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An Isaac Newton Institute Workshop

Quantum Statistics - quantum measurements, estimation and related topics

Maximum likelihood methods in Quantum Mechanics

Author: Zdenek Hradil (Department of Optics )


The principle of Maximum Likelihood (MaxLik) is not a rule that requires justification. It does not need to be proved and nowadays it is widely used in many applications. What makes this technique so attractive and powerful is it efficiency and versatility. MaxLik estimation may be advanatageously applied to the inverse problems in quantum mechanics. Variables, which cannot be directly measured may always be estimated obeying the rules of quantum theory. Such a tight relationship will be demonstrated on the estimation of the phase shift and number-phase uncertainty relations. This motivates the application of the MaxLik for the quantum state reconstruction, the so called quantum tomography. Extremal equation for the MaxLik estimate of quantum state will be found and interpreted as the closure relation for quantum state measurement. Though the operator equation is nonlinear, it may be solved by iterations. The accuracy may be evaluated by means of Fisher information matrix. MaxLik estimation may be easily modified in order to treat the insufficient data. In this sense the MaxLik reconstruction represents an advantageous alternative to the linear reconstruction techniques based for example on the Radon transformation, which are prone to artifacts of various origin. MaxLik estimation will be demonstrated on several examples including the operational phase concepts, reconstruction of spin and entangled spin states, reconstruction of higher dimensional states of photons with angular momentum, reconstruction of photon statistics counted by inefficient detectors or absorption and phase X-ray tomography.