In a quantum which-way eraser experiment, the purpose is to observe the interference pattern associated with wave-like behavior after the "which-way" information has been erased. Let's break down the experiment and address your question.
In the standard double-slit experiment, a particle, such as an electron or a photon, is sent towards a barrier with two slits. On the other side of the barrier, a screen is placed to detect the particle's arrival. When both slits are open, the particle behaves as a wave and exhibits an interference pattern on the screen, indicating the wave-like nature of the particle.
Now, if we introduce a measurement device near one of the slits to determine which path the particle takes, we gain information about the particle's behavior. This information causes the wave function to collapse, and the particle behaves more like a classical particle, showing a particle-like behavior and no interference pattern.
In a quantum which-way eraser experiment, an additional step is taken. After the particle passes through the double slits, the which-path information is obtained by measuring some property of the particle, such as its momentum or position. However, a second measurement, known as the "eraser," is applied to the particle to remove the which-path information.
The eraser measurement introduces an entangled state between the particle and the measurement apparatus, which allows the interference pattern to be restored. This means that even after the which-path information is obtained, if an appropriate eraser measurement is applied, the particle can once again exhibit wave-like behavior and display an interference pattern on the screen.
Now, let's address your question about the second instance of decoherence causing the particle to be in a future state of all possible paths. Decoherence refers to the phenomenon where a quantum system interacts with its environment, causing the system to lose its coherence and behave more classically. Decoherence typically destroys interference patterns.
In the context of a quantum which-way eraser experiment, the eraser measurement is designed to reverse the effects of the initial measurement that obtained the which-path information. It effectively "undoes" the decoherence caused by the first measurement, allowing the particle to regain its wave-like behavior and display interference.
However, it is important to note that the eraser measurement does not cause the particle to be in a future state of all possible paths. It restores the particle's wave-like behavior and allows it to interfere with itself, but it does not determine the specific path the particle took in the past or its future behavior.
In summary, the second instance of decoherence, caused by the initial measurement in a quantum which-way eraser experiment, can be undone by an appropriate eraser measurement, allowing the particle to exhibit wave-like behavior and display interference. However, this does not imply that the particle is in a future state of all possible paths; it simply means that the interference pattern is restored.