Delayed Choice Entanglement Elimination: The Reduced Density Matrix is Not Applicable, The No Communication Theorem Does Not Apply

The application of the reduced density matrix (rdm) to one particle (A) of an entangled pair of particles depends on the existence of states of the other particle of the entangled pair (B) that are related to the entanglement whenever B is measured. The reduced density matrix for particle A is based on a partial trace over these states of B. If this other particle (B) on which the partial trace is done is destroyed before particle detections are made on either particles A or B and no record is left of B including its states, the rdm cannot be applied to A. The particle to which the rdm was to be applied enters a pure state and can show interference. The No Communication Theorem is applicable only where the rdm applies. In the case where B is destroyed before particle detections are made on either particles A or B and no record is left of B including its states, the No Communication Theorem is not applicable. A delayed choice experiment whether to destroy B and leave no record of B (including its states relevant to its entanglement with A) before particle detections are made on either A or B can be done and has been proposed. This experiment can produce distinct distributions showing either interference or no interference with no correlation of A and B detections.