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Optimal covariant measurement of momentum on a half line in quantum mechanics
1.J. von Neumann, Mathematische Grundlagen der Quantumechanik (Springer, Berlin, 1932) [
1.J. von Neumann, J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, 1955)].
2.The “observable” is a technical term. We use this term as a self-adjoint operator á la von Neumann by the Kato–Rellich theorem while the use of this term might be controversial.
6.M. Ozawa, in Squeezed and Nonclassical Light, edited by P. Tombesi and E. R. Pike (Plenum, New York, 1989), pp. 263–286.
7.We often call the detection process a magnification process, which is how to observe a pointed value of measuring devices, e.g., physical processes in a photomultiplier. This process is discussed by many people, e.g., see Ojima (Ref. 42).
8.P. Busch, P. Mittelstaedt, and P. J. Lahti, Quantum Theory of Measurement (Springer-Verlag, Berlin, 1991).
10.Many physicists do not classify symmetric operators into self-adjoint operators and often call a symmetric operator Hermitian and identify a Hermitian operator with an observable without checking a domain of an operator (see, e.g., Refs. 43 and 44).
13.A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 1982).
14.A. S. Holevo, Statistical Structure of Quantum Theory (Springer, Berlin, 2001).
15.N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space (Dover, New York, 1993).
25.T. Fülöp, Ph. D. thesis, University of Tokyo, 2005.
29.M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).
37.A. Arai and Y. Matsuzawa, mp_arc:08-24.
38.M. Reed and B. Simon, Methods of Modern Mathematical Physics II, Fourier Analysis, Self-Adjointness (Academic, New York, 1975).
43.L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1965).
44.L. D. Landau and E. M. Lifschitz, Quantum Mechanics Non-Relativistic Theory, 3rd ed. (Pergamon, New York, 1977).
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We cannot perform the projective measurement of a momentum on a half line since it is not an observable. Nevertheless, we would like to obtain some physical information of the momentum on a half line. We define an optimality for measurement as minimizing the variance between an inferred outcome of the measured system before a measuring process and a measurement outcome of the probe system after the measuring process, restricting our attention to the covariant measurement studied by Holevo [Rep. Math. Phys.13, 379 (1978)]. Extending the domain of the momentum operator on a half line by introducing a two dimensional Hilbert space to be tensored, we make it self-adjoint and explicitly construct a model Hamiltonian for the measured and probe systems. By taking the partial trace over the newly introduced Hilbert space, the optimal covariant positive operator valued measure of a momentum on a half line is reproduced. We physically describe the measuring process to optimally evaluate the momentum of a particle on a half line.
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