# A Uncertainty vs Randomness in Quantum Physics

#### Sasho Andonov

Summary
What is present in Quantum Physics: Randomness or uncertainty...?
In Quanta Magazine there is article on this link:

The article is speaking about generation of random numbers by quantum computer, but I am confused...
I thought that every process in Quantum Physics is full with uncertainty, not with randomness... Does i means that Heisenberg Uncertainty Principle would be Heisenberg Randomness Principle...?
What is difference between randomness and uncertainty in Quantum Physics...?

Related Quantum Physics News on Phys.org

#### StevieTNZ

Randomness is in the sense that if you measured the spin of a quantum system, it is randomly up OR down (there is no cause for it to be up rather than down, unless you're factoring in Bohmian Mechanics), roughly 1/2 the time, when measuring an ensemble of quantum systems.

#### Sasho Andonov

Randomness is in the sense that if you measured the spin of a quantum system, it is randomly up OR down (there is no cause for it to be up rather than down, unless you're factoring in Bohmian Mechanics), roughly 1/2 the time, when measuring an ensemble of quantum systems.
My question is more regarding use of randomness vs uncertainty. Are they same in Quantum Mechanics? They are different in other areas...

#### Aidyan

My question is more regarding use of randomness vs uncertainty. Are they same in Quantum Mechanics? They are different in other areas...
A process is random if its outcomes have equal probability to be realized, With a dice 1/6 , coins 1/2, spins 1/2. Uncertainty refers to a set of possible outcomes which are not necessarily equally probable. A photon going through a double slit will displace itself on a detector screen onto interference fringes, but it remains uncertain which until the time of detection. Heisenberg's uncertainty principle tells us that once you measure one observable, say x, the momentum becomes uncertain, in the sense that you will have to describe it with a value centered over a probability curve (a Gaussian law, which implies not equally probable outcomes). The important point to keep in mind is not to confuse quantum with classical deterministic randomness and uncertainty. If one posits that qp has no hidden variables, then quantum randomness and uncertainty are 'real' or 'a-causal', so to speak. The randomness of the spin outcomes of an electron is not as that of tossing a coin.

#### Mentz114

Gold Member
A process is random if its outcomes have equal probability to be realized, With a dice 1/6 , coins 1/2, spins 1/2. Uncertainty refers to a set of possible outcomes which are not necessarily equally probable.
[]
Would a biased die whose probabilities are not all equal have non-random outcomes ?

#### Sasho Andonov

A process is random if its outcomes have equal probability to be realized, With a dice 1/6 , coins 1/2, spins 1/2. Uncertainty refers to a set of possible outcomes which are not necessarily equally probable. A photon going through a double slit will displace itself on a detector screen onto interference fringes, but it remains uncertain which until the time of detection. Heisenberg's uncertainty principle tells us that once you measure one observable, say x, the momentum becomes uncertain, in the sense that you will have to describe it with a value centered over a probability curve (a Gaussian law, which implies not equally probable outcomes). The important point to keep in mind is not to confuse quantum with classical deterministic randomness and uncertainty. If one posits that qp has no hidden variables, then quantum randomness and uncertainty are 'real' or 'a-causal', so to speak. The randomness of the spin outcomes of an electron is not as that of tossing a coin.
I agree totally with you, but the article in Quanta Magazine gives another picture of randomness and uncertainty.
But I am a little bit sceptical about the statement that the the randomness and uncertainty are different in Quantum Physics then in other areas... If they are different why we are using the same names...?

#### Aidyan

I agree totally with you, but the article in Quanta Magazine gives another picture of randomness and uncertainty.
The article speaks about a more complex version of randomness arising from 50 quantum entangled particles, but essentially it is about the same notion of quantum randomness you have by measuring the spin on two particles. It may sound confusing because there isn't really a commonly agreed rigorous definition of concepts like "randomness" (or pure "chance", "unpredictability", "uncertainty", "indeterminism", etc.). That's why also there are several different randomness tests. For a non-professional audience one uses it in a somewhat loose way or defines a precise one in the context needed. The article speaks of "randomly output a binary number after being given a distribution that specifies the desired probability for each possible 50-bit output string", which, as I understand it, is a 'probability distribution'. As I said, a probability distribution like a Gaussian law is usually not connected to the concept of randomness, or at least not of pure randomness, since one event is more probable than another.

Would a biased die whose probabilities are not all equal have non-random outcomes ?
So, for example, in this case one could say that it is not purely random, since one event is more probable than another. Another analogy could be white noise. If a signal in its spectral decomposition has every frequency as equally likely to show up, then it is 'white noise', per definition. If some part of the spectrum is more or less likely than others, then it is no longer 'white' and one does not consider it purely random. In a certain sense, one could also say that pure randomness is the maximal degree of uncertainty. But, I gave you an intuitive elementary definition of "randomness" (high school concept and which is used in most applications). If you want a more rigorous definition of randomness then you would need complexity theory which works with the entropy measure and chaos theory and that would ultimately lead to Goedel incompleteness theorem. It is much more complicated stuff (though fascinating). As an answer to your question however I believe it suffices to keep it simple.

But I am a little bit sceptical about the statement that the the randomness and uncertainty are different in Quantum Physics then in other areas... If they are different why we are using the same names...?
If one believes, as the supporters of Bohmian mechanics do, that QM has hidden variables, then you are right. At bottom everything could be conceived as a deterministic process and randomness would be the same in classical as quantum physics (even though with strange pilot waves acting non-locally with 'spooky action at a distance'....). . However, I think most physicists do not support it and if one thinks of quantum indeterminism as an 'a-causal' process, then the two things are quite different. They are named with the same nomenclature only for historical reasons. To distinguish the two people speak loosely of 'classical indeterminism' vs. 'quantum indeterminism' (or randomness or uncertainty, etc.). And, most importantly, Heisenberg's uncertainty must not be connected with a measurement uncertainty arising due to the interaction of the "observer", how so many like to put it. This is a common fallacy but wrong interpretation.

#### Sasho Andonov

Dear Aidyan, thank you very much for your efforts. I am happy with the answers... Have a nice day!

#### Mentz114

Gold Member
The question -
Would a biased die whose probabilities are not all equal have non-random outcomes ?
[]

So, for example, in this case one could say that it is not purely random, since one event is more probable than another. Another analogy could be white noise. If a signal in its spectral decomposition has every frequency as equally likely to show up, then it is 'white noise', per definition. If some part of the spectrum is more or less likely than others, then it is no longer 'white' and one does not consider it purely random. In a certain sense, one could also say that pure randomness is the maximal degree of uncertainty. But, I gave you an intuitive elementary definition of "randomness" (high school concept and which is used in most applications). If you want a more rigorous definition of randomness then you would need complexity theory which works with the entropy measure and chaos theory and that would ultimately lead to Goedel incompleteness theorem. It is much more complicated stuff (though fascinating). As an answer to your question however I believe it suffices to keep it simple.
[]
That is not an answer to the question - the answer is that the outcomes are random, by being unpredictable.
It is true that a long run of outcomes for the fair die has a higher entropy than the biased die but I don't see how that addresses the question posed by the OP which mentions the Heisenberg uncertainty principle.

(There are many threads in this forum that give the full analysis if one cares to search).

#### Aidyan

I did address Heisenberg's uncertainty principle distinguishing between randomness and uncertainty.

#### Mentz114

Gold Member
I did address Heisenberg's uncertainty principle distinguishing between randomness and uncertainty.
The OP is content with your reply so I'll leave it there.

#### PeterDonis

Mentor
What is difference between randomness and uncertainty in Quantum Physics...?
Randomness in QM has to do with the results of individual measurements on quantum objects not being predictable; all you can predict are the statistics of large numbers of measurements of the same observable on ensembles of similarly prepared quantum objects.

Uncertainty in QM has to do with the relationship between measurements of non-commuting observables (such as position and momentum). It says nothing by itself about the predictability or lack thereof of measurements of a single observable.

#### microsansfil

It says nothing by itself about the predictability or lack thereof of measurements of a single observable.

/Patrick

#### PeterDonis

Mentor
Randomness in measurements of multiple non-commuting observables, yes. Which in no way contradicts the statement of mine that you quoted.

#### atyy

The article is speaking about generation of random numbers by quantum computer, but I am confused...
I thought that every process in Quantum Physics is full with uncertainty, not with randomness... Does i means that Heisenberg Uncertainty Principle would be Heisenberg Randomness Principle...?
What is difference between randomness and uncertainty in Quantum Physics...?
There is no essential difference. Randomness can be described by equivalently probabilities or expectations. A particular type of expectation is the standard deviation. In the Heisenberg uncertainty principle, the "uncertainty" refers to a relationship between the standard deviation of results obtained from measurements of position on a quantum state and the standard deviation of results obtained from measurements of momentum on the same quantum state.

#### microsansfil

Randomness in measurements of multiple non-commuting observables, yes. Which in no way contradicts the statement of mine that you quoted.
Yep,

The question was: What is the difference between randomness and uncertainty in Quantum Physics...?

You write: Randomness in QM has to do with the results of individual measurements on quantum objects not being predictable;

And thus also, Randomness in QM has to do with measurements of multiple non-commuting observables, isn't it?

which is not the case for a complete set of commuting observables (CSCO) who is a set of commuting operators whose eigenvalues completely specify the state of a system.

/Patrick

#### vanhees71

Gold Member
Uncertainty in QM has to do with the relationship between measurements of non-commuting observables (such as position and momentum). It says nothing by itself about the predictability or lack thereof of measurements of a single observable.
I think it's crucial to formulate this in the following way:

Uncertainty relations describe a statistical property of two observables $A$ and $B$ represented by self-adjoint operators $\hat{A}$ and $\hat{B}$. For ANY (pure or mixed) state you can prepare the system in, always
$$\Delta A \Delta B \geq \frac{1}{2} |\langle [\hat{A},\hat{B}] \rangle|$$
holds.

To make it concrete and choose $x$ and $p_x$ of a particle for the observables. Then the uncertainty relation reads
$$\Delta x \Delta p \geq \frac{\hbar}{2}.$$
This tells you that, however you prepare the particle, you cannot make the product of the standard deviations between $x$ and $p_x$ smaller than $\hbar/2$. If you prepare the particle to be rather sharp in $x$ its momentum is rather unsharp and vice versa.

The state preparation has nothing to do with possible measurements. This is solely a question of how to measure the one or the other observable. Given an appropriate apparatus you can measure either $x$ or $p_x$ as accurately as you like. It is completely independent in which state the particle is prepared.

To get $\Delta x$ and $\Delta p$ you have to measure either $x$ or $p$ on very many equally prepared particles (where equally prepared means it's always described by the same quantum state when the mearurement is done). Then you'll find that the uncertainty relation always holds, and thus it says something about the possibility to prepare particles but nothing about what's measurable or not.

The question, in which sense the measurement disturbs the particles is much more complicated to analyze and subject of current research since to answer this question you have to carefully analyze any concrete experimental setup to find this out.

"Uncertainty vs Randomness in Quantum Physics"

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