1. Quantum Measurement Theory on a Half Line
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

2. The 52nd Condensed Matter Physics Summer School at Wakayama
Outline and Aim
What is Quantum Information?
What is Measurement? (e.g. Measuring Process)
What is Covariant Measurement?
Comments on the Momentum on a Half Line
Optimal 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

3. What is Quantum Information?
Operational Processes in the Quantum System
My Research Field
Object
Initial Conditions
Output Data
Preparation
Measurement
Similar to Information Process
Solve the Schroedinger Equations.
What information do we obtain from this result?
8/7/2007
The 52nd Condensed Matter Physics Summer School at Wakayama

4. 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 optimally?
Question: Is the essence of quantum mechanics the operational concept?
8/7/2007
The 52nd Condensed Matter Physics Summer School at Wakayama

5. Axioms of Quantum Mechanics
Definition of state, state space, observable
Observable is defined as the self-adjoint operator since the operator has real spectrums.
Time evolution of state (Schroedinger Equation)
Born’s probablistic formula
Definition of the combined system
8/7/2007
The 52nd Condensed Matter Physics Summer School at Wakayama

6. The 52nd Condensed Matter Physics Summer School at Wakayama
What is Measurement?
Measured System
Probe System
t = 0 1 3
We can evaluate the “measurement” value t = 0 on the measured system from the measurement value t = ⊿t.
Interaction between the measured system and probe system. 2
We obtain the measurement value on the probe system.
t = ⊿t
This process is called magnification or observation and is different from measurement.
t = ⊿t+⊿T
We obtain the macroscopic value.
(e.g. Photomultiplier)
time
8/7/2007
The 52nd Condensed Matter Physics Summer School at Wakayama

7. To describe the Measuring Process
We have to know the follows to describe the measuring process physically.
Hamiltonian on the combined system between the measured system and probe system.
Evolution operator on the combined system from the Hamiltonian
Measuring time of the measuring process.
Measurement value of the probe system.
8/7/2007
The 52nd Condensed Matter Physics Summer School at Wakayama

8. What is Covariant Measurement?
For any bases “0” on the space,
Measured System 3
This condition is satisfied by the ideal measuring device.
Shifted
Probe System 1 3
Remark
The measure on the measured system, that is POVM, is constrained.
measurement value
measurement value 3 2
Shifted as same!!
8/7/2007
The 52nd Condensed Matter Physics Summer School at Wakayama

9. Formulation of the Covariant Measurement
Definition
transformation of the momentum.
Property of the covariant measurement.
Using the Born formula.
(Axiom 3)
(Holevo 1978,1979,1982)
8/7/2007
The 52nd Condensed Matter Physics Summer School at Wakayama

10. Comments on the Momentum on a Half Line
NOT
SAME
8/7/2007
The 52nd Condensed Matter Physics Summer School at Wakayama

11. Lesson from this example.
is symmetric but not self-adjoint operator. This means that the momentum on a half line is NOT observable.
Lesson:
When we consider the infinite dimensional Hilbert space, e.g. momentum and position in quantum mechanics, we have to check the domain of the operator.
How to classify the operator.
(Weyl 1910, von Neumann 1929, Bonneau et al. 2001)
Deficiency Theorem
8/7/2007
The 52nd Condensed Matter Physics Summer School at Wakayama

12. 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.
position
Copy
Inversion 1 2 3
Combined
8/7/2007
The 52nd Condensed Matter Physics Summer School at Wakayama

13. 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

14. The 52nd Condensed Matter Physics Summer School at Wakayama
Model Hamiltonian
Assume that the measured system alone is coupled to the bulk system at zero temperature.
Evolution operator
8/7/2007
The 52nd Condensed Matter Physics Summer School at Wakayama

15. The 52nd Condensed Matter Physics Summer School at Wakayama
Remarks
Following iεprescription, we can obtain the optimal covariant POVM.
Bulk System
Measured System
Probe System
Instantaneous interaction
t = 0
T=0
Energy Dissipates!
Precise evaluation from the momentum conservation.
Controllable
T=0
Ground State
t = ∞
time
Assumption: Ground Energy = 0
Measurable
8/7/2007
The 52nd Condensed Matter Physics Summer School at Wakayama

16. The 52nd Condensed Matter Physics Summer School at Wakayama
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.
8/7/2007
The 52nd Condensed Matter Physics Summer School at Wakayama

17. The 52nd Condensed Matter Physics Summer School at Wakayama
References
Y. Shikano and A. Hosoya, in preparation
J. von Neumann, Mathematische Grundlagen der Quantmechanik (Springer, Berlin, 1932), [ Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, 1955) ]
A. S. Holevo, Rep. Math. Phys. 13, (1978)
A. S. Holevo, Rep. Math. Phys. 16, (1979)
A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 1982)
H. Weyl, Math. Ann. 68, (1910)
J. von Neumann, Math. Ann. 102, (1929)
G. Bonneau et al. Am. J. Phys. 69, (2001)
8/7/2007
The 52nd Condensed Matter Physics Summer School at Wakayama