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

2

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

3

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

6

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.

9

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

16

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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