Exercise 6.3

The singlet state can be written as \[ \ket{sing} = \frac{1}{\sqrt{2}}( \ket{ud} - \ket{du} ) = \frac{1}{\sqrt{2}} \ket{ud} - \frac{1}{\sqrt{2}} \ket{du} \]

A product state can be written as \[ \ket{product\, state} = \alpha_u \beta_u \ket{uu} + \alpha_u \beta_d \ket{ud} + \alpha_d \beta_u \ket{du} + \alpha_d \beta_d \ket{dd} \]

where the coefficients are products of complex numbers \(\alpha\) and \(\beta\).

In order to write the singlet state in the form of a product state, it follows for the coefficients \[ \begin{flalign*} \alpha_u \beta_d &= \frac{1}{\sqrt{2}} \\ \alpha_d \beta_u &= -\frac{1}{\sqrt{2}} \\ \alpha_u \beta_u &= 0 \\ \alpha_d \beta_d &= 0 \end{flalign*} \]

But if \(\alpha_u \beta_u = 0\), then either \(\alpha_u\) or \(\beta_u\) must be zero. This means that either \(\alpha_u \beta_d\) or \(\alpha_d \beta_u\) must be zero, which leads to a contradiction.