Effective Non-Hermitian Hamiltonian of a pre- and post-selected quantum system

Lev Vaidman

12.7.2015

Conditioned evolution

Weak values and weak measurements

Evolution of pre- and post-selected system

Plan

Two state-vector

Past of a quantum particle

3-box paradox

Correlations of uncorrelated pre- and post-selected particles

( )tUnitary evolution

no click

Non-unitary evolution

no click

Non-unitary evolution

no click

Non-unitary evolution

Collapse of the wave function

What is the evolution conditioned on nondetection?

no click

no click( ) ?t

What is the evolution conditioned on nondetection?

What was time evolution before the particle was detected, given that it was detected?

What is the evolution conditioned on detection?

What is the evolution conditioned on detection?

What was the interaction Hamiltonian for (weak) interaction with other systems?

What is the evolution conditioned on detection?

What is the evolution conditioned on detection?

What was the interaction Hamiltonian for (weak) interaction with other systems?

What was the interaction Hamiltonian for (weak) interaction with other systems?

Where were the pre- and post-selected photons?

B

A

Asking photons where have they been

POWER SPECTRUM

fB0

280 290 300 310 320 330

1

0.8

0.6

0.4

0.2

[ ]f Hz

UP DOWNSIGNAL I I

A. Danan, D. Farfurnik, S. Bar-Ad and L. Vaidman, Phys. Rev. Lett. 111, 240402 (2013)

POWER SPECTRUM

fA fB0

280 290 300 310 320 330

1

0.8

0.6

0.4

0.2

[ ]f Hz

B

A

Asking photons where have they been

[ ]f Hz

UP DOWNSIGNAL I I Photons were on the paths they could pass

B

A

Asking photons where have they been

POWER SPECTRUM

fA fB0

280 290 300 310 320 330

1

0.8

0.6

0.4

0.2

[ ]f Hz

UP DOWNSIGNAL I I Photons were on the paths they could pass

B

A

Asking photons where have they been

POWER SPECTRUM

fA fB0

280 290 300 310 320 330

1

0.8

0.6

0.4

0.2

[ ]f Hz

UP DOWNSIGNAL I I Photons were on the paths they could pass

C

F

E

POWER SPECTRUM

fA fB fC fE fF0

280 290 300 310 320 330

1

0.8

0.6

0.4

0.2

[ ]f HzB

A

Asking photons where have they been

Photons were on the paths they could pass

Asking photons where have they been

B

C

A

F

E

POWER SPECTRUM

fA fB fC fE fF0

280 290 300 310 320 330

1

0.8

0.6

0.4

0.2

[ ]f Hz

Photons were on the paths they could not pass!

How to explain this?

The two-state vector formalism

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

t

t

t

t

t

t

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:

Strong measurements performed on a pre- and post-selected system

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

The weak value

t

2t

1t

3t

w

CC

If the pre- and post-selected system is coupled to other systems through C, then its coupling at time t is described (completely) by the weak value wC

intˆ ˆH gCB

ˆ ˆCig Bdte ˆ ˆ1 Cig Bdt 1C

ig Bdt

ˆ

wigC Bdte

intˆ

wH g BC intˆ ˆH gCB

ˆwigC Bdte

Effective non-Hermitian Hamiltonian

y z x

n z wy x

i

1tx

1x

1y y

2t

1

N

nn

gH

N

1y x x

n x wy x

1y y x

n y wy x

1

N

eff nn w

gH

N

x y zg i

eff x y zH g i

Y. Aharonov, S. Massar, S. Popescu, J. Tollaksen, and L. Vaidman, PRL 77, 983-987 (1996)

Asking photons where have they been

B

C

A

F

E

POWER SPECTRUM

fA fB fC fE fF0

280 290 300 310 320 330

1

0.8

0.6

0.4

0.2

[ ]f Hz

Photons were on the paths they could not pass!

How to explain this?

B

A

The two-state vector formalism explanation

B

A

The two-state vector formalism explanation

B

APOWER SPECTRUM

fA fB0

280 290 300 310 320 330

1

0.8

0.6

0.4

0.2

[ ]f Hz

The two-state vector formalism explanation

B

C

A

F

E

D

The two-state vector formalism explanation

B

C

A

F

E

D

The two-state vector formalism explanation

B

C

A

F

E

D

The two-state vector formalism explanation

B

C

A

F

E

D

The two-state vector formalism explanation

POWER SPECTRUM

fA fB fC fE fF0

280 290 300 310 320 330

1

0.8

0.6

0.4

0.2

[ ]f Hz

w

CC

) 0A w (P ) 0B w (P ) 0C w (P

) 0E w (P ) 0F w (P

B

C

A

F

E

D

The two-state vector formalism explanation

POWER SPECTRUM

fA fB fC fE fF0

280 290 300 310 320 330

1

0.8

0.6

0.4

0.2

[ ]f Hz

w

CC

) 0A w (P ) 0B w (P ) 0C w (P

) 0E w (P ) 0F w (P

B

C

A

F

E

D

The two-state vector formalism explanation

1

3A B C

B

C

A

F

E

D

The two-state vector formalism explanation

1

3A B C

B

C

A

F

E

D

The two-state vector formalism explanation

1

3A B C

1

3A B C

) 1AA w

P

(P

) 1BB w

P

(P

) 1CC w

P

(P

POWER SPECTRUM

fA fB fC fE fF0

280 290 300 310 320 330

1

0.8

0.6

0.4

0.2

[ ]f Hz

K.J. Resch, , J.S. Lundeen, , A.M. Steinberg , PLA 324, 125 (2004)

Experimental realization of the quantum box problem

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 in

A B C

Aalways !

The three box paradox

t

2t

1t

3t

It is always in

B

1

3A B C

1

3A B C

A B C

Two useful theorems:

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

Prob( ) 1i w iC c C c For dichotomic variables:

Prob( ) 1w i iC cC c

Correlation between separable pre- and post-selected particles

( , )corr A B AB A B

t

1t 1x

1y 2t

1x

1y

Aharonov and Cohen, arXiv:1504.03797

A B

A z wi B z w

i

( , )A z B z A z B z A z B zcorr 0A z B z

?A z B z

A z B z A z B zw w w i i 1 1A z B z

( , ) 1A z B zcorr

A z B z A z B z

Failure of the product rule for pre- and post-selected particles

t

1t

1x 2t 1y A B

1A y

1

2

1B x 1A y B x

,A a B b AB ab

Pre- and post-selected quantum systems are described best by two-state vector and weak values of observables

w

CC

Evolution of systems coupled to pre- and post-selected quantum systems is described by non-Hermitian Hamiltonians

Conclusions

B

C

A

F

E

D

The one-state vector formalism explanation

C

F

E

POWER SPECTRUM

fA fB fC fE fF0

280 290 300 310 320 330

1

0.8

0.6

0.4

0.2

[ ]f HzB

A

Photons:Wheeler is right!

jj C cj

w i

c

C c

P

2

2Prob( ) 1 , 0i

j

j

C c

i C c

C cj

C c j i

PP

P

ii C ci

C c P

Prob( ) 1i w iC c C c

For dichotomic variables Prob( ) 1w i iC cC c

1 21 21

C c C cw

c cC c

P P2 1

I - C c C c P P

1 1

1 1 2 1C c C cc c c

P P1 1

1 21 1C c C cc c

P P1 1C c

P

w

CC

2

2Prob( )

i

C c

C ci

C c

P

P

Connection between strong and weak measurements

-i jC c C c

j i

P I P I

Pointer probability distribution

? t

1tx

1x

1y y

2t

1.4w !

strong

weak

Weak measurement of

Pre-selection 1x

2x y

int ( ) MDH g t P 2

22( )Q

MDin Q e

Post-selection 1y

The outcomes of weak measurements are weak values

Pointer probability distribution

Weak Measurement of

t

1t

20

1x i

i

1i x

20 particles pre-selected 1x 20 particles post-selected 1y

1i y 20

1i

iy

20

1

1

20 ii

20

1

1

20 ii

Robust weak measurement on a pre- and post-selected single system

The system of 20 particles

20

1

11.4

20 ii w

!

strong

weak

2t