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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 Hosoya
12/8/2006 Physics Colloquium 2 at Titech
Outline
My Research’s standpoint
Introduction of the Quantum Measurement
Theory
– Various operators
– Projective Measurement and POVM
Our proposed problem setup
Holevo’s works
Summary and Further discussions
2
My Research’s standpoint
Overview of Quantum Information Theory
– Quantum Computing (Deutsch, Shor, Grover,
Jozsa, Briegel)
– Quantum
Communication
(Milburn)
Infinite
Finite
dimensional
dimensional
• Entanglement (Vedral, Nielsen)
Hilbert
Hilbert
Space
Space
– Quantum Cryptography (Koashi)
– Quantum Optics (Shapiro, Hirota)
– Quantum Measurement & Metrology (Ozawa,
Yuen, Fuchs, Holevo, Lloyd)
3
Symmetric Operators v.s.
Self-adjoint Operators
Symmetric Operators
linear operator A : Dom( A) H H : Hilbert space
Ax y x Ay x, y Dom( A)
Bounded Symmetric Operators: Hermitian
Riesz representation theorem
! x H s.t. x Ay x y
Dom( A ) {x H | y x Ay : continuous linear functional}
A : x x
Dom( A) Dom( A )
A x y x Ay ; y Dom( A), x Dom( A )
Dom( A) Dom( A )
Self-adjoint Operators
4
Projective Measurement and
Positive Operator Valued Measure
Measurement
Action to decide the probability distribution.
S : density operator
Projective Measurement
S S
(Von-Neumann Measurement)
B A(U )
{M ( B j )} : orthogonal
Measurement
without error
S ( B) Tr SM ( B)
Positive Operator Valued
M ( B) : resolution of identity
Measure (POVM)
(1) M ( ) 0, M (U ) I
{Measurement
M ( B j )} : Not orthogonal
with error
(2) M ( B) 0
(3)for any at most countable decomposit
ion
POVM
was proposed by E. Davies & J. Lewis
{B j } of B , M(B j ) M(B) weakly convergenc e
5
Relation between Operators and
Measurement
Outlook:
Symmetric
This region is POVM only.
Hermitian
Self-adjoint
Von-Neumann Measurement
POVM
6
Canonical Measurement
Uncertainty relation
y E {M } y (dy)
x
x
Dx {M } y y x (dy)
M {M (dy)}
2
Dx ( A) Tr S x A A , where A Tr S x A
2
d
Dx {M } E x {M }
dx
2
4 Dx ( A)
Canonical Measurement
– To satisfy the minimum uncertainty relation
– proposed by Holevo in 1977
d
Dx {M } E x {M }
dx
2
4 Dx ( A)
7
“Optimal” measurement
Our proposed problem
H L [0, )
2
How do you measure the momentum
( x, p) optimally of particle on a half line?
P
1d
i dx
0
d
Dom( P ) H | (0) 0, dx
0 dx
1d
P
i dx
2
d
Dom( P ) H | dx
0 dx
2
The momentum
operator is symmetric,
but not self-adjoint.
Not Von-Neumann
measurement, but
POVM only.
8
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
measurement and the uncertainty principle
9
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.
10
Our future work
H L [0, )
2
( x, p)
0
Our problem:
How to construct the CP-map from the measure
to satisfy the minimum uncertainty relation.
11
Summary & Further Directions
We propose the problem how you measure the
momentum optimally of particle in infinitedimensional 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
proposed problem set.
12
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
(1973)
P. Carruthers & M. Nieto, Rev. Mod. Phys., 40, 411 (1968)
G. Bonneau, J. Faraut & G. Valent, Am. J. Phys., 69, 322
(2001)
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.:13
Princeton University Press (1955)]
14
Potential Questions
CP-map (Completely Positive map)
: is positive map if 0, 0
H
H
A
1
Detector
A
Output Data
H
A
Final State
H
A
A
Object
To extend from H to H H trivially.
A
A
A
If 1 is a positive map, is called CP - map.
A
16
My Research’s standpoint
Operational Processes in the Quantum System
Preparation
Initial Conditions
Measurement
Object
Output Data
Quantum Measurement
Quantum Operations Quantum Metrology
Y. Okudaira et. al, PRL 96 (2006) 060503
Quantum Estimation
Y. Okudaira et. al, quant-ph/0608039
17
Observable & Self-adjoint operator
An Axiom of the Quantum Mechanics
– “A physical quantity is the observable. The
Observable defines that the operator which
corresponds to the “physical quantity“ is selfadjoint.” proposed by Von-Neumann in 1932
In short
Von-Neumann Measurement:
To measure the physical quantity without error.
POVM:
To measure the physical quantity with error.
18
Bounded Operators
A sup
H
A
19
Uncertainty relation
Self - adjoint operator Y yM (dy)
Heisenberg - Robertson uncertainty relation
1
D (Y ) D ( A) E (iY , A)
4
S exp(iAx) S exp(iAx)
2
x
x
x
x
0
dE (Y )
E (iY , A)
dx
D (Y ) D {M } E (Y ) E {M }
x
x
x
x
x
x
20
Why is the momentum operator
defined on the half symmetric?
P
d
( x) ( x)dx
dx
i0
d
(0) (0) ( x) ( x)dx
i 0 dx
i
d
d
Dom
(
P
)
H
|
(
0
)
0
,
dx
( x) ( x)dx
dx
0 i dx
d
Dom
(
P
)
H
|
dx
P
dx
2
0
2
0
21
Holevo’s solution
M (dx) M (dx) M (dx)
0
1
dx
M (dx) e e expi ( ) x
2
0
expi ( ) x
M (dx) I
1
2
dx
2
( x)dx
22
