In the Copenhagen interpretation, the collapse of the wavefunction is said to take place when a quantum system is measured. Essentially, the Wigner's friend experiment asks the question: at what stage does a "measurement" take place? It posits a friend of Wigner who performs the Schrödinger's cat experiment while Wigner is out of the room. Only when Wigner comes into the room does he himself know the result of the experiment: until this point, was the state of the system a superposition of "dead cat/sad friend" and "alive cat/happy friend", or was it determined at some previous point? Wigner designed the experiment to highlight the inadequacy of the idea (central to the Copenhagen interpretation) that the measurement process is somehow inherently special.
Compare quantum suicide.
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