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 Hosoya12/8/2006 Physics Colloquium 2 at Titech
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Outline
My Research’s standpoint Introduction of the Quantum Measurement Th
eory– Various operators– Projective Measurement and POVM
Our proposed problem setup Holevo’s works Summary and Further discussions
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My Research’s standpoint
Overview of Quantum Information Theory– Quantum Computing (Deutsch, Shor, Grover, Jozs
a, Briegel)– Quantum Communication (Milburn)
• Entanglement (Vedral, Nielsen)
– Quantum Cryptography (Koashi)– Quantum Optics (Shapiro, Hirota)– Quantum Measurement & Metrology (Ozawa, Yue
n, Fuchs, Holevo, Lloyd)
Finite dimensional Hilbert Space
Infinite dimensional Hilbert Space
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Symmetric Operators v.s.Self-adjoint Operators
Symmetric Operators
Self-adjoint Operators
}functionallinear continuous:|{)(Dom AyxyHxA
)(Dom, AyxAyxyAx
spaceHilbert :)(Dom:operator linear HHAA
Bounded Symmetric Operators: Hermitian
yxAyxHx s.t. !
)(Dom),(Dom;
:
AxAyAyxyxA
xxA
Riesz representation theorem
)(Dom)(Dom AA
)(Dom)(Dom AA
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Projective Measurement andPositive Operator Valued Measure
Measurement
Action to decide the probability distribution.
)(Tr )(
)(
operatordensity :
BSMB
UAB
S
S
S
S
econvergenc weakly , of }{
iondecomposit countablemost at any for )3(
0)()2(
)(,0)()1(
identity of resolution:)(
M(B))M(BBB
BM
IUMM
BM
jj
POVM was proposed by E. Davies & J. Lewis
Projective Measurement
(Von-Neumann Measurement)
Positive Operator Valued Measure (POVM)
orthogonal:)}({jBM
orthogonalNot :)}({jBM
Measurement without error
Measurement with error
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Relation between Operators and Measurement
Symmetric
Hermitian
Self-adjoint
Von-Neumann Measurement
POVM
Outlook:
This region is POVM only.
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Canonical Measurement
Canonical Measurement– To satisfy the minimum uncertainty relation– proposed by Holevo in 1977
)(4}{}{
Tr where,Tr )(
)(}{
)(}{
2
2
2
ADEdxd
D
ASAAASAD
dyyyD
dyyEy
xxx
xxx
xx
xx
MM
M
M
)(4}{}{2
ADEdxd
Dxxx
MM
Uncertainty relation
)}({ dyMM
“Optimal” measurement
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Our proposed problem
How do you measure the momentum optimally of particle on a half line?
),0[2 LH
),( px
0
0
2
0
2
|)(Dom
1
,0)0(|)(Dom
1P
dxdx
dHP
dx
d
iP
dxdx
dHP
dx
d
i
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 measurem
ent and the uncertainty principle
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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.
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Our future work
Our problem:
How to construct the CP-map from the measure to satisfy the minimum uncertainty relation.
),0[2 LH
),( px
0
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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 Holevo’s.
We will solve this problem set. I have to find the experiments similar to our pr
oposed 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 (197
3) P. Carruthers & M. Nieto, Rev. Mod. Phys., 40, 411 (1968) G. Bonneau, J. Faraut & G. Valent, Am. J. Phys., 69, 322 (200
1)
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)
0,0 if map positive is :
map.-CP called is map, positive a is 1 If
. trivially to from extend ToAA
A
AHHH
Detector
Object Final State
Output DataA
H A
H
AH
AH
A1
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My Research’s standpoint
Operational Processes in the Quantum System
ObjectInitial Conditions Output Data
Preparation Measurement
Quantum Operations
Y. Okudaira et. al, PRL 96 (2006) 060503
Y. Okudaira et. al, quant-ph/0608039
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 Observ
able defines that the operator which corresponds to the “physical quantity“ is self-adjoint.” proposed by Von-Neumann in 1932
Von-Neumann Measurement:
To measure the physical quantity without error.
POVM:
To measure the physical quantity with error.
In short
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Bounded Operators
AA
H
sup
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Uncertainty relation
}{)(}{)(
),()(
)exp()exp(
),(4
1)()(
relationy uncertaintRobertson -Heisenberg
)(operator adjoint -Self
0
2
MEYEMDYD
AYiEdx
YdE
iAxSiAxS
AYiEADYD
dyyMY
xxxx
x
x
x
xxx
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Why is the momentum operator defined on the half symmetric?
P
dxxxdx
d
i
dxxxdx
d
ii
dxxdx
dx
i
P
0
0
0
)()(
)()()0()0(
)()(
0
2
0
2
|)(Dom
,0)0(|)(Dom
dxdx
dHP
dxdx
dHP
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Holevo’s solution
dxxIdxM
dxxi
dxxieedxM
dxMdxMdxM
)()(
2)(exp
2)(exp)(
)()()(
21
0
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