Transcript 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
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）
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The 52nd Condensed Matter Physics Summer
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What is Quantum Information?
 Operational Processes in the Quantum System
Similar to
Information Process
Preparation
Initial Conditions
My Research
Field
Output Data
Object
Solve the
Schroedinger
Equations.
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Measurement
What information do we obtain
from this result?
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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
optimally?
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 formula
4. Definition of the combined system
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What is Measurement?
Measured System
Probe System
t=0
3
t = ⊿t
t=
⊿t+⊿T
time
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We can evaluate the
“measurement” value t = 0
on the measured system
from the measurement
value t = ⊿t.
This process is called
magnification or
observation and is different
from measurement.
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Interaction between the
measured system and
probe system.
2
We obtain the
measurement value
on the probe system.
We obtain the
macroscopic value.
(e.g. Photomultiplier)
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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.
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What is Covariant Measurement?
For any bases “0” on the space,
Measured System
3
0
Probe System
1
Shifted
3
0
measurement value
measurement value
0
3
This condition is
satisfied by the
ideal measuring
device.
Remark
The measure on the
measured system, that is
POVM, is constrained.
2
Shifted as same!!
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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)
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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
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.
Deficiency Theorem
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(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.
3
position
0
1
Copy
0
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Combined
2
Inversion
position
position
0
3
position
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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
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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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Remarks
Following iεprescription, we can obtain the
optimal covariant POVM.
Bulk System
t=0
time
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Probe System
Instantaneous interaction
T=0
Energy
Dissipates!
t=∞
Measured System
T=0
Precise evaluation from the
momentum conservation.
Controllable
Ground State
Assumption: Ground Energy = 0
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Measurable
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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
Quantmechanik (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
Quantum 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)
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