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

can refer to state

<

 in the spatial representation (which serves as our measuring base) 

described by a Gaussian function   

2

2

( )

exp(

/ 2

)

x

x

V

<

 

 

The pointer movement in that case is connected to the weak value of the operator A defined 

by:  

w

A

A

M \

M \

 

 

where

\

 is the initial (preparation) state of the measured system, and 

M

 the final state 

into which it is projected. 

For a pre-/post-selected ensemble described by the two-state

M \

, the time evolution of 

the total system (measured plus measuring system) is expected to be (

1

 1

 

) [25]:  

int

exp(

)

(1

)

exp(

)

W

d

W

d

i H dt

i A P

i A P

M

\

M \

H

M \

H

< |

<  

<

³

 

which results in 

exp(

) ( )

(

)

w

w

i A p

x

x

A

H

H

<

  <

           

For example, when weakly measuring the spin-z (described by the Pauli matrix 

z

V

) of an 

ensemble of spin-1/2 particles prepared in the X

+

 direction, with coupling strength 

/

N

H O

 

, the time evolution is determined by  

int

1

exp(

)

exp(

/

)

N

z

n

d

n

W

i H dt

i

P

N

O V

 

 

 

¦

³

 

so for a single measurement, the evolution of the spin states becomes entangled with the  

pointer of Eq. 18 (when 

1

V

 

) :  

2

2

(

/

) / 2

(

/

) / 2

2

2

1

[

1

1

]

1

2

[(1

)

1

1 ]

x

N

x

N

z

z

x

x

e

e

Nf

x

x

Nf

N

N

O

O

V

V

O

O

V

V

 

 

|

|

 

 

 

To complete the weak measurement, the pointer itself must be strongly measured. Then, the 

particle's initial state X

+

 changes by only a fraction

2

~

/ N

O

. Statistically, this means that only 

2

~

O

 out of N particles' states have changed. When 

O

 is small enough, this number of 

(23)

(19)

(20)

(21)

(22)

(18)

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“flipped” spins (changed from the initial X

+

 to X

-

) is negligibly small compared to the 

ensemble's size [25]. 

Moreover, if the coupling strength is 

/ N

H O

 

 rather than 

/

N

H O

 

, the weak measurement 

process will most likely end without a single flip.  

In case that all weak measurement are performed on a single particle (without constantly pre- 

and post-selecting it as in [11]), using a single pointer but only on one particle, it was shown 

[15] that the pointers undergoes a biased random walk with a log-normal distribution (see 

Fig. 4). When increasing the number of weak measurements, this procedure tends to a strong 

measurement (see Fig. 5). 

 

Fig. 4. Distribution of the number of measurements until the collapse given a fixed standard  

deviation σ. 

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Fig. 5. Success probability for hypothesis testing with low number of weak measurements. The solid 

curve describes the optimal success probability for projective measurement. 

3.2 Measuring Non-commuting variables of the same particle 

The above technique entails an even more intriguing result: When the two (initial and final) 

strong measurements are made on non-commuting operators, then, for the intermediate 

states, these two operators can coexist with arbitrary precision. 

3.3 Exotic mass and momentum 

With the uncertainty principle thus subtly outsmarted and ordinary temporal order strained, it 

is perhaps not surprising that these between-measurements states revealed by WM display 

other physical oddities as well. Brief examination of Eq. 19 reveals that the weak values may 

not belong to the operator's spectrum. As a consequence, particles with odd mass or 

momentum, at times being even negative, are predicted by TSVF and amenable to isolation 

and measurement by appropriate slicing. Such effects are demonstrated elsewhere

  [25]  

leading, for instance, to violation of Bohr's correspondence principle [13]. 

3.4 Weak Values of Entangled States

It was shown that weak measurements can be performed on an entangled pair (or triplet, 

quartet etc.) without destroying the entanglement. Utilizing this possibility, we showed [12] 

that one can record the results of 

,

,

x

y

z

V V V

of two entangled spins which will later violate 

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