow \downarrow\rangle-|\downarrow \uparrow\rangle) (12.1).
Consider a two-level system, \left|\phi_{a}\right\rangle \text { and }\left|\phi_{b}\right\rangle, \text { with }\left\langle\phi_{i} \mid \phi_{j}\right\rangle=\delta_{i j} \text { (For example, }\left|\phi_{a}\right\rangle might represent spin up and \left|\phi_{b}\right\rangle spin down.) The two-particle state
\alpha\left|\phi_{a}(1)\right\rangle\left|\phi_{b}(2)\right\rangle+\beta\left|\phi_{b}(1)\right\rangle\left|\phi_{a}(2)\right\rangle ,
\text { (with } \alpha \neq 0 \text { and } \beta \neq 0 \text { ) } cannot be expressed as a product
\left|\psi_{r}(1)\right\rangle\left|\psi_{s}(2)\right\rangle ,
for any one-particle states \left|\psi_{r}\right\rangle \text { and }\left|\psi_{s}\right\rangle .
Hint: Write \left|\psi_{r}\right\rangle \text { and }\left|\psi_{s}\right\rangle \text { as linear combinations of }\left|\phi_{a}\right\rangle \text { and } \mid \phi_{b} \text { ) } .