If is an element of reality then If then is an element of reality For dichotomic variables:

t

P 1

1t

2t

P 1

?C

Prob( ) 1jC c

2P 0

iC ci j

If is an element of reality then jC cw jC c

2P 0,

iC c i j P 0,iC c i j

|w

CC

Pii C c

i

C c P

|

ii C ci

c

P

|iC c

ii

c

P

|jC c

jc

I PiC c

i

P

|

iC ci

jc

jc

2 2

2 2 2

P P

P P P

j j

i j i

C c C c

C c C c C ci i j

If then is an element of realityw jC c jC cFor dichotomic variables:

2wC c1 21 2P PC c C cC c c

1 11 2P (I-P )C c C cc c

|w

CC

1 11 2P (I-P )

|C c C cc c

1

2 1 2

P( )

|C cc c c

2c

1P 0C c

2

1 2

2

2 2 2

PProb( )

P P

C c

C c C c

C c

1

Two useful theorems:

If is an element of reality then jC cw jC c

If then is an element of realityw jC c jC 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

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

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

Properties of a quantum system during the time interval between two measurements Y. Aharonov and L. Vaidman PRA 41, 11 (1990)

Another example: superposition of positive shifts yields negative shift

A. Botero

Superposition of Gaussians shifted by small values yields the Gaussian shifted by the large value

http://prola.aps.org/abstract/PRA/v41/i1/p11_1?qid=1d712839cdbd26f9&qseq=18&show=10

http://prola.aps.org/abstract/PRA/v41/i1/p11_1?qid=1d712839cdbd26f9&qseq=18&show=10

Generalized two-state vector

t

1t

2t

?C

1j i iN

j i

j i

i i ii

protection

2

2

P

Prob( )

Pn

i i C c ii

i i C c in i

C c

i ii

i

1jN

j

j

i i ii

wi i i

i

CC

t

1t

2t

1, 1, 1x y z i i ii

protection

PRL 58, 1385 (1987)

1, 1, 1

1, 1, 1

1, 1, 1

x y z

x y z

x y z

What is the past of a quantum particle?

The “past” and the “Delayed Choice” Double-Slit Experiment J.A. Wheeler 1978

The present choice of observation influences what we say about the “past” of the photon; it is undefined and undefinable without the observation.

The “past” of the photon is defined after the observation

Wheeler:

No phenomenon is a phenomenon until it is an observed phenomenon.

My lesson:

Wheeler delayed choice experiment

Wheeler: The photon took the upper pathIt could not come the other way


Wheeler: The photon took both pathsOtherwise, the interference cannot be explained

Interaction-free measurement

Did photon touched the bomb?Wheeler: The photon took the upper pathIt could not come the other way

The past of a quantum particle can be learned by measuring the trace it left


Wheeler: The photon took the upper pathIt could not come the other way

The trace shows Wheeler’s past of the photon


Wheeler: The photon took both pathsOtherwise, the interference cannot be explained

The trace shows Wheeler’s past of the photon


Did photon touched the bomb?

Operational meaning: Nondemolition measurements show NO!

Yes

No

No


Nondemolition measurements show that the photon took the upper path

Operational meaning:

Yes

No

No

Yes

Nondemolition measurements show that the photon took one of the paths


Yes

No

Yes

Where is the photon when it is inside a Mach-Zehnder interferometer?

But nondemolition (strong) measurements disturb the photon

Weak measurementsOperational meaning:

Where is the photon when it is inside a Mach-Zehnder interferometer?

The information is obtained from weak measurements on an ensemble of identically prepared photons

“Half a photon” or half the times the photon passes each path

(no disturbance at the limit)

YesNo

Yes or No

or Half a photon

Yes or No

or Half a photon


Weak measurementsOperational meaning:(no disturbance at the limit)

Yes

No

Yes No

The information is obtained from a pre- and post-selected ensemble



Weak measurementsOperational meaning:The information is obtained from a pre- and post-selected ensemble

Yes No

Yes

No


Yes

No

No

Strong measurementsDid photon touched the bomb?





No

Yes




Weak measurementsOperational meaning:(no disturbance at the limit)


Yes

No




No

Yes



The best measuring device for pre-and post-selected photon is the photon itself

Strong measurements

Yes


Strong measurements

No


Weak measurements

Yes


Weak measurements

No

Wheeler’s argument: “The photon took the upper path because

it could not come the other way”seems to be sound.

The presence of the bomb can be found without anything passing near the bomb

Can we find that the bomb or anything else is not present in a particular place without anything passing near this place?

Hosten,…Kwiat, Nature 439, 949 (2006) Yes!

Its validity is tested in a best way by weak measurements using external system or the photon itself.

Kwiat’s proposal

Kwiat’s proposal

Kwiat’s proposal

Kwiat’s proposal

Kwiat’s proposal

Wheeler: We know that the bomb is not there and the photon was not there since it could not come this way.

Weak measurements: the photon was there!

Kwiat’s proposal


Yes

No

No

But it was not on the path which leads towards it!

Kwiat’s proposal


But it was not on the path which leads towards it! Yes

Kwiat’s proposal


But it was not on the path which leads towards it! No

Kwiat’s proposal


But it was not on the path which leads towards it! No

Kwiat’s proposal

Weak measurements by environment

Kwiat’s proposal

Weak measurements by environment

Kwiat’s proposal


But also in another place

Kwiat’s proposal


But also in another place. The effects are equal! Yes

Kwiat’s proposal


But also in another place. The effects are equal! Yes

t

P 1

1t

2t

P 1

The pre- and post-selected particle is described by the two-state vector

w

CC

The outcomes of weak measurements are weak values

?C

t

The two-state vector formalism expalnation



Where Is the Quantum Particle between Two Measurements?




1

3A B C

A

B

C

The two-state vector formalism explanation

1

3A B C

A

B

C


) 1BB w

P

(P

A

B

C


Yes

) 1AA w

P

(P

A

B

C


Yes

) 1CC w

P

(P

A

B

C


?




In IFM the photon was not near the bomb

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If is an element of reality then If then is an element of reality For dichotomic variables: