# Confusion with quantum superposition

I have been studying QM and I think doing research on quantum superposition got me more confused about the topic. So I have two interpretation that I came across:

1. A quantum state that is in superposition (say) of state 1 and 2 means that it is not in exactly state 1 and 2 but a unique state called superposition that gets influenced by the measurement. (This is the explanation I understood by first lecture of MIT QM 8.04 lecture on youtube by Allan Adams).

2. A quantum state that is in superposition of state 1 and 2 is already in state 1 or 2 but since we don't know until we take a measurement we mathematically write it with these two possibilities.

The difference is that idea 2) says that the state is already fixed between 1 and 2 but we just don't know until we measure it and 1) says that the quantum state behaves something that is neither like state 1 and 2.

I understand that this might be a basic question but even after looking at questions on here at stackexchange and quora, I didn't know which answer to believe because both were presented.

• A "superposition" is just a fancy word for "linear combination". In QM, states form a vector space (linear space), and you can always take a linear combination of states to form another state, e.g. $|\psi\rangle = \alpha|1\rangle + \beta|2\rangle$. Commented Nov 27, 2022 at 13:39

According to quantum mechanics, the answer is interpretation 1: the system is in a state that is neither 1 nor 2, but a superposition.

We can make things a little more concrete by considering a single particle with spin. Let's suppose that that whenever we measure the particle's spin along a particular axis, we always get one of two values, $$+1$$ or $$-1$$, relative to that axis.

Then suppose the particle is in the superposition state $$$$|\psi\rangle = \frac{|+1; z\rangle + |-1; z\rangle}{\sqrt{2}}$$$$ where the notation $$|+1; z\rangle$$ refers to a state in which the particle is measured to have spin $$+1$$ along the $$z$$ axis.

It turns out that this same state, expressed in terms of the outcome of measuring spin along the $$x$$ axis, is $$$$|\psi\rangle = |+1; x \rangle$$$$ With interpretation 1, there is no problem. The state of the system is $$|\psi\rangle$$; there is no extra information that the system has that we don't know (a "hidden variable"). If we try to impose interpretation 2, we will run into a contradiction with quantum mechanics. For interpretation 2 to be true, the system must secretly have a definite spin direction for both the $$x$$ and $$z$$ components of spin. This contradicts the Heisenberg uncertainty principle, which (in this case) states that we cannot measure the $$x$$ and $$z$$ components of spin simultaneously. If we cannot measure a quantity even in principle, we should not ascribe physical meaning to it.

Now, being in contradiction with the Heisenberg uncertainty principle doesn't necessarily mean that the particle really doesn't have definite spin along the $$x$$ and $$z$$ axes; it could be that Nature does not follow the rules of quantum mechanics. However, you can make the above arguments sharper by including multiple entangled particles separated by a spacelike distance -- so far that light cannot travel between them. This leads to the Bell inequalities, which have been shown to hold experimentally, demonstrating that no local hidden variable theory can explain the predictions of quantum mechanics. There are "loopholes" -- ways to explain the predictions of quantum mechanics while having "extra" information we don't directly observe like you are describing -- but as time has gone on and more careful experiments have been done, these loopholes have become smaller and smaller and to date quantum mechanics is consistent with all experiments that have been done to test it.

• Thank you very much for your clear answer. I agree that it might just be the nature itself that a spin can't both be in x and z direction but only in 1 axis chosen. Now what you are saying is a way to prove this is by Bell inequality? I would like to research more to understand why a spin or some observables can't be measured simultaniously. Any suggested readings and topics for this? Commented Nov 26, 2022 at 23:50
• @Mardia (1) There is a bit more to Bell's inequalities than the uncertainty principle; you also need to have multiple entangled particles and you need to appeal to special relativity. Personally I would recommend learning more "basic" quantum mechanics and building up to that, but the wikipedia article I linked to is a good starting point (especially the "GHZ-Mermin" secion). (2) To understand why quantum mechanics implies some observables cannot be simultaneously measured, I recommend reading the wiki article on the Robertson-Schrodinger uncertainty relations. Commented Nov 27, 2022 at 0:06
• "If we cannot measure a quantity even in principle, we should not ascribe physical meaning to it." By this same logic, we should not ascribe physical meaning to superpositions of states between measurements. Is that correct? Commented Nov 27, 2022 at 13:28
• @D.Halsey I think that's a little too extreme. For example, by sending many photons through a double slit device, we can see the interference pattern in the single particle wavefunctions, by essentially sampling from the corresponding probability distribution many times. Commented Nov 27, 2022 at 13:35
• @D.Halsey The older I get, the more I identify with "shut up and calculate" :) Commented Nov 27, 2022 at 17:47

The first statement is correct (see also this important article on the interpretations of quantum mechanics).

A quantum state that is in superposition of state 1 and 2 is already in state 1 or 2

This is poorly worded. Saying that the system is in a superposition and then saying "already in state 1..." is contradictory. It doesn't make much sense to say that the system is in any particular state (prior to measurement).

A quantum state that is in superposition (say) of state 1 and 2 means that it is not in exactly state 1 and 2 but a unique state called superposition

This is the right way to look at it. The system is in a superposition of its possible eigenstates.

The difference is that idea 2) says that the state is already fixed

If this is what the statement implies, then it is clearly wrong. The system is in a superposition, and that is perhaps all that can be said. Saying that the state is fixed (prior to measurement) contradicts the notion of superposition.

1. says that the quantum state behaves something that is neither like state 1 and 2.

Right. It is in a superposition of its possible eigenstates.

The concept of quantum superposition is an idea that never quite "makes sense" when we use classical thinking. What we can do, as you have alluded to, is write down the state of the system in a superposition of its possible eigenstates, and the coefficients$$^1$$ give us a measure of the likelihood of finding the system in a particular state.

$$^1$$ For example, a quantum state $$\Psi = \sqrt{\frac{1}{3}}\ |1\rangle + \sqrt{\frac{2}{3}}\ |2\rangle$$ tells us that if we were to measure the system, we will find the system in the states $$|1\rangle$$ or $$|2\rangle$$ with probability $$\frac 13$$ and $$\frac 23$$ respectively.

• Is there a more theoretic explanation to this quantum superposition idea then (perhaps in QFT? since that is the most complete theory of QM today)? Or is it still accepeted as an axiom/postulate today? Commented Nov 26, 2022 at 23:46
• This is one of the postulates of QM. However, if you wanted a deeper understanding of particle interactions, then you could study QFT. The basic postulates of QM are still accepted. What has changed over time is the interpretations, of which I have provided a link. Cheers. Commented Nov 27, 2022 at 0:00