Can probability amplitudes be used as credences?

Question

Probability amplitudes in quantum mechanics are sometimes called a "generalization" of probabilities. They are complex numbers a + bi. The probability associated with the probability amplitude is |a + bi|^2 = a^2 + b^2.

If probability amplitudes are really a generalization of probabilities, is it possible to use them as credences for everyday statements outside of quantum physics? So that a person might say they assign an 0.5 - 0.3i probability amplitude that it will rain. Is there a way to make such a statement sensible, in the same way that it is sensible to say there is an 80% chance it will rain?

Answer 1

Probability amplitudes in quantum mechanics arise for a particular reason. Our best understanding of the physics of particles involves a wave equation, and this equation delivers complex solutions. The time-dependent Schrödinger equation even has an explicit i in it. These solutions allow us to describe the strange behaviours of quantum experiments such as the double slit experiment. Interpreting the square of the modulus of the amplitude as a probability density function (the Born rule) fits with empirical observation. In particular it yields results that agree with the concept of quantum superposition.

If we try to enquire what is the physical significance of a complex amplitude, it is difficult to understand or explain. It is just part of the weird way the universe behaves within quantum theory. If you favour the "shut up and calculate" approach to interpreting QM then you don't care what complex amplitudes or negative energies or negative probabilities mean, as long as they deliver useful results.

If you prefer an information theoretic interpretation of QM, you might well be content with the observation that using squared magnitudes is the only way to get a measure in state space that conserves probabilities. Since this is a highly desirable property in information theory, that justifies the use of it.

If you hold with the many worlds interpretation, which is more 'realistic', it makes more sense to enquire how the Born rule is justified. Some have claimed that the Born rule can be derived through counting of finite worlds. My understanding is that this claim is controversial, and justifying the Born rule is still a work in progress among theoreticians who favour the MWI.

My point is that without some empirical data that needs to be accounted for, or some feature of credences that we would like to explain, it is not clear why a second imaginary axis would be needed. Probabilities, under the Bayesian interpretation, work fine as a first approximation to rational degrees of credence. This has been justified by the work of Richard Cox and Bruno de Finetti.

The only analog with QM that I can imagine would be some theory that uses probabilities to describe the credences of an agent who entertains inconsistent beliefs but is still able to function properly and make sensible decisions. After all, we all do that. Maybe you could formulate a theory in which credences have complex values and when conflicting beliefs are combined for the purposes of making a decision they yield non-trivial probabilities by a process of belief superposition. Looks like a research project for someone. I'm surprised Kristian Berry hasn't already done it.

Answer 2

In quantum physics in general square amplitudes do not obey the rules of probability because of quantum interference. See Section 2 of this paper for an example:

https://arxiv.org/abs/math/9911150

If you interact with a quantum system and produce a record of its state that can be copied this will suppress interference, an effect called decoherence. See this review article especially Section 2:

https://arxiv.org/abs/1911.06282

The title of your question mentions credences: probabilities that are supposedly attached to theories, but the body of your question doesn't. The body of your question only mentions the probability of a future event: whether it will rain in your location.

There are papers about using quantum probabilities as credences in the light of David Deutsch's explanation of quantum probability in terms of decision theory. See these papers for some examples:

https://arxiv.org/abs/quant-ph/0211104

https://arxiv.org/abs/quant-ph/0312136

Other people have criticised using these probabilities as credences:

https://arxiv.org/abs/0905.0624

There are criticisms of the framing of this controversy. It is notable that Deutsch's original decision theory paper doesn't mention credences:

https://arxiv.org/abs/quant-ph/9906015

The entire paper is about the behaviour of decision theoretic agents in preference ranking quantum states.

There are severe problems with the idea of credences. Defining probabilities requires a space of events and that space of events comes from some explanatory theory, such as quantum theory or whatever. As such the idea of the probability of a theory is gibberish since you have no space of events on which to define it. Another problem is that a theory is either right or wrong so what does the probability mean? There aren't multiple copies of the whole of physical reality on which you can define that probability. Yet another problem is that since a theory is right or wrong it's difficult to see how assigning a probability to it could be possibly be relevant to any real decision. You have to either accept or reject a theory if you're considering using it for some application including understanding some issue. If you want to understand radioactive decay it does you no good to say that quantum chromodynamics is 50% true since you can't half use it to calculate the cross section of a nuclear reaction. For more explanation see

https://criticalfallibilism.com/yes-or-no-philosophy/

https://criticalfallibilism.com/yes-or-no-philosophy-and-score-systems/

The probability of an event like "it will rain in my location at 2pm" can make sense. I wouldn't want to do the quantum calculation required to get that probability but there is no difficulty with doing it if you have vastly more computational resources than anyone has now, but the idea of doing isn't nonsense, unlike credences. The real role of probabilities in the assessment of theories is that if the observed relative frequencies of an event don't match the probabilities when some theory claims they should that creates a problem which has to be solved by either modifying the theory or coming up with a flaw in the experiment, as David Deutsch has pointed out:

https://arxiv.org/abs/1508.02048

Answer 3

1. As you say a probability amplitude in quantum mechanics (QM) is a complex number. Only after taking the square of its modulus one obtains a probability, which is a real number between zero and one. The phase of the complex number is the reason that the probability amplitudes of different states can interfere. This produces the characteristic interference properties from the superposition of quantum states in QM.
2. On the level of classical physics one does not observe any interference of states: The interaction with the environment produces the decoherence of the states. As a consequence any ability of interference fades away during the transition from the QM-level to the classical level.

Hence I do no see an explanatory value when replacing the real numbers of probabilities by the complex numbers of probability amplitudes.

Answer 4

tl;dr: Yes, but why?

If probability amplitudes are really a generalization of probabilities,

What do you mean by "really"? "Probability amplitude" is a man-made technical term for the value of the Schrödinger wavefunction. Being a wave, it has two properties: the amplitude and the phase. This is why you need two real numbers to fully describe it. It was Max Born's observation that it's "power" (i.e. the squared amplitude) corresponds to probability of an event.

is it possible to use them as credences for everyday statements outside of quantum physics? So that a person might say they assign an 0.5 - 0.3i probability amplitude that it will rain

To what end? As far as I know (and may physicists correct me if I'm wrong) probability amplitudes aren't observable in quantum mechanics. Even the probabilities are not observable, but have to be estimated from observing a large number of discrete events.

If you really wish, you can map not only complex numbers, but almost everything - words, sounds, images... - to numbers between 0 and 1 and hence consider them "generalized probabilities". This is, essentially, how many classification algorithms / predictive probabilistic models in machine learning and statistics work. This is in no way special to quantum mechanics and "probability amplitudes".

Answer 5

No, I don't think so. Ordinary probability has a frequentist interpretation and hence can be modelled by frequency of events. Whereas probability amplitudes has no such direct interpretation in the real world.

The generalisation of probability theory to the quantum realm is merely a formal generalisation and not of its interpretation. The closest approach is perhaps QBism which takes an epistemological approach to quantum probabilities. But these to interpret real world events but quantum phenomena.

Answer 6

TL;DR: See last sentence.

Often, physics happens on a sphere. If you extend something past 100 percent, it rotates around and goes back to zero again, and while doing that, the values above 100% are imaginary. Angles work like this.

Spacetime does too, in 4 dimensions. If your velocity exceeds c, you'll find that your velocity is less than c but time is running backwards. Your velocity is complex. This is because physics treats elapsed time as negative distance. 4D distance (in spacetime, not generally) is called "pseudometric."

Note that you convert elapsed time into a spatial distance by multiplying it by c. Hours x miles/hour = miles.

If you go 6 light years in 6 years, your 4-dimensional distance is 0. Yes, really. That's why entanglement happens. Things at 0 distance interact, even though both objects are distinct and separated in both space and time.

If you go 6 light years in 3 years, you've exceeded c, and your 4-dimensional distance is negative. You can represent the total distance as -3 light-years, which doesn't make sense in 3 dimensions (but does in 4), or as (6 + 3i) in strange units.

That is unnecessary in real domains. Sure, you could give the probability of flipping a coin heads as .5 + 0i, but the imaginary part would always be zero.