A Modern Approach to Quantum Mechanics · Problem 5.26

Wave Mechanics in One Dimension — John S. Townsend

Physics — Quantum Mechanics · density-operator-entangled-vs-mixed-state

Problem

Use the density operator formalism to show that the probability that a measurement finds two spin-

\frac{1}{2}

particles in the state

|{+x}, {+x}\rangle

differs for the pure Bell state

|\Phi^{(+)}\rangle = \frac{1}{\sqrt{2}}|{+z}, {+z}\rangle + \frac{1}{\sqrt{2}}|{-z}, {-z}\rangle

for which

\hat{\rho} = |\Phi^{(+)}\rangle\langle\Phi^{(+)}|

and for the mixed state

\hat{\rho} = \frac{1}{2}|{+z}, {+z}\rangle\langle{+z}, {+z}| + \frac{1}{2}|{-z}, {-z}\rangle\langle{-z}, {-z}|

Thus, the disagreement between the predictions of quantum mechanics for the entangled state

|\Phi^{(+)}\rangle

and those consistent with the views of a local realist are apparent without having to resort to Bell inequalities.