Editing Einstein–Podolsky–Rosen paradox (section)

== Mathematical formulation ==
Bohm's variant of the EPR paradox can be expressed mathematically using the [[Spin (physics)|quantum mechanical formulation of spin]]. The spin degree of freedom for an electron is associated with a two-dimensional complex [[vector space]] ''V'', with each quantum state corresponding to a vector in that space. The operators corresponding to the spin along the ''x'', ''y'', and ''z'' direction, denoted ''S<sub>x</sub>'', ''S<sub>y</sub>'', and ''S<sub>z</sub>'' respectively, can be represented using the [[Pauli matrices]]:<ref name=Sakurai />{{rp|9}}
<math display="block">
  S_x = \frac{\hbar}{2} \begin{bmatrix} 0 &  1 \\ 1 &  0 \end{bmatrix}, \quad
  S_y = \frac{\hbar}{2} \begin{bmatrix} 0 & -i \\ i &  0 \end{bmatrix}, \quad
  S_z = \frac{\hbar}{2} \begin{bmatrix} 1 &  0 \\ 0 & -1 \end{bmatrix},
</math>
where <math>\hbar</math> is the [[reduced Planck constant]] (or the Planck constant divided by 2π).

The [[eigenstate]]s of ''S<sub>z</sub>'' are represented as
<math display="block">
  \left|+z\right\rangle \leftrightarrow \begin{bmatrix}1\\0\end{bmatrix}, \quad
  \left|-z\right\rangle \leftrightarrow \begin{bmatrix}0\\1\end{bmatrix}
</math>
and the eigenstates of ''S<sub>x</sub>'' are represented as
<math display="block">
  \left|+x\right\rangle \leftrightarrow \frac{1}{\sqrt{2}} \begin{bmatrix}1\\1\end{bmatrix}, \quad
  \left|-x\right\rangle \leftrightarrow \frac{1}{\sqrt{2}} \begin{bmatrix}1\\-1\end{bmatrix}.
</math>

The vector space of the electron-positron pair is <math> V \otimes V </math>, the [[tensor product]] of the electron's and positron's vector spaces. The spin singlet state is
<math display="block">
  \left|\psi\right\rangle = \frac{1}{\sqrt{2}} \biggl(
    \left|+z\right\rangle \otimes \left|-z\right\rangle -
    \left|-z\right\rangle \otimes \left|+z\right\rang
  \biggr),
</math>
where the two terms on the right hand side are what we have referred to as state I and state II above.

From the above equations, it can be shown that the spin singlet can also be written as
<math display="block">
  \left|\psi\right\rangle = -\frac{1}{\sqrt{2}} \biggl(
    \left|+x\right\rangle \otimes \left|-x\right\rangle -
    \left|-x\right\rangle \otimes \left|+x\right\rangle
  \biggr),
</math>
where the terms on the right hand side are what we have referred to as state Ia and state IIa.

To illustrate the paradox, we need to show that after Alice's measurement of ''S<sub>z</sub>'' (or ''S<sub>x</sub>''), Bob's value of ''S<sub>z</sub>'' (or ''S<sub>x</sub>'') is uniquely determined and Bob's value of ''S<sub>x</sub>'' (or ''S<sub>z</sub>'') is uniformly random. This follows from the principles of [[measurement in quantum mechanics]]. When ''S<sub>z</sub>'' is measured, the system state <math>|\psi\rangle</math> collapses into an eigenvector of ''S''<sub>z</sub>. If the measurement result is +''z'', this means that immediately after measurement the system state collapses to
<math display="block"> \left| +z \right\rangle \otimes \left| -z \right\rangle = \left| +z \right\rangle \otimes \frac{\left| +x \right\rangle - \left| -x \right\rangle}{\sqrt2}.</math>

Similarly, if Alice's measurement result is −''z'', the state collapses to
<math display="block"> \left|-z\right\rangle \otimes \left|+z\right\rangle = \left| -z \right\rangle \otimes \frac{\left| +x \right\rangle + \left| -x \right\rangle}{\sqrt2}.</math>
The left hand side of both equations show that the measurement of ''S''<sub>z</sub> on Bob's positron is now determined, it will be −''z'' in the first case or +''z'' in the second case. The right hand side of the equations show that the measurement of ''S<sub>x</sub>'' on Bob's positron will return, in both cases, +''x'' or −''x'' with probability 1/2 each.