Quantum mechanics becomes massively simpler to interpret once you recognize that the wave function is just a compressed list of expectation values for the observables of a system. An expectation value is like a weighted probability. They can be negative because the measured values can be negative, such as for qubits, the measured values can be either +1 or -1, and if you weight by -1 then it can become negative. For example, an expectation value of -0.5 means there is a 25% chance of +1 and a 75% of -1.
If I know for certain that X=+1 but I have no idea what Y is, and the physical system interacts with something that we know will have the effect of swapping its X and Y components around, then this would also swap my uncertainty around so now I would know that Y=+1 without knowing what X is. Hence, if you don’t know the complete initial conditions of a system, you can represent it with a list of all of possible observables and assign each one an expectation value related to your certainty of measuring that value, and then compute how that certainty is shifted around as the system evolves.
The wave function then just becomes a compressed form of this. For qubits, the expectation value vector grows at a rate of 4^N where N is the number of qubits, but the uncertainty principle limits the total bits of information you can have at a single time to 2^N, so the vector is usually mostly empty (a lot of zeros). This allows you to mathematically compress it down to a wave function that also grows by 2^N, making it the most concise way to represent this.
But the notation often confuses people, they think it means particles are in two places at once, that qubits are 0 and 1 at the same time, that there is some “collapse” that happens when you make a measurement, and they frequently ask what the imaginary components mean. But all this confusion just stems from notation. Any wave function can be expanded into a real-valued list of expectation values and you can evolve that through the system rather than the wave function and compute the same results, and then the confusion of what it represents disappears.
When you write it out in this expanded form, it’s also clear why the uncertainty principle exists in the first place. A measurement is a kind of physical interaction between a record-keeping system and the recorded system, and it should result in information from the recorded system being copied onto the record-keeping system. Physical interactions are described by an operator, and quantum theory has certain restrictions on what qualifies as a physically valid operator: it has to be time-reversible, preserve handedness, be completely positive, etc, and these restrictions prevent you from constructing an operator that can copy a value of an observable from one system onto another in a way that doesn’t perturb its other observables.
Most things in quantum theory that are considered “weird” are just misunderstandings, some of which can even be reproduced classically. Things like double-slit, Mach–Zehnder interferometer, the Elitzur–Vaidman “paradox,” the Wigner’s friend “paradox,” the Schrodinger’s cat “paradox,” the Deutsch algorithm, quantum encryption and key distribution, quantum superdense coding, etc, can all be explained entirely classically just by clearing up some confusion about the notation.
This narrows it down to only a small number of things that genuinely raise an eyebrow, those being cases that exhibit what is sometimes called quantum contextuality, such as violations of Bell inequalities. It inherently requires a non-classical explanation for this, but I don’t think that also means it can’t be something understandable.
The simplest explanation I have found in the literature is that of time-symmetry. It is a requirement in quantum mechanics that every operator is time-symmetric, and that famously leads to the problem of establishing an arrow of time in quantum theory. Rather than taking it to be a problem, we can instead presume that there is a good reason nature demands all its microscopic operators are time-symmetric: because the arrow of time is a macroscopic phenomena, not a microscopic one.
If you have a set of interactions between microscopic particles where A causes B and B causes C, if I played the video in the reverse, it is mathematically just as valid to say that C causes B and B causes A. Most people then introduce an additional postulate that says “even though it is mathematically valid, it’s not physically valid, we should only take the evolution of the system in a single direction of time seriously.” You can’t derive that postulate from quantum theory, you just have to take it on faith.
If we drop that postulate and take the local evolution of the system seriously in both its time-forwards evolution and its time-reversed evolution, then you can explain violations of Bell inequalities without having to add anything to the theory at all, and interpret it completely in intuitive local realist terms. You do this using the Two-State Vector Formalism where all you do is compute the evolution of the wave function (or expectation values) from both ends until they meet at an intermediate point, and that gives you enough constraints to deterministically derive a weak value at that point. The weak value is a physical variable that evolves locally and deterministically with the system and contains sufficient information to generate its expectation values when needed.
You still can’t always assign a definite value, but these expectation values are epistemic, there is no contradiction with there being a definite value as the weak value contains all the information needed for the correct expectation values, and therefore the correct probability distribution, locally within the particle.
In terms of computation, it’s very simple, because for the time-reverse evolution you just treat the final state as the initial state and then apply the operators in reverse with their time-symmetric equivalents (Hermitian transpose) and then the weak value equation looks exactly like the expectation value equation except rather than having the same wave function on both ends of the observable, you have the reverse-evolved wave function on one end of the observable and the forwards-evolved wave function on the other. (You can also plug the expectation value vectors on both ends and it works as well.)
Nothing about this is hard to visualize because you just imagine playing a moving forwards and also playing it in the reverse, and in both directions you get a local causal chain of interactions between the particles. If A causes B and B causes C in the time-forwards movie, playing the movie in reverse you will see C cause B which then causes A. That means B is both caused by A and C, and thus is influenced by both through a local chain of interactions.
There is nothing “special” going on in the backwards evolution, the laws of physics are symmetrical so, visually, it is not distinguishable from its forwards evolution, so you visualize it the exact same way, so you can pretty much still maintain a largely classical picture in your head, just with the caveat that you have to consider both directions in order to place enough constraints on the system to explain the observed results. All the “paradoxes” suddenly evaporate away because you can just compute how the system locally evolves in any “weird” situation and look at exactly what is going on.
That is enough to explain QM in local realist terms, doesn’t require any modifications to the theory, and has been well-established in the literature for decades, is easy to visualize, but people often seem to favor explanations that are impossible to visualize, like treating the wave function as a literal object despite the wave function being, at times, even infinite-dimensional for continuous observables, or even believing we all live in an infinite-dimensional multiverse. And then they all complain it’s impossible to visualize and so confusing and “no one understands quantum mechanics”… I don’t understand why people seem to prefer to think about things in a way that they themselves admit just leads to endless confusion.
Quantum mechanics becomes massively simpler to interpret once you recognize that the wave function is just a compressed list of expectation values for the observables of a system. An expectation value is like a weighted probability. They can be negative because the measured values can be negative, such as for qubits, the measured values can be either +1 or -1, and if you weight by -1 then it can become negative. For example, an expectation value of -0.5 means there is a 25% chance of +1 and a 75% of -1.
If I know for certain that X=+1 but I have no idea what Y is, and the physical system interacts with something that we know will have the effect of swapping its X and Y components around, then this would also swap my uncertainty around so now I would know that Y=+1 without knowing what X is. Hence, if you don’t know the complete initial conditions of a system, you can represent it with a list of all of possible observables and assign each one an expectation value related to your certainty of measuring that value, and then compute how that certainty is shifted around as the system evolves.
The wave function then just becomes a compressed form of this. For qubits, the expectation value vector grows at a rate of 4^N where N is the number of qubits, but the uncertainty principle limits the total bits of information you can have at a single time to 2^N, so the vector is usually mostly empty (a lot of zeros). This allows you to mathematically compress it down to a wave function that also grows by 2^N, making it the most concise way to represent this.
But the notation often confuses people, they think it means particles are in two places at once, that qubits are 0 and 1 at the same time, that there is some “collapse” that happens when you make a measurement, and they frequently ask what the imaginary components mean. But all this confusion just stems from notation. Any wave function can be expanded into a real-valued list of expectation values and you can evolve that through the system rather than the wave function and compute the same results, and then the confusion of what it represents disappears.
When you write it out in this expanded form, it’s also clear why the uncertainty principle exists in the first place. A measurement is a kind of physical interaction between a record-keeping system and the recorded system, and it should result in information from the recorded system being copied onto the record-keeping system. Physical interactions are described by an operator, and quantum theory has certain restrictions on what qualifies as a physically valid operator: it has to be time-reversible, preserve handedness, be completely positive, etc, and these restrictions prevent you from constructing an operator that can copy a value of an observable from one system onto another in a way that doesn’t perturb its other observables.
Most things in quantum theory that are considered “weird” are just misunderstandings, some of which can even be reproduced classically. Things like double-slit, Mach–Zehnder interferometer, the Elitzur–Vaidman “paradox,” the Wigner’s friend “paradox,” the Schrodinger’s cat “paradox,” the Deutsch algorithm, quantum encryption and key distribution, quantum superdense coding, etc, can all be explained entirely classically just by clearing up some confusion about the notation.
This narrows it down to only a small number of things that genuinely raise an eyebrow, those being cases that exhibit what is sometimes called quantum contextuality, such as violations of Bell inequalities. It inherently requires a non-classical explanation for this, but I don’t think that also means it can’t be something understandable.
The simplest explanation I have found in the literature is that of time-symmetry. It is a requirement in quantum mechanics that every operator is time-symmetric, and that famously leads to the problem of establishing an arrow of time in quantum theory. Rather than taking it to be a problem, we can instead presume that there is a good reason nature demands all its microscopic operators are time-symmetric: because the arrow of time is a macroscopic phenomena, not a microscopic one.
If you have a set of interactions between microscopic particles where A causes B and B causes C, if I played the video in the reverse, it is mathematically just as valid to say that C causes B and B causes A. Most people then introduce an additional postulate that says “even though it is mathematically valid, it’s not physically valid, we should only take the evolution of the system in a single direction of time seriously.” You can’t derive that postulate from quantum theory, you just have to take it on faith.
If we drop that postulate and take the local evolution of the system seriously in both its time-forwards evolution and its time-reversed evolution, then you can explain violations of Bell inequalities without having to add anything to the theory at all, and interpret it completely in intuitive local realist terms. You do this using the Two-State Vector Formalism where all you do is compute the evolution of the wave function (or expectation values) from both ends until they meet at an intermediate point, and that gives you enough constraints to deterministically derive a weak value at that point. The weak value is a physical variable that evolves locally and deterministically with the system and contains sufficient information to generate its expectation values when needed.
You still can’t always assign a definite value, but these expectation values are epistemic, there is no contradiction with there being a definite value as the weak value contains all the information needed for the correct expectation values, and therefore the correct probability distribution, locally within the particle.
In terms of computation, it’s very simple, because for the time-reverse evolution you just treat the final state as the initial state and then apply the operators in reverse with their time-symmetric equivalents (Hermitian transpose) and then the weak value equation looks exactly like the expectation value equation except rather than having the same wave function on both ends of the observable, you have the reverse-evolved wave function on one end of the observable and the forwards-evolved wave function on the other. (You can also plug the expectation value vectors on both ends and it works as well.)
Nothing about this is hard to visualize because you just imagine playing a moving forwards and also playing it in the reverse, and in both directions you get a local causal chain of interactions between the particles. If A causes B and B causes C in the time-forwards movie, playing the movie in reverse you will see C cause B which then causes A. That means B is both caused by A and C, and thus is influenced by both through a local chain of interactions.
There is nothing “special” going on in the backwards evolution, the laws of physics are symmetrical so, visually, it is not distinguishable from its forwards evolution, so you visualize it the exact same way, so you can pretty much still maintain a largely classical picture in your head, just with the caveat that you have to consider both directions in order to place enough constraints on the system to explain the observed results. All the “paradoxes” suddenly evaporate away because you can just compute how the system locally evolves in any “weird” situation and look at exactly what is going on.
That is enough to explain QM in local realist terms, doesn’t require any modifications to the theory, and has been well-established in the literature for decades, is easy to visualize, but people often seem to favor explanations that are impossible to visualize, like treating the wave function as a literal object despite the wave function being, at times, even infinite-dimensional for continuous observables, or even believing we all live in an infinite-dimensional multiverse. And then they all complain it’s impossible to visualize and so confusing and “no one understands quantum mechanics”… I don’t understand why people seem to prefer to think about things in a way that they themselves admit just leads to endless confusion.