Download Weak-measurement elements of reality

‫חוק המכפלה‬
A
A  a,
Bb
A  a,
B  AB

Bb
AB  ab

AB  ab
The 3-boxes paradox
t3
 
1
3

A  B  C

t2

It is always in A !
It is always in B
t
PA  1

t1
 
A
1
3

A  B  C
B
C

PB  1
PA PB  0
Peculiar example: a failure of the product rule
x
A
z
B
x 1
z
 zA  ?
t
1
2
t1

   

B

A

B

 zA  1
 xB  1
 xB  ?
 zA  xB  1

B
A
Prob( zA  xB  1) 
A
z 1
t2
x

1

2

 x  z P( zA  xB 1)     

 x  z P( zA  xB 1)     

2

2

  x  z P( zA  xB 1)     

2
1
t3
 
1
3

A  A  B 

t2

PA  1
t
PA PA  0

t1
B
A

PA  1

1
3

A  A  B

‫פרדוקס של חרדי‬
e
Y
P
e
Y
P
e
Y
e
Y
P P
1
1
0
O
E
F
C
D
D'
C'
A
B
B'
A'
F'
E'
O'
PD  1
PD '  1
PD PD '  0
Weak Measurements
Quantum measurement of
t2
Q fin  ci
C
H int  g (t ) PMDC
 MD (Q )
t
t1
Qin  0
0
c1
c2 c3
Q
Weak quantum measurement of
t2
H int  g (t ) PMDC
PMD  0, PMD small
t
t1
C
Qin  0
 MD (Q )
0
 H int is small
c1
c2 c3
Q
Weak quantum measurement of
t2
H int  g (t ) PMDC
PMD  0, PMD small
t
t1
C
Q 0
 MD (Q )
0
 H int is small
c1
c2 c3
Q
Weak quantum measurement of
t2
H int  g (t ) PMDC
Q fin  C
PMD  0, PMD small
t
t1
C
Q 0
 MD (Q )
0
 H int is small
c1
C c2 c3
Q
The outcomes of weak measurements are weak values
Weak value of a variable C of a pre- and post-selected system
described at time t by the two-state vector  
t2

P  1
C?
t

t1
P  1
C 
Cw 

Weak measurement of
t2
with post-selection
H int  g (t ) PMDC
PMD  0, PMD small
t
t1
C
Q 0
 MD (Q )
0
 H int is small
c1
c2 c3
Q
Weak measurement of
Q fin  Cw
t2
Q 0
t1
with post-selection
H int  g (t ) PMDC
P  1
t
C
PMD  0, PMD small
 MD (Q )
 H int is small
P  1
0
Cw
c1
c2 c3
Q
The outcomes of weak measurements are weak values
Weak value of a variable C of a pre- and post-selected system
described at time t by the two-state vector  
t2

P  1
C?
t
C 
Cw 


t1
P  1
 A  Bw  Aw  Bw
 AB w  Aw Bw
The outcomes of weak measurements are weak values
Weak value of a variable C of a pre- and post-selected system
described at time t by the two-state vector  
t2
y
y 1
  ?
t
x
t1
C 
Cw 

x 1
 
 
 w

x y
 y  x
 y x
2

y
x y
2
 y x
x
 2
Weak measurements performed on a pre- and post-selected ensemble
x y
Pointer probability distribution
Weak Measurement of   
strong
2
H int  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
The outcomes of weak measurements are weak values

z

x
  1
t2
x
z  ?
weak
H int  g (t ) z z
t

t1
 z  w 
How the result of a measurement of a component of
the spin of a spin-1/2 particle can turn out to be 100
Y. Aharonov, D. Albert, and L. Vaidman PRL 60, 1351 (1988)
 x  z 
 x 
strong
H int  g (t ) x x
x 1
 tan

2
Realization of a measurement of a ``weak value''
N. W. M. Ritchie, J. G. Story, and R. G. Hulet
Phys. Rev. Lett. 66, 1107-1110 (1991)
Science 8 February 2008:
Amplifying a Tiny Optical Effect
K. J. Resch
“In the first work on weak measurement (AAV), it was speculated that the
technique could be useful in amplifying and measuring small effects. Now, 20
years later, this potential has finally been realized.”
Observation of the Spin Hall Effect of Light via Weak Measurements
O. Hosten and P. Kwiat
Weak-measurement elements of reality
If we can infer that the quantum wave of the pointer of the measuring device
which measures C will, in the limit of weak interaction, be shifted without distortion
by the value c, then there is a weak element o reality C  c.
w
Weak-measurement elements of reality
If we can infer that the quantum wave of the pointer of the measuring device
which measures C will, in the limit of weak interaction, be shifted without distortion
by the value c, then there is a weak element o reality C  c.
w
 MD (Q )
0
c1
c2 c3
Q
Weak-measurement elements of reality
If we can infer that the quantum wave of the pointer of the measuring device
which measures C will, in the limit of weak interaction, be shifted without distortion
by the value c, then there is a weak element o reality C  c.
w
 MD (Q )
0
c1
c2 c3
Q
Weak-measurement elements of reality
If we can infer that the quantum wave of the pointer of the measuring device
which measures C will, in the limit of weak interaction, be shifted without distortion
by the value c, then there is a weak element o reality C  c.
w
 MD (Q )
c
0
Cw
c1
c2 c3
Q
Two useful theorems:
If C  ci is an element of reality then Cw  ci
For dichotomic variables:
If Cw  ci
then C  ci 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