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In quantum mechanics, the POVM element is associated with the measurement outcome , such that the probability of obtaining it when making a measurement on the quantum state is given by Prob ( i ) = tr ( ρ F i ) {\displaystyle {\text{Prob}}(i)=\operatorname {tr} (\rho F_{i})} ,
The Born rule is a postulate of quantum mechanics that gives the probability that a measurement of a quantum system will yield a given result. In one commonly used application, it states that the probability density for finding a particle at a given position is proportional to the square of the amplitude of the system's wavefunction at that position.
In quantum mechanics, the measurement problem is the problem of definite outcomes: quantum systems have superpositions but quantum measurements only give one definite result. [ 1 ] [ 2 ] The wave function in quantum mechanics evolves deterministically according to the Schrödinger equation as a linear superposition of different states.
In quantum mechanics, the expectation value is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the most probable value of a measurement; indeed the expectation value may have zero probability of occurring (e.g. measurements which ...
In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The square of the modulus of this quantity represents a probability density . Probability amplitudes provide a relationship between the quantum state vector of a system and the results of observations of that system, a link was first ...
The probability of any outcome of a measurement upon a quantum system must be a real number between 0 and 1 inclusive, and in order to be consistent, for any individual measurement the probabilities of the different possible outcomes must add up to 1.
This difference is called the measurement problem of quantum mechanics. To predict measurement outcomes from quantum solutions, the orthodox interpretation of quantum theory postulates wave function collapse and uses the Born rule to compute the probable outcomes. [9]
In quantum mechanics, the key property of a POVM is that it determines a probability measure on the outcome space, so that can be interpreted as the probability of the event when measuring a quantum state | .