Honoring Epimenides of Crete \(±Δx\): From Quantum Paradoxes, through Weak Measurements, to the Nature of Time

1

(

)

2

i Z

Z

\

J

 

The boxes are transparent for the photon but opaque for the atom. Now let the atom’s Z

−

box

be positioned across the photon’s v path in such a way that the photon can pass through the 

box and interact with the atom inside in a 100% efficiency. 

Next let the photon be transmitted by BS

1

: 

1

(

)(

)

2

i u

v

i Z

Z

\  

Discarding all these cases of the photon’s absorption by the atom (25% of the experiments): 

1

(

)

2

u Z

i u Z

i v Z

\

 

Next, reunite the photon by BS

2

:

2

1

(

)

2

BS

v

d

i c

o

2

1

(

)

2

BS

u

d

i c

o

So that:

3

[

(

2

)

]

2

i

c i Z

Z

d Z

\

 

After the photon reaches one of the detectors, the atom’s Z boxes are joined and a reverse 

magnetic field –B is applied to bring it to its final state. Measuring this state's spin yields: 

1

1

(

)

( 3

)

4

4

d

i X

X

c

X

i X

\

 

In 1/16 of the cases, the photon hits detector D, while the atom is found in a final spin state 

of X

-

 rather than its initial state X

+

. In every such a case, both particles performed IFM on 

one another; they both destroyed each other’s interference. Nevertheless, the photon has not 

been absorbed by the atom, so no interaction seems to have taken place. 

Hardy’s analysis stressed a striking aspect of this result: The atom can be regarded as EV’s 

“bomb” as long as it is in superposition, and its interaction with the photon can end up with 

one out of three consequences: 

• The atom absorbs the photon – this is analogous to the explosion in EV’s original device.

• The atom remains superposed – this is analogous to the no-explosion outcome. 

• The atom does not absorb the photon but loses its superposition – a third possibility that 

does not exist with the classical bomb and amounts to a delicate form of explosion. 

Hence, when the last case occurs, it appears that the photon has traversed the u arm of the 

MZI, while still affecting the atom on the other arm by forcing it to assume (as measurement 

will indeed reveal) a Z

+

spin! 

(1)

(2)

(3)

(4)

(5)

(6)

ICNFP 2013

00028-p.5

---

2.5 EPR Effects between Particles that never interacted in the past 

Another elegant experiment by Hardy [21] brings together nearly all the famous quantum 

experiments, such as the double-slit, the delayed choice, EPR and IFM – all in one simple 

setup (Fig. 2).

Fig. 2. Entangling two atoms that never interact.

Let again a single photon traverse a MZI. Now let two Hardy atoms be prepared as in Sec. 

2.3, each atom superposed in two boxes that are transparent for the photon but opaque for the 

atom. Then let the two atoms be positioned on the MZI’s two arms such that atom 1’s Z

+

 box 

lies across the photon’s v path and 2’s Z

−2

box is positioned across the photon’s u path. On 

both arms, the photon can pass through the box and interact with the atom inside in 100% 

efficiency. Now let the photon be transmitted by BS

1

: 

1

1

2

2

3

1

(

)(

)(

)

2

i u

v

i Z

Z

i Z

Z

\

 

Once the photon was allowed to interact with the atoms, we discard the cases in which 

absorption occurred (50%), to get: 

1

2

1

2

1

2

1

2

3

1

(

)

2

i u Z

Z

u Z

Z

i v Z

Z

v Z

Z

\

 

Now, let photon parts u and v pass through BS

2

: 

2

1

(

)

2

BS

v

d

i c

o

2

1

(

)

2

BS

u

c

i d

o

(7)

(8)

(9)

EPJ Web of Conferences

00028-p.6

---

giving 

1

2

1

2

1

2

1

2

1

(

2

)

4

d Z

Z

d Z

Z

i c Z

Z

c Z

Z

\

 

If we If we now post-select only the experiments in which the photon was surely disrupted 

on one of its two paths, thereby hitting detector D, we get: 

1

2

1

2

1

(

)

4

d

Z

Z

Z

Z

\

 

Consequently, the atoms, which never met in the past, become entangled in an EPR-like 

relation. In other words, they would violate Bell’s inequality [22]. Unlike the ordinary EPR, 

where the two particles have interacted earlier or emerged from the same source, here the 

only common event in the two atoms’ past is the single photon that has “visited” both of 

them. 

2.6 EPR Upside-down 

Hardy’s abovementioned experiment [21] inspired Elitzur and Dolev propose a simpler 

version [23] that constitutes an inverse EPR. Let two coherent photon beams be emitted from 

two distant sources as in Fig. 3. Let the sources be of sufficiently low intensity such that, on 

average, one photon is emitted during a given time interval. Let the beams be directed 

towards an equidistant BS. Again, two detectors are positioned next to the BS: 

1

0

u

u

u

p

q

J

I

 

1

0

v

v

v

p

q

J

I  

1

1

1

1

(

)

2

A

i Z

Z

\

 

2

2

2

1

(

)

2

A

i Z

Z

\

 

where 

1

denotes a photon state (with probability p

2

),

0

denotes a state of no photon (with 

probability q

2

), p

≪

 1, and p

2

+ q

2

= 1. 

Since the two sources’ radiation is of equal wavelength, a static interference pattern will be 

manifested by different detection probabilities in each detector. Adjusting the lengths of the 

photons’ paths v and u will modify these probabilities, allowing a state where one detector, 

D, is always silent due to destructive interference, while all the clicks occur at the other 

detector, C, due to constructive interference. Notice that each single photon obeys these 

detection probabilities only if both paths u and v, coming from the two distant sources, are 

open. We shall also presume that the time during which the two sources remain coherent is 

long enough compared to the experiment’s duration, hence we can assume the above phase 

relation to be fixed. 

Next, let two Hardy atoms be placed on the two possible paths such that atom 1’s Z

+ 1 

box

lies across the photon’s u path and 2’s Z

−2

box is positioned across the photon’s v path. After 

the photon was allowed to interact with the atoms, we discard the cases in which absorption 

occurred (50%), getting 

(10)

(11)

(12)

ICNFP 2013

00028-p.7

---