A Modern Approach to Quantum Mechanics · Problem 5.23

Wave Mechanics in One Dimension — John S. Townsend

Physics — Quantum Mechanics · density-matrix-mixed-photon-bell-state

Problem

An attempt to perform a Bell-state measurement on two photons produces a mixed state, one in which the two photons are in the entangled state 12∣x,x⟩+12∣y,y⟩\frac{1}{\sqrt{2}}|x, x\rangle + \frac{1}{\sqrt{2}}|y, y\rangle with probability pp and with probability (1−p)/2(1-p)/2 in each of the states ∣x,x⟩|x, x\rangle and ∣y,y⟩|y, y\rangle. Determine the density matrix for this ensemble using the linear polarization states of the photons as basis states.
Intuitive explanation

Bell-state measurements are designed to project onto maximally entangled states, but imperfect apparatus can fail and leave you with a statistical mixture instead. The key is to recognize that you have three possible outcomes—each occurring with its own probability—and build the density matrix by summing the outer product of each state weighted by its probability: ρ=∑ipi∣ψi⟩⟨ψi∣\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i|. This captures both the coherence from the entangled term and the incoherence from the separable states.

Intuitive explanation

Bell-state measurements are designed to project onto maximally entangled states, but imperfect apparatus can fail and leave you with a statistical mixture instead. The key is to recognize that you have three possible outcomes—each occurring with its own probability—and build the density matrix by summing the outer product of each state weighted by its probability: ρ=∑ipi∣ψi⟩⟨ψi∣\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i|. This captures both the coherence from the entangled term and the incoherence from the separable states.

Related theory

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