What does it mean when we say that the outcome of a quantum measurement is a random variable (or a quantum ensemble) such as in renes notes (page 51) or in this paper (page 4 B-1) ? Does it mean that we have as many copies of the state , on which we can perform the POVMs and obtain the random variable by counting frequencies ? Because when I hear measuring with a POVM, I understand using one element of the list of POVMs , getting one outcome and altering the initial state, so we should not be allowed to use a second element from the list of the POVMs because we lost the initial state. Can someone help me clarify this confusion ?
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A random variable can be interpretted with ensembles as you say, but it can also be intrinsic: you measure something, and it can actually produce any of the outcomes, just each with different probabilities
Quantum Mechanic
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When we speak about the mean value of an observable in a quantum state, we are referring to the average of many measurements of that observable. This means we assume the original state has been recreated many times, not measured in a state that has been altered by a previous measurement. If a measurement is realized previous on a system, subsequent measurements will all yield the same result. So, the answer for the following question is YES, IT DOES:
Does it mean that we have as many copies of the state , on which we can perform the POVMs and obtain the random variable by counting frequencies ?
Hadamard
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