# "Randomness" versus "uncertainty"

Highly rated PhysicsSE contributor @CuriousOne regularly makes the following claim about quantum mechanics (e.g. here):

There is no randomness in quantum mechanics, there is only uncertainty.

I want to know what this is supposed to mean.

• I think you misquoted? Apr 7, 2016 at 1:42
• A source can emit one photon in a random direction but turn of the energy and the isotropic nature will become more apparent. Apr 7, 2016 at 2:40
• The "no randomness" presumably means he believes in many worlds via decoherence. Judging from your posts, you already know that -- are you looking for a defense of many worlds instead? Or what he specifically thinks about it? If it's the latter you could just chat him... Apr 7, 2016 at 2:55
• If you have a random/probabilistic process, its dynamic can't be reversed. A quantum mechanical process, e.g. a spin system, can be reversed as long as you don't perform a measurement on it. Take a look at spin echos and compare that to the dynamic properties of truly probabilistic systems. It's a day and night difference. Apr 7, 2016 at 3:27
• this question relates to the problem of nonlocality/ "spooky action at a distance" studied by a long string of physicists eg Einstein, Bohm, Bell etc.
– vzn
May 4, 2016 at 15:58

## 2 Answers

There is no randomness in quantum mechanics, there is only uncertainty. , as stated, whoever may have said it.

The fields of mathematics, probability, and statistics use formal definitions of randomness. In statistics, a random variable is an assignment of a numerical value to each possible outcome of an event space. This association facilitates the identification and the calculation of probabilities of the events.

So by this definition, mathematically, randomness is defined wherever probability distributions can be assigned to expected outcomes.

As quantum mechanics is par excellence a probabilistic theory, i.e. probability distributions are assigned to measurable variables from solutions of relevant differential equations, this mathematical definition of randomness is not the one used in the statement. It must be the everyday concept in the beginning of the link:

Randomness is the lack of pattern or predictability in events. A random sequence of events, symbols or steps has no order and does not follow an intelligible pattern or combination.

So the statement There is no randomness can be applied to any event that can be predicted/estimated from a probability distribution, and not only in quantum mechanics. When a mathematical probability distribution exists, there is only uncertainty for a specific measurement , given in terms of the probability for that outcome.

So the statement should be :

There is no randomness where a mathematical probability distribution exists, there is only uncertainty .

So the original is a statement valid in a discussion where deterministic classical theories which predict a single measurement value as the solutions of the appropriate equations are being discussed, whereas quantum mechanics ( or any probabilistic theory) predicts an uncertain value with a given probability.

The above exposition goes to prove that the context where a statement is made is as important as boundary values for physics solutions.

• Please take a look at the central results of probability theory, which are a number of central limit theorems. Where, in quantum mechanics, do you find central limit theorems to be useful? Apr 7, 2016 at 3:50
• @CuriousOne can you give a link? I can give one central limit theorem for quantum mechanics by googling arxiv.org/abs/quant-ph/0608198 Apr 7, 2016 at 3:52
• Serious mathematics textbooks on probability theory should do. See e.g. mathworld.wolfram.com/CentralLimitTheorem.html for an introduction to the most trivial aspects of it. Apr 7, 2016 at 3:59
• @CuriousOne It seems though that they exist for quantum mechanics also , which you think it does not Apr 7, 2016 at 4:04
• No one is trying to squash anything here (I hope). The central limit theorem is the most famous example of universality, where coarse-graining (implemented through averaging) leads to a unique scaling limit. In quantum systems, the "coarse-graining" is implemented differently, but involves grouping subsystems together and tracing over microscopic degrees of freedom. The scaling limit is a density matrix, or field theory (though canonically the ground state/DM is most important). (Also, the Cauchy distribution doesn't converge to a Gaussian).
– TLDR
Apr 7, 2016 at 4:30

The core of your question is subtle, so I'll be careful in how I set up my answer.

In my understanding of quantum mechanics, wave function collapse is the closest a physical process can be to the mathematical idealization of a random variable. However, before the collapse, a complicated many-body process, the wave function evolution of the system is deterministic (unitary). One could say that uncertainty is what characterizes the wave function before measurement (measurement being viewed as a kind of dissipative process), while "randomness" characterizes the probabilities of making various measurements, implicitly after any decoherence necessary for measurement has occured.

[Addendum: In the context of quantum computing, it's important to find a set of measurements (quantum numbers used to read the output of the computation) that align as much as possible with the set of (meaningful) final states that the algorithm could produce. If there is some misalignment between the measurement axis and these final states, noise is generated when reading the output.]

• "Wave function collapse" (which us really just a very unfortunate name for the Born rule) requires the system to be open, i.e. exposed to many unknown degrees of freedom. It is "random" because it is physically made to be random. Apr 7, 2016 at 3:31
• I agree. But measurements need to be made, and the collapse of the wave function is what our limited senses are able to observe. Also, all physical systems are open.
– TLDR
Apr 7, 2016 at 3:32
• One can observe spin echos just fine. A strong measurement is such a crude operation on a quantum system that many forget how much really fun non-destructive things one can do on them. Part of the problem with QM 101 is that it is being taught in a phenomenological vacuum, whereas a short look into a few textbooks on quantum optics, spin resonance and solid state physics could teach that there is a whole universe of phenomenally useful quantum measurement techniques out there to which strong measurement is but one tiny piece (and in hep it's not used, at all). Apr 7, 2016 at 3:36
• All I am saying is that physical randomness that is not a consequence of the observer's ignorance comes from wave function collapse. (You may argue that randomness from wave function collapse is a consequence of ignorance of the many-body wave function of system+environment.)
– TLDR
Apr 7, 2016 at 3:42
• Indeed, "wave function collapse" is the observer's chosen ignorance. Not every system is, automatically, a good measurement apparatus. Only those systems that have a sufficient number of unknown degrees of freedom to store the result of an unbiased measurement for "eternity" are. Apr 7, 2016 at 3:47