_{1}

^{*}

In this paper the following information interpretation of uncertainty relation is proposed: if one bit of information was extracted from the system as a result of the measurement process, then the measurement itself adds an additional uncertainty (chaos) into the system equaled to one bit. This formulation is developed by calculating of the Shannon information entropy for the classical N-slit interference experiment. This approach allows looking differently at several quantum phenomena. Particularly, the information interpretation is used for explanation of entangled photons diffraction picture compression.

Heisenberg uncertainty relation is one of the fundamental principles of Quantum Mechanics. On the other hand, an information approach to Quantum Mechanics is popular now (for example, see [

Let’s consider the N-slit interference experiment (

where θ is the angle between the wave vector and the optical axis, δ = k・d・sinθ, d is the space between slits. Multiplying (1) by its complex conjugate the interference picture intensity on the screen M as a function of angle θ is obtained:

I ( θ ) = I 0 sin 2 ( N δ 2 ) sin 2 ( δ 2 ) . (2)

This intensity can also be considered as a probability distribution. Let’s calculate the information entropy H of this distribution using Shannon’s formula [

H = ∫ I ( θ ) ⋅ log 2 ( 1 I ( θ ) ) d θ , (3)

where the normalization condition is: ∫ I ( θ ) d θ = 1 . The curve presenting dependence H from N in a semilogarithmic scale for kl = 10 is shown in _{2}(N) which is demonstrated by high value of correlation coefficient equaled to −0.9996.

Second, the equation of straight line, which describes the best point’s distribution on the curve (

Н = 2.2586 − 0.9865 ⋅ log 2 ( N ) . (4)

It is called the regression equation. As it can be seen from this equation, reduction of slits amount by a factor of 2 increases the distribution entropy by one bit. As it was demonstrated in the popular lections by R. Feynman [

Since the obtained regularity does not depend on the specific experimental parameters, this regularity can be generalized for any quantum system. Let’s call this: information interpretation of uncertainty relation. The uncertainty relation itself can be written in the information form. In fact, the uncertainty relation for a harmonic oscillator is: ΔX・ΔP = ћ/2 (expression 16.8 from [_{2}(ΔX) − 1 bit + log_{2}(ΔP) + 1 bit = const. Here: 1 bit = log_{2}(2), const = log_{2}(ћ/2), log_{2}(ΔX) and log_{2}(ΔP) − amounts of information during determination of coordinate and pulse, respectively (up to additional constant, which is equaled to log_{2}(2πe)^{0.5} for the normal distribution [

As an example let’s show how the information interpretation of uncertainty relation can be used for explanation of entangled photons diffraction picture compression. In [_{2}O_{4} crystal. Photons generated in the crystal have frequencies equaled to the half of falling photon frequencies. Let’s describe the principle scheme of this experiment (

To understand this result let’s notice that in ordinary photons diffraction case we can always distinguish one photon from the other as one comes to detector earlier than the other. However, this information is lost in the experiment described above. The entangled photons are born simultaneously and come to

detector also at the same time. As a result, the system obtains an additional uncertainty corresponding to impossibility distinguishing photons. The system entropy increases by one bit. However, the corresponding diffraction picture shrinks in half. It is easy to show that the diffraction picture entropy decreases by one bit. Let’s assume that P(х) and G(х) are a normalized distributions of diffraction picture intensity before and after shrinking, respectively. So, ∫ P ( x ) d x = 1 . The entropy of this distribution is by definition equaled:

H 1 = ∫ P ( x ) ⋅ log 2 ( 1 P ( x ) ) d x . (5)

The entropy of diffraction picture distribution after shrinking is equaled:

H 2 = ∫ G ( x ) ⋅ log 2 ( 1 G ( x ) ) d x . (6)

An experimental factor of shrinking is equaled to 2. So, G ( x ) = constant × P ( 2 x ) . Then, we can obtain from normalization ∫ G ( x ) d x = 1 : G ( x ) = 2 × P ( 2 x ) . Then:

H 2 = ∫ 2 P ( 2 x ) ⋅ log 2 ( 1 2 P ( 2 x ) ) d x = ∫ P ( y ) ⋅ log 2 ( 1 P ( y ) ) d y + log 2 ( 1 2 ) ∫ P ( y ) d y (7)

Here we used change of variable y = 2x inside the integral. Then, form ∫ P ( y ) d y = 1 and log_{2}(1/2)= -1 bit we can obtain:

H 2 = H 1 − 1 bit (8)

Thus the system compensates loosing of photons identification possibility by diffraction spot reduction. This allows conservation of the balance between the system uncertainty and the amount of information extracted from the system.

The information entropy for the N-slit interference experiment was calculated. A formulation of the information interpretation of the uncertainty relation was proposed. It was applied for an explanation of the entangled photons diffraction picture compression.

Oleg Petrov deceased in 2009. His unpublished original work was in Russian. It was translated to English by A.M. Smolovich with help of D.A. Oulianov. Figures were edited by A.P. Orlov and P.A. Smolovich.

The author declares no conflicts of interest regarding the publication of this paper.

Petrov, O.V. (2019) Information Interpretation of Heisenberg Uncertainty Relation. Optics and Photonics Journal, 9, 9-14. https://doi.org/10.4236/opj.2019.92002