חוק המכפלה. It is always in ! The 3-boxes paradox.
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Transcript of חוק המכפלה. It is always in ! The 3-boxes paradox.
t
2t
1t
3t
It is always in
B
1
3A B C
1
3A B C
A B C
It is always in A !
The 3-boxes paradox
1A P
1B P
0A B P P
?zA ?xB
Peculiar example: a failure of the product rule
1zA
1xB
1xBzA
2
( 1)
2 2
( 1) ( 1)
PProb( 1) 1
P P
zA xB
zA xB zA xB
x z
x z x z
xBzA
1z
x
1
2
z
1x
A B
t
1t
2t
1
2x zA B A B A B
Q0 1c 2c 3c
Weak quantum measurement of C
t
1t
2t
0inQ
int ( ) MDH g t P C
( )MD Qint
0, small
is smallMD MDP P
H
Q0 1c 2c 3c
Weak quantum measurement of C
t
1t
2tint ( ) MDH g t P C
( )MD Qint
0, small
is smallMD MDP P
H
0Q
Q0 1c 2c 3c
CfinQ C int ( ) MDH g t P C
( )MD Q
C
Weak quantum measurement of
t
1t
2t
int
0, small
is smallMD MDP P
H
0Q
t
P 1
1t
2t
P 1
?C
The outcomes of weak measurements are weak values
Weak value of a variable C of a pre- and post-selected systemdescribed at time t by the two-state vector
w
CC
Q0 1c 2c 3c
Weak measurement of with post-selectionC
t
1t
2t
( )MD Q0Q
int ( ) MDH g t P C
int
0, small
is smallMD MDP P
H
Q0 1c 2c 3c
fin wQ Cint ( ) MDH g t P C
( )MD Q
wC
t
1t
2t
int
0, small
is smallMD MDP P
H
0Q
Weak measurement of with post-selectionC
P 1
P 1
t
P 1
1t
2t
P 1
?C
The outcomes of weak measurements are weak values
Weak value of a variable C of a pre- and post-selected systemdescribed at time t by the two-state vector
w
CC
w wwA B A B
w wwAB A B
Weak value of a variable C of a pre- and post-selected systemdescribed at time t by the two-state vector
The outcomes of weak measurements are weak values
2 2
x yy x
y x
wy x y x
t
1tx
?
1x
1y y
2t
w
CC
2x y
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
How the result of a measurement of a component ofthe spin of a spin-1/2 particle can turn out to be 100Y. Aharonov, D. Albert, and L. Vaidman PRL 60, 1351 (1988)
?z t
1t
1x
1 x
2t
tan2
x z
z wx
Realization of a measurement of a ``weak value''N. W. M. Ritchie, J. G. Story, and R. G. HuletPhys. Rev. Lett. 66, 1107-1110 (1991)
weak
int ( ) zH g t z strongint ( ) xH g t x
x
z
The outcomes of weak measurements are weak values
Observation of the Spin Hall Effect of Light via Weak Measurements
Science 8 February 2008: Amplifying a Tiny Optical EffectK. J. Resch
O. Hosten and P. Kwiat
“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.”
Weak-measurement elements of reality
If we can infer that the quantum wave of the pointer of the measuring devicewhich measures C will, in the limit of weak interaction, be shifted without distortionby the value c, then there is a weak element o reality .wC c
Q0 1c 2c 3c
( )MD Q
Weak-measurement elements of reality
If we can infer that the quantum wave of the pointer of the measuring devicewhich measures C will, in the limit of weak interaction, be shifted without distortionby the value c, then there is a weak element o reality .wC c
Q0 1c 2c 3c
( )MD Q
Weak-measurement elements of reality
If we can infer that the quantum wave of the pointer of the measuring devicewhich measures C will, in the limit of weak interaction, be shifted without distortionby the value c, then there is a weak element o reality .wC c
Q0 1c 2c 3c
( )MD Q
wC
Weak-measurement elements of reality
If we can infer that the quantum wave of the pointer of the measuring devicewhich measures C will, in the limit of weak interaction, be shifted without distortionby the value c, then there is a weak element o reality .wC c
c
Two useful theorems:
If is an element of reality then iC cw iC c
If then is an element of realityw iC c iC 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