Quantum theory of incompatible observations

Abstract

Maximum likelihood principle is shown to be the best measure for relating the experimental data with the predictions of quantum theory.Quantum theory describes events on the most fundamental level currently available. The synthesis of information from mutually incompatible quantum measurements plays the key role in testing the structure of the theory. The purpose of this Letter is to show a unique relationship between quantum theory and the mathematical statistics used to obtain optimal information from incompatible observations: Quantum theory prefers the relative entropy (maximum likelihood principle) as the proper measure for evaluation of the distance between measured data and probabilities defined by quantum theory.In the standard textbooks [1], a quantum measurement is represented by a hermitian operatorÂ, whose spectrum determines the possible results of the measurement A|a = a|a .(1)In the following, the Dirac notation will be used and for the sake of simplicity a discrete spectrum will be assumed. Eigenstates are orthogonal a|a ′ = δ aa ′ and the corresponding projectors provide the closure relation a |a a| =1.Projectors predict the probability for detecting a particular value of the q-variable a represented by the operator A as p a = a|ρ|a , provided that the system has been prepared in a quantum state ρ. This mathematical picture corresponds to the experimental reality in the following sense: When the measurement represented by the operator is repeated N times on identical copies of the system, the number a particular output a is collected N a times. The relative frequencies f a = Na N will sample the true probability as f a → p a fluctuating around them. The exact values are reproduced only in the asymptotical limit N → ∞. Experimentalist's knowledge may be expressed in the form of a diagonal density matrixprovided that error bars of the order 1/N are associated with the sampled relative frequencies. This should be understood as mere rewriting of the experimental data {N, N a }. Similar knowledge may be obtained by observations, which can be parameterized by operators diagonal in the |a basis, i.e. by operators commuting with operatorÂ. But the possible measurement of non-commuting operators yields new information, which cannot be derived from the measurement ofÂ. From this viewpoint it seems to be advantageous to consider the sequential synthesis of various noncommuting observables. In this case several operatorŝ A j , j = 1, 2, . . . will be measured by probing of the system N times together. Now, one expects to gain more than just the knowledge of the diagonal elements of the density matrix in some a priori given basis. This sequential measurement of non-commuting observables should be distinguished from the similar problem of approximate simultaneous measurement of non-commuting observables [2]. As in the case of the measurement of a hermitian operators, the result of sequential measurements of noncommuting operators may be represented by a series of projectors |y i y i |. This should be accompa...

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