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

1. One atom is positioned in the intersecting box. 

2. It has not absorbed any photon. 

3. Still, the fact that the other atom’s spin is affected by this atom’s position means that 

something has traveled the path blocked by the first atom. To prove that, let another object 

be placed after the first atom on the virtual photon’s path. No nonlocal correlations will show 

up. 

Thus, the very fact that one atom is positioned in a place that seems to preclude its 

interaction with the other atom is affected by that other atom. This is logically equivalent to 

the statement “this sentence has never been written”. We are unaware of any other quantum 

mechanical experiment that demonstrates such inconsistency. 

2.8 Concluding Remarks 

It thus seems quite obvious that the quantum realm is unique mainly because the time-

evolution it presents allows events to affect one another in both time directions. This has so 

far been shown with the aid of ordinary quantum measurements. A new type of 

measurements, more delicate and sensitive, will be employed next. 

3 Weak Measurements

  

Superposition is quantum mechanics’ most intrinsic concept, an emblem of its uniqueness. 

An unmeasured particle's state is not only unknown but indeterminate, co-sustaining 

mutually-exclusive states. Equally crucial (and even less understood) is “measurement” or 

“collapse,” upon which one of these states is realized, inflicting uncertainty on conjugate 

variables. In view of these limitations, can there be any reason to make quantum 

measurement less precise?  

It is, surprisingly, weak measurement (WM) [24-26] that overcomes these limitations as well 

as many others [12,13,25]. Moreover, the Two-State-Vector-Formalism (TSVF), within 

which WM has been conceived, predicts several peculiar phenomena occurring between 

measurements, which only WM can reveal. Consider the question “What is a particle's state 

between two measurements?” Obviously, measuring such a state would change it into a state 

upon measurement, rendering the question meaningless. Not so with WM: The state, almost 

without being disturbed, can be made known with great accuracy, moreover manifesting a 

host of new peculiarities. 

This, however, is a non-trivial task since most projective measurements performed on the 

system would change its dynamics. To overcome this challenge, weak measurement was 

introduced [24]. 

Weak measurement of a quantum system enables studying it without changing its wave-

function. An intuitive explanation of this feat is given in [11]. In a nutshell, strong 

measurement is composed of a quantum pointer and an amplification mechanism making the 

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reading macroscopic. In order to provide an accurate result, the pointer must have a certain 

momentum. This way, when our particle interacts with it, the reading (in terms of 

momentum change) would be unambiguous. Unfortunately, the amplified interaction with 

this pointer results in an irreversible change of the measured system – the so-called collapse. 

As opposed to this ordinary "strong" measurement, weak measurement creates a loose 

coupling to a quantum pointer whose momentum is highly uncertain, and again the pointer 

reading is being amplified by the same mechanism. This combination of weak coupling and 

noisy reading naturally gives a very small amount of information, but also a negligible 

change of its dynamics. It is on the ensemble level that weak measurement gains the desired 

precision, overcoming its inherent inaccuracy to the extent of even surpassing the limits of 

ordinary quantum measurement. By the Large Numbers Law, if x

i

 (the different 

measurement outcomes) are independent and identically distributed random variables with a 

finite second moment, their average goes to their expectation value:

.

a s

n

x

P

o

. Furthermore, 

since the variance (noise) is proportional to N, the relative error diminishes. We showed [11] 

that an ensemble, can be a horde of states of a single particle undergoing cyclic weak 

measurements rather than an ensemble of particles undergoing a weak measurement. 

Weak measurements were proven to be an important tool, not only for better answering 

fundamental questions but also for solving practical problems such as utilization of quantum 

amplification, cross-correlations between quantum signals [14], and increasing the signal-to-

noise ratio [27]. 

In the language of quantum information, weak measurements were used to construct the 

quantum weak channel and the weak analogies of several bounds such as Holevo's [28]. 

3.1 Mathematical description 

Using von Neumann's arguments as in [25], a quantum measurement of the observable A is 

defined by the interaction:  

 

where the momentum P

d

 is canonically conjugated to Q

d

, representing the pointer's position 

on the measuring device. The coupling g(t) differs from zero only at 

0

t

T

d d

and normalized 

according to  

         

 

i.e. the measurement lasts no longer than T.  

In weak measurement, the coupling Hamiltonian of Eq. 16 is small in comparison to the 

pointer's standard deviation, i.e., the measuring device is prepared in a symmetric quantum 

state with standard deviation  

V

H

!!

 and zero expectation. Without loss of generality we 

int

( )

( )

d

H

t

g t AP

H

 

0

( )

1

T

g t dt

 

³

(16)

(17)

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