חוק המכפלה

A B AB

,A a B b AB ab

,A a B b AB ab

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

t

2t

1t

3t

1

3A A B

1

3A A B

A B

1A P 1

A P

0A A P P

פרדוקס של חרדי

1eY

P

1eY

P

0e eY Y

P P

A B 'B 'A

C D 'D 'CO 'O

E F 'F 'E

1D P

' 1D P

' 0D D P P

Weak Measurements

Q0 1c 2c 3c

Quantum measurement of C

t

1t

2t

0inQ

fin iQ cint ( ) MDH g t P C

( )MD Q

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