t

P 1

1t

2t

P 1

The two-state vector

t

t

P 1

1t

2t

P 1

The two-state vector

t

t

1t P 1

t

2t P 1

?

t

1t

z t

2t

z?

At time t:

1z

1z

Prob( ) 1z Prob( ) 1z z

2

Prob( ) cos2

Prob( ) ?

t

1t

z t

2t

z?

At time t:

1z

1z

Prob( ) 1z Prob( ) 1z z

2

Prob( ) cos2

Prob( ) 0

1

1t

t

1t

z t

2t

z

At time t:

1z

1z

Prob( ) 1z Prob( ) 1z z

2

Prob( ) cos2

Prob( ) ?

known1t

unknown

t

1t

z t

2t

z

At time t:

1z

1z

Prob( ) 1z Prob( ) 1z z

2

Prob( ) cos2

unknown1t

unknown

2

Prob( ) cos2

t

1t

z t

2t

z

1z

1z

1t

unknown

Erasing the past

1

t

1t

z t

2t

z

1z

1z

1t

unknown

Erasing the past

1

Alternative past for the same present

1

2

Hint:

t

1t

z t

2t

z

1z

1z

1t

unknown

1

2

protection

unknown

unknown

Erasing the pastprotection

t

1t P 1

t

2t P 1

unknown

unknown

A=? A=?t

1t P 1

t

2t P 1

Are there any differences between what can be done to and ?

Nondemolition (von Neumann) measurements

?

U Ut

1t P 1

t

2t P 1

Unitary transformation

Are there any differences between what can be done to and ?

?

The two-state vector is a complete description of a system at time t

t

2t P 1

1t P 1

?

3tThe two-state vector is what we can say now ( )about the pre- and post-selected system at time t

3t

The two-state vector describes a single pre- and post-selected system, but to test predictions of the two-state vector we need a pre- and post-selected ensemble

t

2tP 1

1tP 1

3t

P 1

P 1

??

U U U

P 1 P 1 P 1

P 0

P 0

?

P 0

P 0

?

P 0

P 1

?

P 1

P 0

?

The pre- and post-selected ensemble

t

2tP 1

1tP 1

3t

P 1

P 1

??

U U U

P 1 P 1 P 1

P 0

P 0

?

P 0

P 0

?

P 0

P 1

?

P 1

P 0

?

The pre- and post-selected ensemble

A single pre- and post-selected system

t

2t P 1

1t P 1

?

3t

A single pre- and post-selected system

t

2t P ?

1t P 1

?

3tThe two-state vector in the framework of the many-worldsinterpretation of quantum mechanics

A single pre- and post-selected system

t

2t P 0

1t P 1

?

3tThe two-state vector in the framework of the many-worldsinterpretation of quantum mechanics

P 0 The other world

A single pre- and post-selected system

t

2t P 1

1t P 1

?

3tThe two-state vector in the framework of the many-worldsinterpretation of quantum mechanics

P 1 This world

The two-state vector is a complete description of a system at time t

t

2t P 1

1t P 1

?

3tThe two-state vector is what we can say now ( )about the pre- and post-selected system at time t

3t

So, what can we say?

t

P 1

1t

2t

P 1

The Aharonov-Bergmann-Lebowitz (ABL) formula:

?C

2

2

PProb( )

Pi

C c

C ci

C c

described by the two-state vector:

Measurements performed on a pre- and post-selected system

The Aharonov-Bergmann-Lebowitz (ABL) formula:

t

P 1

1t

2t

P 1

?C

2

2

PProb( )

Pi

C c

C ci

C c

P ( , ) P ( ) P ( | ) P ( ) P ( | )c c cc P ( ) P ( | )

P ( | )P ( )

c cc

P ( ) P ( | )Prob( ) P ( | )

P ( )

c cC c c

2

P ( ) PC cc 2

1P ( | ) P

P C cC c

c

2P ( ) P ( | ) PC cc c

P ( ) P ( ) P ( | )i ii

c c 2

PiC c

i

2

2

P

Pi

C c

C ci

Measurements performed on a pre- and post-selected system

described by the two-state vector:

The Aharonov-Bergmann-Lebowitz (ABL) formula:

2

2

PProb( )

Pi

C c

C ci

C c

t

1t

2t

?z

1x

1z z

x

2

2 2

PProb( ) 1

P P

z

z z

x z

z

x z x z

Measurements performed on a pre- and post-selected system

described by the two-state vector:

The Aharonov-Bergmann-Lebowitz (ABL) formula:

2

2

PProb( )

Pi

C c

C ci

C c

t

1t

2t

?x

1x

1z z

x

2

2 2

PProb( ) 1

P P

z

z z

x z

z

x z x z

2

2 2

PProb( ) 1

P P

x

x x

x z

x

x z x z

1x 1z

At time t:

Measurements performed on a pre- and post-selected system

described by the two-state vector:

The Aharonov-Bergmann-Lebowitz (ABL) formula:

2

2Prob( )

i

C c

C ci

C c

P

Pt

1t

2t

?x

1x

1z z

x

2

2 2Prob( ) 1z

z z

x z

z

x z x z

P

P P

2

2 2

PProb( ) 1

P P

x

x x

x z

x

x z x z

1x 1z

Can we arrange at time t:

1y ?

3 CUPS GAME

NEW RULES

You win when you do not find the ball

You can look only under one of the two cups

The dealer does not see your action, but he can look at the ball later and cancel a particular run of the game

Quantum dealer can win without cheating!

The ball is a pre- and post-selected system

t

2t P 1

1t P 1

?

The two-state vector

t

2t

1t

3t

Where is the ball?

?

1

3A B C

1

3A B C

A B C

Aharonov and Vaidman, JPA 24, 2315 (1991) 

Aharon and Vaidman, PRA 77, 052310 (2008)

The 3-boxes paradox Vaidman, Found. Phys.  29, 865 (1999)

The three box paradox

t

2t

1

3A B C

1t 1

3A B C

3t

It is

A B C

Aalways in !

2

2 2Prob( 1) 1A

A

A B C

A B C A B C

A B C A B C A B C A B C

PP

P P

The three box paradox

t

2t

1t

3t

It is always in

B

1

3A B C

1

3A B C

A B C

2

2 2Prob( 1) 1B

B

B A C

A B C A B C

A B C A B C A B C A B C

PP

P P

A single photon “sees” two balls

t

2t

1t

It scatters exactlyas if there weretwo balls

1

3A B C

1

3A B C

A B C

Y. Aharonov and L. Vaidman Phys. Rev. A 67, 042107 (2003) 

חורים עם פקק אחדNאיך לסגור

0

The outcome is 0. The computer was not running!

Simple Counterfactual Computation with Outcome 0.

Kwiat has to be right!

0

The computer was running!

Kwiat cannot be right!

A

B

C

Kwiat’s scheme = 3-boxes paradox

A

B

C

1

3A B C

Kwiat’s scheme = 3-boxes paradox

A

B

C

0

1

3A B C

Kwiat’s scheme = 3-boxes paradox

0

A

B

C

1

3A B C 1

3A B C

Kwiat cannot be right!

There is no difference between A and B:counterfactually it is in A as it is in Bthe same quantum descriptions in A and in B

Kwiat’s scheme = 3-boxes paradox