The delayed choice quantum eraser experiment is a thought experiment in quantum mechanics that explores the nature of wave-particle duality and entanglement. It can be a bit counterintuitive, so let's break it down to understand why FTL (faster-than-light) communication is not possible.
In the experiment, a pair of entangled particles are created and separated. Let's call them Particle A and Particle B. Particle A is sent through a double-slit apparatus, which creates an interference pattern on a screen. Meanwhile, Particle B is kept away from any interference. The key aspect here is that the experimenter can choose to either measure or erase the information about which path Particle A took.
If the experimenter chooses to measure the path of Particle A, the interference pattern on the screen disappears. This measurement collapses the wave function of Particle A into a definite state and, due to entanglement, also affects the state of Particle B instantaneously. However, it is important to note that the measurement on Particle A does not convey any useful information about Particle B's state.
On the other hand, if the experimenter chooses to erase the path information of Particle A by using a quantum eraser setup, the interference pattern reappears on the screen. Again, the state of Particle B is instantaneously affected, but no useful information about Particle B can be extracted.
Here's the crucial point: even though the state of Particle B is influenced instantaneously by the measurement or erasure of Particle A, it does not allow for faster-than-light communication or information transfer. The reason is that the outcome of the measurement on Particle A is probabilistic. Each measurement result occurs with a certain probability, and it is not possible to predict in advance which result will be obtained for a given measurement. Therefore, one cannot use this setup to transmit information in a deterministic and meaningful way faster than the speed of light.
In summary, while the entanglement between the particles allows for instantaneous correlation, the probabilistic nature of quantum measurements prevents FTL communication. The outcomes of measurements are random, and it is not possible to control or predict them reliably to transmit meaningful information.