EPR Paradox - Mathematical Formulation

Mathematical Formulation

The above discussion can be expressed mathematically using the quantum mechanical formulation of spin. The spin degree of freedom for an electron is associated with a two-dimensional complex Hilbert space H, 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_x, S_y, and S_z respectively, can be represented using the Pauli matrices:

$S_x = \frac{\hbar}{2} \begin{bmatrix} 0&1\\1&0\end{bmatrix}, \quad S_y = \frac{\hbar}{2} \begin{bmatrix} 0&-i\\i&0\end{bmatrix}, \quad S_z = \frac{\hbar}{2} \begin{bmatrix} 1&0\\0&-1\end{bmatrix}$

where stands for Planck's constant divided by 2π.

The eigenstates of S_z are represented as

$\left|+z\right\rang \leftrightarrow \begin{bmatrix}1\\0\end{bmatrix}, \quad \left|-z\right\rang \leftrightarrow \begin{bmatrix}0\\1\end{bmatrix}$

and the eigenstates of S_x are represented as

$\left|+x\right\rang \leftrightarrow \frac{1}{\sqrt{2}} \begin{bmatrix}1\\1\end{bmatrix}, \quad \left|-x\right\rang \leftrightarrow \frac{1}{\sqrt{2}} \begin{bmatrix}1\\-1\end{bmatrix}$

The Hilbert space of the electron pair is, the tensor product of the two electrons' Hilbert spaces. The spin singlet state is

$\left|\psi\right\rang = \frac{1}{\sqrt{2}} \bigg (\left|+z\right\rang \otimes \left|-z\right\rang - \left|-z\right\rang \otimes \left|+z\right\rang \bigg)$

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

$\left|\psi\right\rang = \frac{-1}{\sqrt{2}} \bigg (\left|+x\right\rang \otimes \left|-x\right\rang - \left|-x\right\rang \otimes \left|+x\right\rang \bigg)$

where the terms on the right hand side are what we have referred to as state Ia and state IIa.

To illustrate how this leads to the violation of local realism, we need to show that after Alice's measurement of S_z (or S_x), Bob's value of S_z (or S_x) is uniquely determined, and therefore corresponds to an "element of physical reality". This follows from the principles of measurement in quantum mechanics. When S_z is measured, the system state ψ collapses into an eigenvector of S_z. If the measurement result is +z, this means that immediately after measurement the system state undergoes an orthogonal projection of ψ onto the space of states of the form

For the spin singlet, the new state is

Similarly, if Alice's measurement result is −z, the system undergoes an orthogonal projection onto

which means that the new state is

This implies that the measurement for S_z for Bob's electron is now determined. It will be −z in the first case or +z in the second case.

It remains only to show that S_x and S_z cannot simultaneously possess definite values in quantum mechanics. One may show in a straightforward manner that no possible vector can be an eigenvector of both matrices. More generally, one may use the fact that the operators do not commute,

along with the Heisenberg uncertainty relation

$\left\lang (\Delta S_x) ^2 \right\rang \left\lang (\Delta S_z) ^2 \right\rang \ge \frac{1}{4} \left|\left\lang \left \right\rang\right|^2$

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