Quantum Measurement Theory for Systems with Finite Dimensional State Spaces

Published on Oct 7, 2021in arXiv: Quantum Physics
Masanao Ozawa32
Estimated H-index: 32
(Chubu University)
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Abstract
In this paper, we present a general theory of finite quantum measurements, for which we assume that the state space of the measured system is a finite dimensional Hilbert space and that the possible outcomes of a measurement is a finite set of the real numbers. We develop the theory in a deductive manner from the basic postulates for quantum mechanics and a few plausible axioms for general quantum measurements. We derive an axiomatic characterization of all the physically realizable finite quantum measurements. Mathematical tools necessary to describe measurement statistics, such as POVMs and quantum instruments, are not assumed at the outset, but we introduce them as natural consequences of our axioms. Our objective is to show that those mathematical tools can be naturally derived from obvious theoretical requirements.
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#1Masanao Ozawa (Tohoku University)H-Index: 32
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#1Masanao Ozawa (Tohoku University)H-Index: 32
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#1Masanao Ozawa (Tohoku University)H-Index: 32
Abstract Heisenberg鈥檚 uncertainty relation for measurement noise and disturbance is commonly understood to state that in any measurement the product of the position measurement noise and the momentum disturbance is not less than Planck鈥檚 constant divided by 4蟺. However, it has been shown in many ways that this relation holds only for a restricted class of measuring apparatuses in the most general formulation of measuring processes. Here, Heisenberg鈥檚 uncertainty relation is generalized to a rela...
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#1Masanao Ozawa (Nagoya University)H-Index: 32
Quantum mechanics predicts the joint probability distributions of the outcomes of simultaneous measurements of commuting observables, but the current formulation lacks the operational definition of simultaneous measurements. In order to provide foundations of joint statistics of local general measurements on entangled systems in a general theoretical framework, the question is answered as to under what condition the outputs of two measuring apparatuses satisfy the joint probability formula for s...
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#1Masanao Ozawa (Nagoya University)H-Index: 32
Every measurement on a quantum system causes a state change from the system state just before the measurement to the system state just after the measurement conditional upon the outcome of measurement. This paper determines all the possible conditional state changes caused by measurements of nondegenerate discrete observables. For this purpose, the following conditions are shown to be equivalent for measurements of nondegenerate discrete observables: (i) The joint probability distribution of the...
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