quantum measurement theory on a half line
DESCRIPTION
PS-09. Quantum Measurement Theory on a Half Line. Yutaka Shikano Department of Physics, Tokyo Institute of Technology Collaborator: Akio Hosoya. Y. Shikano and A. Hosoya, in preparation. Outline and Aim. What is Quantum Information? What is Measurement? (e.g. Measuring Process) - PowerPoint PPT PresentationTRANSCRIPT
Quantum Measurement Theory on a Half Line
Yutaka ShikanoDepartment of Physics,
Tokyo Institute of TechnologyCollaborator: Akio Hosoya
Y. Shikano and A. Hosoya, in preparation
PS-09
8/7/2007 The 52nd Condensed Matter Physics Summer School at Wakayama
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Outline and AimWhat is Quantum Information?What is Measurement? (e.g. Measuring
Process)What is Covariant Measurement?Comments on the Momentum on a Half LineOptimal Covariant Measurement Model
Why need we consider Quantum Measurement?
Why need we consider the Half Line system?（ Details: Y. Shikano and A. Hosoya, in preparation ）
8/7/2007 The 52nd Condensed Matter Physics Summer School at Wakayama
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ObjectInitial Conditions Output Data
Preparation Measurement
What is Quantum Information?
Operational Processes in the Quantum System
My Research Field
Solve the Schroedinger Equ
ations. What information do we obtain from this result?
Similar to Information Process
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Aim of Quantum Information
Solve the Schroedinger Equation.Understand how to obtain Information from
Quantum System. How much information can we get? What method can we obtain information optimal
ly?
Question: Is the essence of quantum mechanics the operational concept?
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Axioms of Quantum Mechanics
1. Definition of state, state space, observable
Observable is defined as the self-adjoint operator since the operator has real spectrums.
2. Time evolution of state (Schroedinger Equation)
3. Born’s probablistic formula4. Definition of the combined system
8/7/2007 The 52nd Condensed Matter Physics Summer School at Wakayama
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What is Measurement?
time
We obtain the measurement value on the probe system.
We can evaluate the “measurement” value t = 0 on the measured system from the measurement value t = t.⊿
Measured System Probe System
Interaction between the measured system and probe system.
This process is called magnification or observation and is different from measurement.
1
2
3
t = 0
t = t⊿
t = t+ T⊿ ⊿
We obtain the macroscopic value.
(e.g. Photomultiplier)
8/7/2007 The 52nd Condensed Matter Physics Summer School at Wakayama
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To describe the Measuring Process
We have to know the follows to describe the measuring process physically.
1. Hamiltonian on the combined system between the measured system and probe system.
2. Evolution operator on the combined system from the Hamiltonian
3. Measuring time of the measuring process.
4. Measurement value of the probe system.
8/7/2007 The 52nd Condensed Matter Physics Summer School at Wakayama
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What is Covariant Measurement?
Measured System
Probe System
0 3
0 3
measurement value
0
measurement value
3
Shifted1
2
Shifted as same!!
For any bases “0” on the space,
This condition is satisfied by the ideal measuring device.
The measure on the measured system, that is POVM, is constrained.
Remark
8/7/2007 The 52nd Condensed Matter Physics Summer School at Wakayama
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Formulation of the Covariant Measurement
Using the Born formula.
(Axiom 3)
Definition
Property of the covariant measurement.
transformation of the momentum.
(Holevo 1978,1979,1982)
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Comments on the Momentum on a Half Line
NOT
SAME
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Lesson from this example. is symmetric but not self-adjoint operat
or. This means that the momentum on a half line is NOT observable.
Lesson: When we consider the infinite dimensional Hilbe
rt space, e.g. momentum and position in quantum mechanics, we have to check the domain of the operator.
How to classify the operator.
Deficiency Theorem (Weyl 1910, von Neumann 1929, Bonneau et al. 2001)
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Prescription: Naimark Extension
Naimark Extension Theorem: When we extend the domain of any symmetric
operators, the symmetric operators become the self-adjoint operator on the extended domain.
position0
position0
Copy
0
position
Inversion
position
12
3
3
Combined
0
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Optimal Covariant Measurement Model
Aim: We find the Hamiltonian to satisfy the optimal
covariant POVM to minimize the variance between the measurement value on the probe system at t = t and the evaluated ⊿“measurement” value on the measured system at t = 0.
Optimal Covariant POVM
8/7/2007 The 52nd Condensed Matter Physics Summer School at Wakayama
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Model Hamiltonian
Assume that the measured system alone is coupled to the bulk system at zero temperature.
Evolution operator
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RemarksFollowing iεprescription, we can obtain the
optimal covariant POVM.
time
Measured System Probe System
t = 0
Bulk System
T=0
T=0t = ∞
Instantaneous interaction
Energy Dissipates!
Ground State
Controllable
Measurable
Precise evaluation from the momentum conservation.
Assumption: Ground Energy = 0
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Concluding Remarks
We have explained the overview of quantum information and the measurement theory of quantum system.
We have shown the strange example of the half line system.
We have obtained the optimal covariant measurement model.
Thank you for your attention although my poster presentation may be out of place in this session.
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References
Y. Shikano and A. Hosoya, in preparation J. von Neumann, Mathematische Grundlagen der Quant
mechanik (Springer, Berlin, 1932), [ Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, 1955) ]
A. S. Holevo, Rep. Math. Phys. 13, 379-399 (1978) A. S. Holevo, Rep. Math. Phys. 16, 385-400 (1979) A. S. Holevo, Probabilistic and Statistical Aspects of Qua
ntum Theory (North-Holland, Amsterdam, 1982) H. Weyl, Math. Ann. 68, 220-269 (1910) J. von Neumann, Math. Ann. 102, 49-131 (1929) G. Bonneau et al. Am. J. Phys. 69, 322-331 (2001)