Abstract
State change, quantum probability, and information gain in the operational phasespace measurement are formulated by means of positive operator-valued measure (POVM) and operation. The properties of the operational POVM and its marginal POVM which yield the quantum probability distributions of the measurement outcomes obtained by the operational phase-space measurement are investigated. The Naimark extension of the operational POVM can be expressed in terms of the relative-position states and the relative-momentum states in the extended Hilbert space. An observable quantity measured in the operational phase-space measurement becomes a fuzzy or unsharp observable. The state change of a physical system caused by the operational phase-space measurement is described by the operation which is obtained explicitly for the position and momentum measurements and for the simultaneous measurement of position and momentum. Using the results, the entropy change of the measured physical system and the information gain in the operational phase-space measurement are investigated. It is found that the average value of the entropy change is equal to the Shannon mutual information extracted from the outcomes exhibited by the measurement apparatus.
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Abramowitz, M., and Stegun, I. A. eds. (1970).Handbook of Mathematical Functions, Dover, New York.
Aharonov, Y., Albert, D. Z., and Au, C. K. (1981).Physical Review Letters,47, 1029.
Agarwal, G. S., and Wolf, E. (1970a).Physical Review D,2, 2161.
Agarwal, G. S., and Wolf, E. (1970b).Physical Review D,2, 2187.
Agarwal, G. S., and Wolf, E. (1970c).Physical Review D,2, 2206.
Arthurs, E., and Kelly, J. L., Jr., (1965).Bell System Technical Journal,44, 725.
Ballan, R., Vénéroni, M., and Balazs, N. (1986).Europhysics Letters,1, 1.
Ban, M. (1991).Journal of Mathematical Physics,32, 3077.
Ban, M. (1992a).Journal of the Optical Society of America 9, 1189.
Ban, M. (1992b).Optics Communications,94, 231.
Ban, M. (1993a).Physical Review A,48, 3452.
Ban, M. (1993b).Physics, Letters A,176, 47.
Ban, M. (1994a).Physical Review A,50, 2758.
Ban, M. (1994b).Physica A,212, 327.
Ban, M. (1996).International Journal of Theoretical Physics,35, 1947.
Banaszek, K., and Wódkiewicz, K. (1997).Physical Review A,55, 3117.
Belavkin, V. P., Hirota, O., and Hudson, R. L., eds. (1995).Quantum Communication and Measurement, Plenum Press, New York.
Bendjaballah, C., Hirota, O., and Reynaud, S., eds. (1991).Quantum Aspects of Optical Communications, Springer-Verlag, Berlin.
Bialynicki-Birula, I., and Mycielski, J. (1975).Communications on Mathematical Physics,44, 129.
Boiteux, M., and Levelut, A. (1973).Journal of Physics A,6, 589.
Braunstein, S. L., Caves, C. M., and Milburn, G. J. (1991).Physical Review A,43, 1153.
Brillouin, L. (1956).Science and Information, Academic Press, New York.
Burak, D., and Wódkiewicz K. (1992).Physical Review A,46, 2744.
Busch, P. (1985).International Journal of Theoretical Physics,24 63.
Busch, P., Grabowski, M., and Lahti, P. J. (1995).Operational Quantum Physics, Springer-Verlag, Berlin.
Bužek, V., Keital, C. H., and Knight, P. L. (1995a).Physical Review A,51, 2575.
Bužek, V., Keital, C. H., and Knight, P. L. (1995b).Physical Review A,51, 2594.
Cahill, K. E., and Glauber, R. J. (1969a).Physical Review,177, 1857.
Cahill, K. E., and Glauber, R. J. (1969b).Physical Review,177, 1882.
Caves, C. M., and Drummond, P. D. (1994).Review of Modern Physics,66, 481.
Cover, T. M., and Thomas, J. A. (1991).Elements of Information Theory, Wiely, New York.
Crawford, J. A. (1958).Nuovo Cimento,10, 698.
Davies, E. B. (1976).Quantum Theory of Open Systems, Academic Press, New York.
De Oliveira, F. A. M., Kim, M. S., and Knight, P. L. (1990).Physical Review A,41, 2645.
Englert, B., and Wódkiewicz, K. (1995).Physical Review A 51, R2661.
Fano, U. (1957).Review of Modern Physics,29, 74.
Glauber, R. J. (1963a).Physical Review,130, 2529.
Glauber, R. J. (1963b).Physical Review,131, 2766.
Hall, M. J. B. (1995).Physical Review Letters,74, 3307.
Harriman, J. E. (1994).Journal of Chemical Physics,100, 3651.
Helstrom, C. W. (1976).Quantum Detection and Estimation Theory, Academic Press, New York.
Hillery, M., O'Connell, R. F., Scully, M. O., and Wigner, E. P. (1984).Physics Report,106, 121.
Hirota, O., Holevo, A. S., and Caves, C. M. eds. (1997).Quantum Communication, Computation and Measurement, Plenum Press, New York.
Holevo, A. S. (1982).Probabilistic and Statistical Aspects of Quantum Theory, North-Holland, Amsterdam.
Hongi-yi, F., and Klauder, J. R. (1994).Physical Review A,49, 704.
Hongi-yi, F., and Xiong, Y. (1995).Physical Review A,51, 3343.
Hongi-yi, F., and Yue, F. (1996).Physical Review A,54, 958.
Husimi, K. (1940).Proceedings of the Physical and Chemical Society of Japan,22, 264.
Jaynes, E. T. (1957a).Physical Review,106, 620.
Jaynes, E. T. (1957b).Physical Review,108, 171.
Kano, Y. (1965).Journal of Mathematical Physics,6, 1913.
Kraus, K. (1983).States, Effects, and Operations, Springer-Verlag, Berlin.
Lai, Y., and Haus, H. A. (1989).Quantum Optics,1, 99.
Lalović, D., Davidović, D. M., and Bijedić, N. (1992).Physical Review A,46, 1206.
Lee, H. (1995).Physical Review A,50, 2746.
Maassen, H., and Uffink, J. B. M. (1988).Physical Review Letters,60, 1103.
Mahran, M. H., and Satyanarayana, M. V. (1986).Physical Review A,34, 640.
Majernik, V. (1970).Acta Physica Austriaca,31, 271.
Majernik, V. (1973).Acta Physica Austriaca,34, 83.
Mayer, J. E., and Mayer, M. G. (1977).Statistical Mechanics, Wiley, New York.
Mehta, C. L., and Sudarshan, E. G. C. (1965).Physical Review,138, B274.
Moller, K. B., Jorgesen, T. G., and Torres-Vega, G. (1997).Journal of Chemical Physics,106, 7228.
Morato, L. M. (1977).International Journal of Theoretical Physics,16, 707.
O'Connell, R. F., and Rajagopal, A. K. (1982).Physical Review Letters,48, 525.
O'Connell, R. F., and Wigner, E. P. (1981).Physics Letters A,85, 121.
Orlowski, A., and Wünsche, A. (1993).Physical Review A,48, 4697.
Ozawa, M. (1984).Journal of Mathematical Physics,25, 79.
Ozawa, M. (1993).Journal of Mathematical Physics,34, 5596.
Partovi, M. H. (1983).Physical Review Letters,50, 1883.
Peres, A. (1993).Quantum Theory: Concepts and Methods, Kluwer, Dordrecht.
Prigogine, I., George, C., Henin, F., and Rosenfeld, L. (1973).Chemica Scripta,4, 5.
Prugovečki, E. (1973).Foundations of Physics,3, 3.
Prugovečki, E. (1974).Foundations of Physics,4, 9.
Prugovečki, E. (1975).Foundations of Physics,5, 557.
Prugovečki, E. (1976a).Journal of Mathematical Physics,17, 517.
Prugovečki, E. (1976b).Journal of Mathematical Physics,17, 1673.
Prugovečki, E. (1977).International Journal of Theoretical Physics,16, 321.
Prugovečki, E. (1982).Physical Review Letters,48, 1065.
Royer, A. (1977).Physical Review A,15, 449.
Schmutz, M. (1978).Zeitschrift für Physik B,30, 97.
Shannon, C. E. (1948a).Bell System Technical Journal,27, 379.
Shannon, C. E. (1948a).Bell System Technical Journal,27, 623.
Shapiro, J. H., Yuen, H. P., and Machado Mata, J. A. (1979).IEEE Transactions on Information Theory,IT-25, 179.
Smith, T. B. (1988).Physics Letters A,127, 75.
Stenholm, S. (1992).Annals of Physics,218, 233.
Sudarshan, E. G. C. (1963).Physical Review Letters,10, 277.
Torres-Vega, Go. (1993a).Journal of Chemical Physics,98, 7040.
Torres-Vega, Go. (1993b).Journal of Chemical Physics,99, 1824.
Torres-Vega, Go., and Frederick, J. H. (1990).Journal of Chemical Physics,93, 8862.
Torres-Vega, Go., and Frederick, J. H. (1993).Journal of Chemical Physics,98, 3103.
Twareque Ali, S., and Emch, G. G. (1974).Journal of Mathematical Physics,15, 176.
Twareque Ali, S., and Doebner, H. D. (1976).Journal of Mathematical Physics,17, 1105.
Twareque Ali, S., and Prugovečki, E. (1977a).Journal of Mathematical Physics,18, 219.
Twareque Ali, S., and Prugovečki, E. (1977b).International Journal of Theoretical Physics,16, 689.
Umezawa, H. (1993).Advanced Field Theory, American Institute of Physics, New York.
Umezawa, H., Matsumoto, H., and Tachiki, M. (1982).Thermo Field Dynamics and Condensed States, North-Holland, Amsterdam.
Von Neumann, J. (1955).Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, New Jersey, [Die Mathematische Grundlagen der Quantenmechanik, Springer-Verlag, Berlin (1932)].
Walls, D. F., and Milburn, G. J. (1994).Quantum Optics, Springer-Verlag, Berlin.
Wehrl, A. (1978).Review of Modern Physics,50, 221.
Wehrl, A. (1979).Report on Mathematical Physics,16, 353.
Wigner, E. P. (1932).Physical Review,40, 749.
Wlodarz, J. J. (1994).Journal of Chemical Physics,100, 7476.
Wódkiewicz, K. (1984).Physical Review Letters,52, 1064.
Wódkiewicz, K. (1986).Physics Letters A,115, 304.
Wódkiewicz, K. (1987).Physics Letters A,124, 207.
Yuen, H. P. (1976).Physical Review A,13, 2226.
Yuen, H. P., and Shapiro, J. H. (1979).IEEE Transaction on Information Theory,IT-24, 657.
Yuen, H. P., and Shapiro, J. H. (1980).IEEE Transaction on Information Theory,IT-26, 78.
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Ban, M. State change, quantum probability, and information in operational phase-space measurement. Int J Theor Phys 36, 2583–2638 (1997). https://doi.org/10.1007/BF02435703
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DOI: https://doi.org/10.1007/BF02435703