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t
P 1
1t
2t
P 1
?C
Prob( ) 1jC c
2P 0
iC ci j
If is an element of reality then jC cw jC c
2P 0,
iC c i j P 0,iC c i j
|w
CC
Pii C c
i
C c P
|
ii C ci
c
P
|iC c
ii
c
P
|jC c
jc
I PiC c
i
P
|
iC ci
jc
jc
2 2
2 2 2
P P
P P P
j j
i j i
C c C c
C c C c C ci i j
If then is an element of realityw jC c jC cFor dichotomic variables:
2wC c1 21 2P PC c C cC c c
1 11 2P (I-P )C c C cc c
|w
CC
1 11 2P (I-P )
|C c C cc c
1
2 1 2
P( )
|C cc c c
2c
1P 0C c
2
1 2
2
2 2 2
PProb( )
P P
C c
C c C c
C c
1
Two useful theorems:
If is an element of reality then jC cw jC c
If then is an element of realityw jC c jC cFor dichotomic variables:
1 1A A w P P
The three box paradox
1 1B B w P P
1 1A B C A B C w P P P P P P
1A B Cw w w P P P
1C w P
t
2t
1t
1
3A B C
1
3A B C
A B C
Tunneling particle has (weak) negative kinetic energy
Pointer probability distribution
?
Weak measurements performed on a pre- and post-selected ensemble
t
1tx
1x
1y y
2t
1.4w !
strong
weak
Weak Measurement of
The particle pre-selected 1x
2x y
int ( ) MDH g t P 2
22( )Q
MDin Q e
The particle post-selected 1y
Pointer probability distribution
Weak Measurement of
t
1t
20
1x i
i
1i x
20 particles pre-selected 1x 20 particles post-selected 1y
1i y 20
1i
iy
20
1
1
20 ii
20
1
1
20 ii
Robust weak measurement on a pre- and post-selected single system
The system of 20 particles
20
1
11.4
20 ii w
!
strong
weak
2t
Properties of a quantum system during the time interval between two measurements Y. Aharonov and L. Vaidman PRA 41, 11 (1990)
Another example: superposition of positive shifts yields negative shift
A. Botero
Superposition of Gaussians shifted by small values yields the Gaussian shifted by the large value
Generalized two-state vector
t
1t
2t
?C
1j i iN
j i
j i
i i ii
protection
2
2
P
Prob( )
Pn
i i C c ii
i i C c in i
C c
i ii
i
1jN
j
j
i i ii
wi i i
i
CC
t
1t
2t
1, 1, 1x y z i i ii
protection
PRL 58, 1385 (1987)
1, 1, 1
1, 1, 1
1, 1, 1
x y z
x y z
x y z
What is the past of a quantum particle?
The “past” and the “Delayed Choice” Double-Slit Experiment J.A. Wheeler 1978
The present choice of observation influences what we say about the “past” of the photon; it is undefined and undefinable without the observation.
The “past” of the photon is defined after the observation
Wheeler:
No phenomenon is a phenomenon until it is an observed phenomenon.
My lesson:
Wheeler delayed choice experiment
Wheeler: The photon took the upper pathIt could not come the other way
Wheeler delayed choice experiment
Wheeler: The photon took both pathsOtherwise, the interference cannot be explained
Interaction-free measurement
Did photon touched the bomb?Wheeler: The photon took the upper pathIt could not come the other way
The past of a quantum particle can be learned by measuring the trace it left
Wheeler delayed choice experiment
Wheeler: The photon took the upper pathIt could not come the other way
The trace shows Wheeler’s past of the photon
Wheeler delayed choice experiment
Wheeler: The photon took both pathsOtherwise, the interference cannot be explained
The trace shows Wheeler’s past of the photon
Interaction-free measurement
Did photon touched the bomb?
Operational meaning: Nondemolition measurements show NO!
Yes
No
No
Wheeler delayed choice experiment
Nondemolition measurements show that the photon took the upper path
Operational meaning:
Yes
No
No
Yes
Nondemolition measurements show that the photon took one of the paths
Operational meaning:
Yes
No
Yes
Where is the photon when it is inside a Mach-Zehnder interferometer?
But nondemolition (strong) measurements disturb the photon
Weak measurementsOperational meaning:
Where is the photon when it is inside a Mach-Zehnder interferometer?
The information is obtained from weak measurements on an ensemble of identically prepared photons
“Half a photon” or half the times the photon passes each path
(no disturbance at the limit)
YesNo
Yes or No
or Half a photon
Yes or No
or Half a photon
Wheeler delayed choice experiment
Weak measurementsOperational meaning:(no disturbance at the limit)
Yes
No
Yes No
The information is obtained from a pre- and post-selected ensemble
Interaction-free measurement
Did photon touched the bomb?
Weak measurementsOperational meaning:The information is obtained from a pre- and post-selected ensemble
Yes No
Yes
No
Interaction-free measurement
Yes
No
No
Strong measurementsDid photon touched the bomb?
Operational meaning:
Interaction-free measurement
Did photon touched the bomb?
Weak measurementsOperational meaning:
No
Yes
(no disturbance at the limit)
The information is obtained from a pre- and post-selected ensemble
Wheeler delayed choice experiment
Weak measurementsOperational meaning:(no disturbance at the limit)
The information is obtained from a pre- and post-selected ensemble
Yes
No
Interaction-free measurement
Did photon touched the bomb?
Weak measurementsOperational meaning:
No
Yes
(no disturbance at the limit)
The information is obtained from a pre- and post-selected ensemble
The best measuring device for pre-and post-selected photon is the photon itself
Strong measurements
Yes
The best measuring device for pre-and post-selected photon is the photon itself
Strong measurements
No
The best measuring device for pre-and post-selected photon is the photon itself
Weak measurements
Yes
The best measuring device for pre-and post-selected photon is the photon itself
Weak measurements
No
Wheeler’s argument: “The photon took the upper path because
it could not come the other way”seems to be sound.
The presence of the bomb can be found without anything passing near the bomb
Can we find that the bomb or anything else is not present in a particular place without anything passing near this place?
Hosten,…Kwiat, Nature 439, 949 (2006) Yes!
Its validity is tested in a best way by weak measurements using external system or the photon itself.
Kwiat’s proposal
Kwiat’s proposal
Kwiat’s proposal
Kwiat’s proposal
Kwiat’s proposal
Wheeler: We know that the bomb is not there and the photon was not there since it could not come this way.
Weak measurements: the photon was there!
Kwiat’s proposal
Weak measurements: the photon was there!
Yes
No
No
But it was not on the path which leads towards it!
Kwiat’s proposal
Weak measurements: the photon was there!
But it was not on the path which leads towards it! Yes
Kwiat’s proposal
Weak measurements: the photon was there!
But it was not on the path which leads towards it! No
Kwiat’s proposal
Weak measurements: the photon was there!
But it was not on the path which leads towards it! No
Kwiat’s proposal
Weak measurements by environment
Kwiat’s proposal
Weak measurements by environment
Kwiat’s proposal
Weak measurements: the photon was there!
But also in another place
Kwiat’s proposal
Weak measurements: the photon was there!
But also in another place. The effects are equal! Yes
Kwiat’s proposal
Weak measurements: the photon was there!
But also in another place. The effects are equal! Yes
t
P 1
1t
2t
P 1
The pre- and post-selected particle is described by the two-state vector
w
CC
The outcomes of weak measurements are weak values
?C
t
The two-state vector formalism expalnation
The two-state vector formalism expalnation
The two-state vector formalism expalnation
Where Is the Quantum Particle between Two Measurements?
The two-state vector formalism expalnation
The two-state vector formalism expalnation
The two-state vector formalism expalnation
1
3A B C
A
B
C
The two-state vector formalism explanation
1
3A B C
A
B
C
The two-state vector formalism explanation
) 1BB w
P
(P
A
B
C
The two-state vector formalism explanation
Yes
) 1AA w
P
(P
A
B
C
The two-state vector formalism explanation
Yes
) 1CC w
P
(P
A
B
C
The two-state vector formalism explanation
?
Interaction-free measurement
Interaction-free measurement
Interaction-free measurement
In IFM the photon was not near the bomb
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