Transcript Weak Values with Decoherence

Weak Values with Decoherence
(Typo in Program)
Yutaka Shikano and Akio Hosoya
Department of Physics, Tokyo Institute of Technology
1.
2.
3.
4.
Aim
Brief Review on Weak Values
Quantum Operations for Weak Operators
Conclusions
Based on arXiv:0812.4502. (Typo in Abstract)
1. Aim
Aim of This Talk

To construct a mathematical 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 promote
the weak value analysis.

I hope that you will consider the new and
strange fashion “weak values” by the way to
going your home.
6/26/2009
ICSSUR'09 and the Feynman Festival
3
2. Brief Review on Weak Values
Review of Weak Values
•Y. Aharonov and D. Rohrlich, “Quantum Paradoxes”
(Wiley-VCH, Weibheim, 2005).
•Introduction Part of YS and A. Hosoya, arXiv:0812.4502.
Definition of Weak Values
Def: Weak values of observable A
pre-selected state
post-selected state
To measure the weak value…
Def: Weak measurement is called if a coupling constant
with a probe interaction is very small.
(Y. Aharonov, D. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988))
We have demonstrated some experiments to obtain weak
values in optical systems.
6/26/2009
ICSSUR'09 and the Feynman Festival
5
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
State of the probe after measurement
Taylor expansion
6/26/2009
ICSSUR'09 and the Feynman Festival
6
Probe system
Target system
the pointer operator
(position of the pointer) is
q and its conjugate
operator is p.
Observable A
Since the weak value of A is complex in general,
We assume the probe wave function
for the position be real-valued.
: 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))
6/26/2009
ICSSUR'09 and the Feynman Festival
7
Strong Measurement
Quantum State
Projection
“in vitro” experiment
6/26/2009
ICSSUR'09 and the Feynman Festival
8
Weak Measurement
Cover
Slightly Seeing
“in vivo” experiment
6/26/2009
ICSSUR'09 and the Feynman Festival
9
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
6/26/2009
ICSSUR'09 and the Feynman Festival
10
1st step: Check the postselected state !!
Creating
superposition of
initial state
Shifting the phase for
each path.
Changeable
6/26/2009
ICSSUR'09 and the Feynman Festival
From the
interference
pattern, we can
construct the postselected state.11
2nd step: See the image of
CCD camera.
Fixed
Weak
Measurement
6/26/2009
ICSSUR'09 and the Feynman Festival
12
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
6/26/2009
ICSSUR'09 and the Feynman Festival
13
Perform weak measurement on rail C.
Post-selection: rail
A+B-C (negative shift)
Post-selection: rail C
(positive shift)
6/26/2009
Post-selection: rail A and B
14
ICSSUR'09 and the Feynman Festival (No shift)
Experimental Realization
Prepare the initial state
Post-selected state
0
0
1
-1
6/26/2009
ICSSUR'09 and the Feynman Festival
15
3. Quantum Operations
for Weak Operators
Could we construct the general framework
analogous to the conventional quantum
operations?
Kraus Representation (Conventional)
: Completely positive map (CP map)
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???
6/26/2009
ICSSUR'09 and the Feynman Festival
17
Weak Operator
(YS and A. Hosoya, arXiv:0812.4502)

To define the quantum operations
associated with the weak values,
Weak Operator
6/26/2009
ICSSUR'09 and the Feynman Festival
18
Properties of Weak Operator (1)
Relationship to Weak Value
Analogous to the expectation value
6/26/2009
ICSSUR'09 and the Feynman Festival
19
Properties of Weak Operator (2)
Forward time evolution for the density operator
Backward time evolution for the density operator
The weak operator describes the entire history of the state
from the past (ti) to the future (tf ) and measurement performed
at the time t.
6/26/2009
ICSSUR'09 and the Feynman Festival
20
Quantum Operations for Weak Operators
Key points of Proof:
1. Polar decomposition for the weak operator
2. Complete positivity of the quantum operation
Roughly speaking,
Kraus operator for the density operator on forward time
Kraus operator for the density operator on backward time
6/26/2009
ICSSUR'09 and the Feynman Festival
21
Weak operator describes
the entire history of the
state evolution.
environment
system
Post-selected state
Possible history
Impulsive Weak
Measurement
Pre-selected state
6/26/2009
environment
ICSSUR'09 and the Feynman Festival
22
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
where
6/26/2009
ICSSUR'09 and the Feynman Festival
23
5. Conclusions
Conclusions


In analogous to the quantum operation for density
operator, we construct the quantum operation for the
weak operator associated with the weak values.
We show that the probe shifts in weak measurement
is given by the weak value defined by the quantum
operation due to the environment.
6/26/2009
ICSSUR'09 and the Feynman Festival
25
Thank you very much for your attention.
6/26/2009
Please be careful
getting
home after the final talk.
26
ICSSUR'09 and the
Feynman Festival