Download Weak Values in Quantum Measurement Theory

“Master Thesis’ Presentation”
Weak Values
in Quantum Measurement Theory
- Concepts and Applications Yutaka Shikano
07M01099
Department of Physics,
Tokyo Institute of Technology
Outline
1.
2.
3.
4.
5.
Aim
Conventional Quantum Measurement
Concepts of Weak Values
Quantum Operations for Weak Operators
Conclusions and Discussions
2/19/2009
Master Thesis' Prsentation at Tokyo Tech
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1. Aim
Motivations


Measurement and state changes are highly
non-trivial in quantum mechanics.
In conventional quantum measurement
theory, we have only obtained the probability
distribution.


Experimentalists obtain the probability distribution
from the experimental data to show quantum
phenomena.
However, is the representation of the
measurement outcome only the probability
distribution?
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Aim


To construct the general framework of the
weak values advocated by Aharonov and his
collaborators, which are experimentally
accessible by the shift of the probe wave
function in weak measurement.
To show the efficiency of our proposed
framework.
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2. Conventional Quantum Measurement
Quantum Measurement Theory
(M. Ozawa, J. Math. Phys. 25, 79 (1984))
Target system
Probe system
t=0
1
3
t = ⊿t
We can evaluate the
“measurement” outcome t =
0 on the measured system
from the measurement
outcome t = ⊿t.
Interaction between the
target and probe systems.
2
We obtain the
measurement outcome
on the probe system.
time
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Representation of Quantum Measurement
Probe observable associated with the measured observable is
Target state to obtain the measurement outcome “m” is
Kraus operator
Positive operator valued measure (POVM)
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What information is obtained?
Experimentalist’s task
histogram
x
x
eigenvalues
Projective measurement (more generally
speaking, POVM measurement) only gives
information of the probability distribution.
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3. Concepts of Weak Values
Could we construct another representation
of the measurement outcome?
Definition of Weak Values
Def: Weak values of observable A
pre-selected state
post-selected state
In order to measure the weak value…
Def: Weak measurement is called if a coupling constant
with a probe interaction is very small and a measurement
back action is also very small.
(Y. Aharonov, D. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988))
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In order to Measure Weak Values
Probe system
Target system
the pointer operator
(position of the pointer) is
q and its conjugate
operator is p.
Observable A
Probe state after measurement
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Probe state before measurement
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Probe system
Target system
Observable A
the pointer operator
(position of the pointer) is
q and its conjugate
operator is p.
Since the weak value of A is complex in general,
: Initial probe variance for the momentum
Weak values are experimentally accessible by the
shifts of expectation values for the probe observables.
(R. Jozsa, Phys. Rev. A 76, 044103 (2007))
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Master Thesis' Prsentation at Tokyo Tech
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Experimental Realization
(K. Resch, J. S. Lundeen and A. Steinberg, Phys. Lett. A 324, 125 (2003))
Prepare the initial state
Post-selected state
0
0
1
-1
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Creating
superposition of
initial state
Creating the postselected state.
Weak
Measurement
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Master Thesis' Prsentation at Tokyo Tech
Measuring the polarization.
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Weak Measurement by Slide Glass
(N. M. W. Ritchie, J. G. Story, and R. G. Hulet, Phys. Rev. Lett. 66, 1107 (2003))


Use transverse position of each photon as pointer
Weak measurement can be performed by tilting a
glass optical flat, where effective
Probe
Mode C
q
Flat
gt
CCD camera
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Perform weak measurement on rail C.
Post-selection: rail
A+B-C (negative shift)
Post-selection: rail C
(positive shift)
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Post-selection: rail A and B
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Master Thesis' Prsentation at Tokyo Tech(No shift)
Experimental Realization
Prepare the initial state
Post-selected state
0
0
1
-1
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4. Quantum Operations
for Weak Operators
Could we construct the general framework
analogous to the conventional quantum
measurement?
CP map for Quantum Operations
Positive map
Arbitrary
extension of
Hilbert space
When
is positive map,
is called a completely positive map (CP map).
(M. Ozawa, J. Math. Phys. 25, 79 (1984))
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Kraus Representation
Any quantum state change can be described
as the operation only on the target system
via the Kraus operator
.
In the case of Weak Values???
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Weak Operator
(YS and A. Hosoya, arXiv:0812.4507)

In order to define the quantum operations
associated with the weak values,
Weak Operator
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Properties of Weak Operator
Relationship to Weak Value
Analogous to the expectation value
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Quantum Operations for Weak Operators
Key points of Proof:
1. Polar decomposition for the weak operator
2. Complete positivity of the quantum operation
The properties of the quantum operation are
1. Two Kraus operators
2. Partial trace for the auxiliary Hilbert space
3. Mixed states for the weak operator
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Weak operator describes
the entire history of the
state evolution.
environment
Post-selected state
Possible history
Pre-selected state
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system
environment
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Weak Measurement with Decoherence
Environment
Target system
Observable A
No noisy operations with
impulsive weak measurement
The shifts of the expectation values of the probe are
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5. Conclusions and Discussions
Conclusions


We have introduced the weak values and reviewed
the experimental realization in the optical system.
In analogous to the quantum operation for density
operator, we construct the quantum operation for the
weak operator associated with the weak values.
Probability Distribution
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Phase Information
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Discussions

To construct the (differential) geometrical
structure for the weak operator. (<--> the
Bloch sphere representation for the density
operator.)

To extend the concept of the observable. The
weak values can be defined for non-selfadjoint operators (e.g., phase operator and
time operator.).
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Thank you for your attention!
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