How to measure the momentum on a half line'

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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
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Outline

• My Researchs standpoint
• Introduction of the Quantum Measurement Theory
• Various operators
• Projective Measurement and POVM
• Our proposed problem setup
• Holevos works
• Summary and Further discussions

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My Researchs 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)

Finite dimensional Hilbert Space
Infinite dimensional Hilbert Space
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Symmetric Operators v.s.Self-adjoint Operators

• Symmetric Operators

• Bounded Symmetric Operators Hermitian

• Riesz representation theorem

• Self-adjoint Operators

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Projective Measurement andPositive Operator
Valued Measure

• Measurement
• Action to decide the probability distribution.

Measurement without error
Measurement with error
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Relation between Operators and Measurement
Outlook This region is POVM only.
Hermitian
Von-Neumann Measurement
POVM
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Canonical Measurement

• Uncertainty relation

• Canonical Measurement
• To satisfy the minimum uncertainty relation
• proposed by Holevo in 1977

Optimal measurement
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Our proposed problem

• How do you measure the momentum optimally of
  particle on a half line?

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The momentum operator is symmetric, but not
self-adjoint.
Not Von-Neumann measurement, but POVM only.
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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

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Holevos 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 dont know how to measure it
  since Holevo didnt give CP-map.

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

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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
  Holevos.
• We will solve this problem set.
• I have to find the experiments similar to our
  proposed problem set.

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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)

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Potential Questions
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CP-map (Completely Positive map)
Detector
Output Data
Final State
Object
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My Researchs standpoint

• Operational Processes in the Quantum System

Quantum Measurement Quantum Metrology Quantum
Estimation
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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.
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Bounded Operators
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Uncertainty relation
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Why is the momentum operator defined on the half
symmetric?
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Holevos solution