P  1
t2
If C  c j is an element of reality then Cw  c j

C ?
t

t1

Prob(C  c j )  1 
P  1

i j
 PC c j 

 PC ci 
2
0 
 PC ci 
2
|

 PC c j 
 PC c j 

2
2
   PC ci 
2
i j
 PCci   0, i  j
I   PC ci
i


 0, i  j
C   ci PC ci
Cw 
2
i
 PC ci 
C 
2
i
c P
i C ci
i
|

  ci
i
 PC ci 
|
 cj
 PC c j 
|

 cj
P
C ci
i
|

 cj
For dichotomic variables:
If Cw  c j
then C  c j is an element of reality
C  c1PC c1  c2PC c2  c1PC c1  c2 (I-PC c1 )
Cw  c2
Cw 
C 
|

 c1PC c1  c2 (I-PC c1 ) 
 c2  (c1  c2 )
|
 PC c1 
|

Prob(C  c2 ) 
 PC c2 
 PC c1 
2
2
  PC c2 
2
1
 c2
 PCc1   0
Two useful theorems:
If C  c j is an element of reality then Cw  c j
For dichotomic variables:
If Cw  c j
then C  c j is an element of reality
The three box paradox
 
1
 A B C
3

t2
PA  1 
PA w  1
PB  1 
PB w  1
PA  PB  PC  1 
t
t1
 
A
1
A  B C
3
B
C



PA  PB  PC w  1
 PA w  PB w  PC w  1
 PC w  1
Tunneling particle has (weak) negative kinetic energy
Weak measurements performed on a pre- and post-selected ensemble
x y
Pointer probability distribution
Weak Measurement of   
strong
2
Hint  g (t )PMD
inMD (Q)  e
The particle post-selected
y 1
y
  ?
t
x
t1
 
 w
x 1
1.4
Q
2
2
2
x 1
y 1
The particle pre-selected
t2

!
weak
Robust weak measurement on a pre- and post-selected single system
The system of 20 particles
1 20
 i

20 i 1
Weak Measurement of
20 particles pre-selected
20 particles post-selected
t2
20

i 1
t
i y  1
y
strong
x 1
y 1
weak
i
1 20
 i

20 i 1
20

t1
Pointer probability distribution
i1
x i
i x  1
 1 20


1.4
 20  i 
i 1

w
!
Superposition of Gaussians shifted by small values yields
the Gaussian shifted by the large value
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
Generalized two-state vector
1
N

i
i i
i
j
j

j
1
N


j
j
j
t2
i
protection
C ?
t
t1

i
i i
i

Prob(C  c) 
2
i
 i PC c  i
i
i
  i i PC c  i
n
n
i
  C 

  
2
Cw
i
i
i
i
i
i
i
i
i i
PRL 58, 1385 (1987)
t2
protection
t i i i
 x  1,  y  1,  z  1
i
t1
 x  1,  y  1,  z  1
 x  1,  y  1,  z  1
 x  1,  y  1,  z   1
What is the past of
a quantum particle?
Wheeler:
The present choice of observation influences what we say about the
“past” of the photon; it is undefined and undefinable without the
observation.
No phenomenon is a phenomenon until it is an observed phenomenon.
The “past” and the “Delayed Choice” Double-Slit Experiment
J.A. Wheeler 1978
My lesson:
The “past” of the photon is defined after the observation
Wheeler delayed choice experiment
Wheeler: The photon took the upper path
It could not come the other way
Wheeler delayed choice experiment
Wheeler: The photon took both paths
Otherwise, the interference cannot be explained
Interaction-free measurement
Did photon touched the bomb?
Wheeler: The photon took the upper path
It 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 path
It could not come the other way
The trace shows Wheeler’s past of the photon
Wheeler delayed choice experiment
Wheeler: The photon took both paths
Otherwise, the interference cannot be explained
The trace shows Wheeler’s past of the photon
Interaction-free measurement
No
Yes
No
Did photon touched the bomb?
Operational meaning:
Nondemolition measurements show NO!
Wheeler delayed choice experiment
Yes
No
Yes
No
Operational meaning:
Nondemolition measurements show that the
photon took the upper path
Where is the photon when it is inside a Mach-Zehnder interferometer?
Yes
No
Yes
Operational meaning:
Nondemolition measurements show that the
photon took one of the paths
But nondemolition (strong) measurements disturb the photon
Where is the photon when it is inside a Mach-Zehnder interferometer?
Yes
or
No
or
Yes
or
No
Half a photon
or
Half a photon
Operational meaning: Weak measurements
No
(no disturbance at the limit)
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
Yes
Wheeler delayed choice experiment
Yes
No
Operational meaning: Weak measurements
Yes
(no disturbance at the limit)
The information is obtained from a pre- and post-selected ensemble
No
Interaction-free measurement
Yes
No
Did photon touched the bomb?
Operational meaning: Weak measurements
Yes
The information is obtained from a pre- and post-selected ensemble
No
Interaction-free measurement
No
Yes
No
Did photon touched the bomb?
Operational meaning: Strong measurements
Interaction-free measurement
No
Yes
Did photon touched the bomb?
Operational meaning: Weak measurements
(no disturbance at the limit)
The information is obtained from a pre- and post-selected ensemble
Wheeler delayed choice experiment
No
Yes
Operational meaning: Weak measurements
(no disturbance at the limit)
The information is obtained from a pre- and post-selected ensemble
Interaction-free measurement
No
Yes
Did photon touched the bomb?
Operational meaning: Weak measurements
(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.
Its validity is tested in a best way by weak
measurements using external system or
the photon itself.
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, Nature439 , 949 (2006)
Yes!
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
No
Yes
No
Weak measurements: the photon was there!
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
The two-state vector formalism expalnation
The pre- and post-selected particle is described
by the two-state vector
t
P  1
t2

t
t1

C ?


C 
Cw 

P  1
The outcomes of weak measurements are weak values
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
The two-state vector formalism explanation
C
B
A
 
1
A  B  C

3

The two-state vector formalism explanation
C
B
A
 
1
3

A  B  C

The two-state vector formalism explanation
C
B
A
 PB 
(PB ) w 
1

Yes
The two-state vector formalism explanation
C
B
A
 PA 
(PA )w 
1

Yes
The two-state vector formalism explanation
C
B
A
 PC 
(PC ) w 
 1

?
Interaction-free measurement
Interaction-free measurement
Interaction-free measurement
In IFM the photon was not near the bomb