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How to measure the momentum on a half line.

How to measure the momentum on a half line. Yutaka SHIKANO Dept. of Phys., Tokyo Institute of Technology Theoretical Astrophysics Group Instructed by Akio Hosoya. 12/8/2006 Physics Colloquium 2 at Titech. Outline. My Research ’ s standpoint Introduction of the Quantum Measurement Theory

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How to measure the momentum on a half line.

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  1. How to measure the momentum on a half line. Yutaka SHIKANO Dept. of Phys., Tokyo Institute of Technology Theoretical Astrophysics Group Instructed by Akio Hosoya 12/8/2006 Physics Colloquium 2 at Titech

  2. Outline • My Research’s standpoint • Introduction of the Quantum Measurement Theory • Various operators • Projective Measurement and POVM • Our proposed problem setup • Holevo’s works • Summary and Further discussions

  3. My Research’s standpoint • Overview of Quantum Information Theory • Quantum Computing (Deutsch, Shor, Grover, Jozsa, Briegel) • Quantum Communication (Milburn) • Entanglement (Vedral, Nielsen) • Quantum Cryptography (Koashi) • Quantum Optics (Shapiro, Hirota) • Quantum Measurement & Metrology (Ozawa, Yuen, Fuchs, Holevo, Lloyd) Infinite dimensional Hilbert Space Finite dimensional Hilbert Space

  4. Symmetric Operators v.s.Self-adjoint Operators • Symmetric Operators • Bounded Symmetric Operators: Hermitian • Riesz representation theorem • Self-adjoint Operators

  5. Projective Measurement (Von-Neumann Measurement) • Positive Operator Valued Measure (POVM) POVM was proposed by E. Davies & J. Lewis Projective Measurement andPositive Operator Valued Measure • Measurement Action to decide the probability distribution. Measurement without error Measurement with error

  6. Symmetric Self-adjoint Relation between Operators and Measurement Outlook: This region is POVM only. Hermitian Von-Neumann Measurement POVM

  7. Canonical Measurement • Uncertainty relation • Canonical Measurement • To satisfy the minimum uncertainty relation • proposed by Holevo in 1977 “Optimal” measurement

  8. Our proposed problem How do you measure the momentum optimally of particle on a half line? 0 The momentum operator is symmetric, but not self-adjoint. Not Von-Neumann measurement, but POVM only.

  9. Motivations • In Physics • Quantum wells • Carbon nanotubes • M. Fisher & L. Glazman, cond-mat/9610037 • M. Bockrath et. al, Nature, 397, 598 (1999) • In Quantum Information • To establish the quantum measurement theory • To clarify the relation between quantum measurement and the uncertainty principle

  10. Holevo’s work • To motivate to establish a time-energy uncertainty relation. • Time v.s. Momentum • Energy v.s. Coordinate • Energy is lowly bounded. v.s. Half line • To solve the optimal POVM of the time operator. • Experimentalists don’t know how to measure it since Holevo didn’t give CP-map.

  11. Our future work Our problem: How to construct the CP-map from the measure to satisfy the minimum uncertainty relation. 0

  12. Summary & Further Directions • We propose the problem how you measure the momentum optimally of particle in infinite-dimensional Hilbert space on a half line. • Our proposed problem set is similar to the Holevo’s. • We will solve this problem set. • I have to find the experiments similar to our proposed problem set.

  13. References • A. Holevo, Rept. on Math. Phys., 13, 379 (1977) • A. Holevo, Rept. on Math. Phys., 12, 231 (1977) • C. Helstrom, Int. J. Theor. Phys., 11, 357 (1974) • E. Davies & J. Lewis, Commun. math. Phys., 17, 239 (1970) • S. Ali & G. Emch, J. Math. Phys., 15, 176 (1974) • H. Yuen & M. Lax, IEEE Trans. Inform. Theory, 19, 740 (1973) • P. Carruthers & M. Nieto, Rev. Mod. Phys., 40, 411 (1968) • G. Bonneau, J. Faraut & G. Valent, Am. J. Phys., 69, 322 (2001) • A. Holevo, “Probabilistic and Statistical Aspects of Quantum Theory”, Elsevier (1982) • M. Nielsen & I. Chuang, “Quantum Computation and Quantum Information”, Cambridge University Press (2000) • J. Neumann, “Mathematische Grundlagen der Quantenmechanik”, Springer Verlag (1932) [English transl.: Princeton University Press (1955)]

  14. Potential Questions

  15. CP-map (Completely Positive map) Detector Output Data Final State Object

  16. Preparation Measurement Initial Conditions Output Data Object Quantum Operations Y. Okudaira et. al, PRL 96 (2006) 060503 Y. Okudaira et. al, quant-ph/0608039 My Research’s standpoint • Operational Processes in the Quantum System Quantum Measurement Quantum Metrology Quantum Estimation

  17. Observable & Self-adjoint operator • An Axiom of the Quantum Mechanics • “A physical quantity is the observable. The Observable defines that the operator which corresponds to the “physical quantity“ is self-adjoint.” proposed by Von-Neumann in 1932 In short Von-Neumann Measurement: To measure the physical quantity without error. POVM: To measure the physical quantity with error.

  18. Bounded Operators

  19. Uncertainty relation

  20. Why is the momentum operator defined on the half symmetric?

  21. Holevo’s solution

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