A Quantum Mechanical Interpretation of Quantum Mechanics
A Dissolution of the Problem of Measurement
The measurement problem is sometimes touted as one of the most profound and unsolved problems in all of physics. This problem is meant to illustrate that quantum theory is fundamentally internally inconsistent due to the problem of measurement, and thus modifications to the theory need to be made in order to reconcile contradictions. These reconciliations involve the introduction of new entities: hidden variables, a universal wave function, an objective collapse, so on and so forth.
What will be presented here is not some new theory, nor an introduction of new entities to “solve” the problem of the measurement in quantum mechanics. What is presented here is the position that quantum mechanics simply does not have a problem of measurement, and that the supposed “paradoxes” arise from misunderstanding the mathematics of the theory.
This is not a “solution” to the measurement problem, but a rejection that there even is such a problem to be “solved,” and thus notions like hidden variable theories, many worlds theories, or objective collapse theories, which all purport to “solve” the measurement problem, are all on poor grounding, as they are “solving” a pseudoproblem.
The Origins of Quantum Theory
Fundamental to quantum theory is the principle of complementarity. This principle tells us that knowledge regarding certain properties of a system inherently contradicts with knowledge of complementary parts of the system. For example, a particle’s position is complementary to its momentum, so knowledge of its momentum contradicts with knowledge of its position. This “contradiction” implies both are not knowable simultaneously, only one can be known at a single time. These parts of a system which can be measured are called its observables.
Now, imagine you design an experiment whereby, at the very beginning, you measure a particle’s momentum, and thus its position becomes unknown. Then, the experiment is set up in such a way that after you measure its momentum, the particle interacts with many other particles in a way that depends upon its position. The other particles in the system then become more and more altered as the experiment progresses, but the precise way in which they are altered depends, again, upon that initial particle’s position.
What we end up with is an evolving system, but one evolving according to an initial parameter which we cannot know. How could we possibly model such a system? The answer, is of course, statistics. Even if we cannot predict something ahead of time with certainty, we could still repeat the experiment multiple times and learn how it behaves statistically, allowing us to model systems like this in terms of evolving probabilities.
This is where things get weird, however. It turns out that, in quantum theory, using traditional, classical probability theory simply gives the wrong answers. To model a quantum system like this, one needs to use complex-valued probabilities, known as probability amplitudes, which evolve as the system evolves according to a law known as the Schrodinger equation.
Let’s say you run the experiment for a bit and compute the evolution of probability amplitudes according to the Schrodinger equation, and then at the very end, you want to actually measure the final results. How can you predict the final results ahead of time if probability amplitudes are complex-valued quantities? What does it even mean to say a particular outcome has a -70.7i% chance of occurring? On its face, that seems entirely meaningless.
This is where a second law comes into play: the Born rule. The Born rule states that the system evolves according to the Schrodinger equation up until you measure it, and then when you perform your measurement, you compute the square magnitude of the probability amplitude. In the case of a probability amplitude of -70.7i%, its square magnitude is 50%, and so we know that corresponds to a 50% chance of that particular outcome being measured. Hence, the Born rule effectively transforms quantum probabilities into classical probabilities.
The probability amplitudes are usually kept track of in a list denoted ψ. In mathematics, a list of numbers is just called a vector, and so ψ is known as the state vector as it describes the state of the system statistically using probability amplitudes. It is sometimes misleadingly called the wave function, however, ψ is simply not a function. It is, again, a vector.
The wave function instead refers to the function ψ(x)=⟨x|ψ⟩. This function effectively allows you to “choose” a desired probability amplitude from the state vector. ψ holds a list of probability amplitudes associated with all possible outcomes. If you are wondering what the probability amplitudes associated with a particular possible configuration of the system is, you can pass that configuration in as the value of x and ψ(x) will give you that singular probability amplitude associated with it.
If you then compute its square magnitude in the form |ψ(x)|² we get the classical probability associated with that particular outcome. In physics, x is often associated with position, and similarly, the wave function is often associated with getting the probability amplitude associated with a particular position in space for a particle. Computing |ψ(x)|² would thus tell you the likelihood of it being there if you were to go measure it.
When only a single element in ψ is 1 and the rest are all 0, then this is known as an eigenstate as it represents a particular outcome being certain to have occurred. If the elements of ψ are anything else, then there must be quantum uncertainty, and this is known as a superposition of states.
The Origin of the Measurement Problem
Physical systems can be represented by ψ, which is a list of quantum probability amplitudes, and ψ evolves according to the Schrodinger equation up until you make a measurement, and at the very moment of measurement, you apply the Born rule to compute |ψ|² which gives you the classical probabilities of what you might observe.
Quantum and classical probabilities have genuinely different behaviors. A system that is evolving according to classical probabilities only evolves with probabilities between 0 and 1, whereas quantum probabilities are complex-valued quantities, and thus can be negative or even imaginary. If a negative and a positive probability overlap, they cancel each other out, something that cannot happen classically. This is known as interference, and is famously demonstrated by the double-slit experiment whereby dark bands can be seen in a pattern of light. These dark bands are caused by locations where the probabilities for a photon landing there have canceled out to 0%.
If you measure a system, you, again, predict what you will measure by computing the square magnitude of the probability amplitudes: |ψ|², which gives you classical probabilities. Yet, classical probabilities, again, behave differently than quantum probabilities. This means if a system is evolving without you measuring it, it evolves according to the Schrodinger equation applied to ψ, but if you measure it, then since you now “know” its values will occur statistically distributed as predicted by the Born rule, |ψ|², then your measurement must necessarily alter the evolution of the system as it transitions from ψ → |ψ|².
With the double-slit experiment, the photons have a certain probability of passing through one narrow slit or the other, and when they leave that slit, their trajectories are statistically spread out. Since there are two possible slits to enter, there are two possible statistically spread out trajectories to leave, and it is those trajectories that overlap and interfere with each other, giving you the dark bands where some cancel out to zero.
However, if you measure the slit the photon goes through, then you would compute the probabilities of it showing up on a particular slit as |ψ|². This would reduce the probabilities of it being at that slit to classical probabilities. The specific spread out trajectory the particle takes, therefore, would be classical, and not quantum, and thus the two trajectories would not interfere with each other.
What happens is that, if you measure the slit the photon goes through, the experiment gets altered such that the dark bands in the pattern of light no longer show up. Below, you can see what the pattern looks like if you were to not measure the slit that the photon passes through at the bottom. If you were to measure it, you get something more like the light pattern at the top, basically the same thing but without interference. Note that you do not get two distinct blobs as depicted in some misleading diagrams.
Now, we can start to understand why the measurement problem came into being. First, we have claimed that physical systems can be described with ψ which evolves according to the Schrodinger equation if left alone, but if measured, we predict what we will measure by computing |ψ|² which is the Born rule. Second, we have shown that this transition from ψ→|ψ|² is not just a mathematical formalism, but a real physical transition, as making these measurements alters the behavior of the system.
In both cases, measurement seems to play a fundamental role in the theory. The physical sciences try to describe reality starting from its microscopic fundamental constituents, and then derive the behavior of all macroscopic objects from that basis. Yet, a measuring apparatus is a macroscopic object, so it does not make much sense for it to play a fundamental role in the theory.
This transition from quantum to classical probabilities, which genuinely alters the evolution of the system, is a discontinuous jump. The system evolves continuously as long as it is following the Schrodinger equation, but the moment it is measured and the Born rule is applied, it jumps to classical probabilities in a way that is discontinuous.
Classical probabilities, only involving real numbers, is obviously also a smaller set than quantum probabilities, involving complex numbers. If a larger set is mapped onto a smaller set, then this is a many-to-one function, and such functions are not reversible as there is not sufficient information on the output to predict the input as many different inputs can map to the same output. Hence, when a measurement is made, not only does the Born rule have to be applied which leads to a discontinuous jump, but that jump is not time-reversible. You cannot take the state of a system at the present moment in time and run it backwards to predict where it started.
On top of that, there is no rigorous description of what even constitutes a measurement. A measurement is clearly a form of physical interaction, yet, we do not apply the Born rule when particles are physically interacting, only when we measure them. Thus, it would seem imperative that the theory explain in more detail what even constitutes a measurement.
It would seem that the theory is exclusively concerned about ‘results of measurement’, and has nothing to say about anything else. What exactly qualifies some physical systems to play the role of ‘measurer’? Was the wavefunction of the world waiting to jump for thousands of millions of years until a single-celled living creature appeared? Or did it have to wait a little longer, for some better qualified system . . . with a PhD? If the theory is to apply to anything but highly idealised laboratory operations, are we not obliged to admit that more or less ‘measurement-like’ processes are going on more or less all the time, more or less everywhere? Do we not have jumping then all the time?
— John Bell, “Against ‘Measurement’”
Problems of Defining Measurement
Let us assume that we do come up with some rigorous definition of what constitutes a measurement. If we do, then ψ would evolve according to the Schrodinger equation up until its configuration fits the definition of measurement, and then it would undergo a jump from quantum to classical probabilities. If, however, we change our definition of measurement to something else, then ψ would not undergo such a jump at the same moment in time. If the definition is stricter, it would undergo the jump earlier in the evolution of ψ, but if it is more broad, it would undergo the jump later in the evolution of ψ.
Again, recall that these jumps actually do imply a physical change to the statistical behavior of the system. Hence, your definition here is not simply for purposes of clarity, but it has real physical implications. Wherever you draw the line as to what constitutes a measurement, you are inherently placing certain boundaries whereby if ψ crosses, it has to undergo one of these jumps, and thus you would expect different statistical behavior depending upon where you place the boundary.
Any definition of measurement requires setting a boundary, which requires modifying the mathematics of the theory to specify the boundary. On top of this, wherever the boundary is drawn would give you different predictions as to the statistical behavior of the system. Indeed, many theories with such boundaries have been proposed, such as Ghirardi–Rimini–Weber theory and the Diósi–Penrose model, but these ultimately constitute different theories as they do not make the same statistical predictions as orthodox quantum theory, which does not have a boundary specified at all.
Hence, this seems to suggest that quantum theory is a broken theory, one that is incomplete and needs to have its mathematics modified in order to specify what this boundary is, what qualifies as “measurement,” and then go out and test it to see if its predictions that deviate from orthodox quantum theory are indeed correct. This is the essence of the so-called “measurement problem.”
However, let us assume for a moment that quantum theory is not broken. Maybe there is a good reason that a definition of measurement is not specified.
Interaction, not Measurement
A measurement is a form of interaction, but if the theory does not specify what kind of interaction qualifies as a measurement, then the only possible conclusion, if the theory is not broken, is that these jumps (the transition from quantum to classical probabilities) must occur for all physical interactions.
At first, this seems obviously wrong. Recall that we said ψ evolves according to the Schrodinger equation up until we make a measurement, and that while ψ is evolving, it can interact with tons of particles. As long as it does not interact with the measuring device, then the jump does not occur. Indeed, if every physical interaction constituted a measurement, then the jump would occur with every interaction, and thus the system could always be described in terms of classical probabilities, and thus the description in terms of ψ would be useless.
However, this argument implicitly assumes that the “jump” is non-contextual. That is to say, if the jump occurs, then the occurrence of the jump is a universal fact which all systems will agree upon. If the jump only occurs contextually, that is to say, from particular relative perspectives, then this might explain why the jump appears to occur only from your perspective only when you physically interact with the system by measuring it, because when you interact with it, it occurs relative to you. When particles are interacting prior to you measuring them, the jump is occurring relative to each other, but not relative to you, since you are not participating in those interactions.
There is a famous thought experiment known as the Wigner’s friend paradox that is relevant here. Suppose Wigner and his friend place a qubit (which can only have a value of 0 or 1) into a superposition of states where |ψ₀⟩ = 1/√2(|0⟩ + |1⟩). This means there is an equal quantum probability of the qubit being in the 0 state as there is in a 1 state. Then, the friend measures the particle and finds it to be an eigenstate, |ψ₁⟩ = |1⟩. That just means it is certainly in the 1 state, because that is what the friend observed.
Wigner knows he is doing this but has left the room. Since his friend’s memory state (what he remembers seeing) would be statistically correlated with the actual state of the qubit (since that’s what he saw), Wigner would have to describe his friend and the particle in an entangled superposition of states (entanglement just means a quantum statistical correlation). We can use another qubit to represent the friend’s memory state, so Wigner would describe the qubit and his friend as |ψ₂⟩ = 1/√2(|00⟩ + |11⟩), meaning, there as an equal probability that either the friend will remember seeing 0 and he measured 0, or he remembers seeing 1 and indeed measured 1. There is no possibility, for example, of him measuring 0 and remembering seeing 1, because we are assuming he is not insane and his measurement equipment is not faulty.
The paradox? ψ₁ ≠ ψ₂. Wigner and his friend describe what occurred from their perspectives differently.
However, if we are assuming that the jumps from quantum to classical probabilities are contextual, as stated before, and since we know that these jumps genuinely affect the evolution of ψ, then it logically follows that the evolution of ψ must also be contextual. Hence, ψ differing between different perspectives is not a “paradox,” but is to be expected.
If this solution is correct, then there should be a way to apply a transformation to transition from one perspective to the next. For example, imagine that there is a moving train, a moving car, and a person on a bench. The person in the car and the person on the bench will describe the velocity of the train differently relative to themselves. This does not lead to a problem, or confusion, because Galilean relativity predicts these disagreements.
If the person in the bench later compares notes with the person in the car regarding the velocity of the train, they should not be surprised their notes would disagree, because they could have applied a Galilean transformation to compute precisely what the disagreement would have been ahead of time. Such a transformation effectively changes your perspective to someone else’s, allowing you to know what they see.
One should also not confuse contextuality with subjectivity. The fact two observers may describe the same situation differently does not demonstrate it is subjective or has anything to do with “consciousness.” Certain physical properties in the real physical world do indeed just differ from perspective to perspective, and it is objectively physically real that those properties do indeed differ. The difference occurs because the two observers occupy different contexts, and that the properties which they are describing are context-dependent. The properties thus differ due to a change in context, not due to a change in observers. “Observer-dependence” is a misnomer.
This is, however, where we run into a bit of a conundrum. I keep talking about a jump that transitions from quantum to classical probabilities, but ψ cannot actually represent classical probabilities. It can only represent quantum probabilities and eigenstates. In the Wigner’s friend paradox, when the friend measures ψ₀, it does not jump to classical probabilities, but it jumps to the eigenstate ψ₁.
If a mathematical transformation existed to transform Wigner’s perspective to his friends such that he could predict what his friend would describe from his own perspective, then there would need to be a transformation from ψ₂ → ψ₁, but ψ₁ is an eigenstate and ψ₂ is a superposition of states. This presents a problem because if Wigner could take his superposition of states, which represents uncertainty, and predict what his friend would see with certainty, then he would also be certain of the particle’s state from his own perspective as well, and thus could reduce his own ψ₂ to an eigenstate.
This violates the principle of complementarity as Wigner would be able to know the outcome ahead of time with certainty, despite the principle of complementarity stating it should be fundamentally unknowable. Does that kill off a contextual interpretation of ψ? We already have shown that a non-contextual version of ψ does not work without specifying a definition for measurement, which must also change the mathematics and predictions of theory, and thus cannot possibly qualify as a consistent interpretation of the theory if it demands an entirely new theory. If a contextual ψ is not possible, then we could not avoid defining measurement, either, thus bringing us back full circle to the measurement problem.
ρ > ψ
There is a bias among many physicists to presume that because we discovered this fanciful way particles behave — following quantum probabilities that evolve according to the Schrodinger equation — that therefore all of nature must solely behave in this way. Unequivocally, that statement contradicts with the Born rule, which tells us that systems sometimes have discontinuous jumps from quantum to classical probabilities. However, those physicists dismiss this as just something due to error caused by measurement which we do not currently understand.
This all leads to an obsession over ψ. It is common to treat ψ as fundamental and fully descriptive of nature, and the Born rule is simply a blemish that we should not think too hard about. However, what if it is not a blemish? What if it is just as fundamental of a law of nature as the Schrodinger equation? What if the jumps from quantum to classical probabilities is a genuine physical event and not simply due to measurement error?
If we assume this, then ψ clearly cannot be treated as the full picture. Why? Because recall we said ψ can only represent quantum probabilities and eigenstates. It cannot represents classical probabilities which the Born rule provides. It, thus, would only be a useful statistical tool in very particular domains where Born rule jumps are not occurring.
Rather than ψ, we instead need ρ. This is the density matrix. With it, we can represent all three categories of quantum probabilities, eigenstates, and classical probabilities, and even mixtures of them. Interestingly, with ρ, we do not even have to ever calculate the Born rule from it, because it always carries the Born rule probabilities across its diagonal elements so they can just be read off. ρ also can be evolved continuously just like ψ can, so you can make all the same predictions with ρ.
Recall that it would be impossible to have a transformation of Wigner’s perspective ψ₂ that brings us into the perspective of ψ₁ because ψ₁ in this case is an eigenstate, and that would be equivalent to predicting the outcome with certainty ahead of time. However, there would be nothing stopping us from having a transformation from ρ₂ to ρ₁ where ρ₁ contains probabilistic eigenstates and thus we know the system is in an eigenstate but still do not know which particular one. This is the same thing as classical probabilities given by the Born rule.
When you adopt the perspective of something as the basis of a coordinate system, it effectively disappears from the picture. For example, if you tare a scale with a bowl on it and place an object in the bowl, the measurement reflects the object’s mass alone, as if the bowl isn’t there. Hence, for Wigner to transform his perspective to that of his friend, he would need to perform an operation on ρ₁ called a partial trace to “trace out” his friend, leaving him with just the friend’s particle.
What he would get is a ρ₁ which is in a probabilistic eigenstate, in other words, it would contain classical probabilities. Thus, he would know his friend is looking at a particle in an eigenstate, even if he can’t predict ahead of time what it is because it’s fundamentally random.
Now, suppose we have two qubits in state |0⟩. We apply a Hadamard gate to the least significant qubit, putting it into the superposition of states 1/√2(|0⟩ + |1⟩), then apply a controlled-NOT gate using it as the control. The controlled-NOT gate records the state of one qubit onto another, provided the target starts in |0⟩. It flips the target to |1⟩ only if the control is |1⟩, so the target ends up matching the control.
The result is an entangled Bell state: 1/√2(|00⟩ + |11⟩). If we use the density matrix form, ρ, we can apply a perspective transformation. Tracing out the most significant qubit leaves us its perspective on the least significant qubit, and if we do that, we get a ρ that represents a probabilistic eigenstate of 50 percent |0⟩, 50 percent |1⟩.
This brings us back to the supposed “problem” that allowing every physical interaction to constitute a “measurement” would disallow particles from being entangled. In this case, what we find is that from the observer’s perspective not interacting with the two particles, he would describe them in a superposition of states, but if we apply a perspective transformation to one of the particles themselves, we find that relative to each other, the other particle is in an eigenstate.
There is no contradiction! That is why there is no definition for measurement in quantum mechanics, because it is a relative theory whereby every physical interaction leads to a reduction of ψ, but only from the perspective of the objects participating in the interaction. The mathematics of the theory not only guarantees consistency between perspectives, but even allows for transformations into different perspectives to predict, at least statistically, what other observers would perceive.
Side Notes
You may have to also apply a unitary operation to transform ρ after the partial trace if the measurement bases differ between yourself and the perspective you are trying to apply this transformation to. There is no preferred basis as the basis is also relative, so whenever you apply a perspective transformation, there is also the implicit question being answered of in relation to what measurement basis? The measurement basis is also a contextual consideration and there is no preferred basis.
Hilbert space is also a constructed space, unlike a something like Minkowski space or Euclidean space that are background spaces. The difference is that background spaces are defined independently of the objects they contain, whereas constructed spaces are defined in terms of the objects they contain. Technically, the density matrix is not an element of Hilbert space but of operator space, but that operator is an operator on Hilbert space, so it still is a constructed space by extension.
The reason this matters is because in Minkowski space, you can apply a perspective transformation to any arbitrary perspective, even those that do not contain objects. Whereas in operator space, since it is defined in terms of the objects it contains, you can only apply a perspective transformation to objects within that operator space, i.e., only to the perspectives of objects you have defined. This means that you can only apply perspective transformations to a subsystem within the ρ that you have defined.
The “jumps" from quantum to classical probabilities also turn out to not be instantaneous jumps. In order for there to be a complete transition from a pure state to a maximally mixed state, the particle has to become perfectly correlated with what it is interacting with. However, there is a limit to how fast a particle can actually change its state, known as the quantum speed limit.
It thus takes time for the correlation to form. You can model the transition from quantum to classical probabilities continuously rather than discontinuously using Kraus operators.
Conclusion
The reduction of ψ to an eigenstate when a measurement is made occurs statistically according to the Born rule, and the Born rule is just as physically fundamental as the Schrodinger equation. Hence, to fully model a physical system’s statistical evolution in all cases, we cannot use ψ but must use ρ. ψ is only applicable in what are known as pure states which is when a system has not undergone such a Born rule transition.
“Measurement” is not a special kind of physical interaction; all interactions constitute measurements. Since all physical interactions constitute measurements, then the Born rule is applicable to all physical interactions. This is not inconsistent because in quantum theory, the reduction of ψ to an eigenstate only occurs from the perspective of the systems involved, not from the perspectives of systems not participating in the interaction.
These are all inevitable consequences of just taking the math seriously at face-value. To deny it inherently requires a modification of the mathematics. These modifications may not necessarily change the predictions, but it would at minimum require introducing new mathematical entities, such as Ψ (the universal wave function), λ (hidden variables), or ξ(t) (an objective collapse). If we just stick to the mathematics of the theory as written without modifying it, the conclusions presented here, of ψ being contextual as well as the reduction of ψ occurring contextually only for systems participating in an interaction, are the only conclusions possible to reach.
If we just accept that conclusion and move on, rather than denying the inevitable conclusions of quantum theory, then none of those additional entities are necessary. There is no need for hidden variable theories, objective collapse theories, or many worlds theories, as all are attempting to “solve” a problem which is not a problem in the first place, but arises from a misunderstanding of the mathematics.
Indeed, you could conceive of pausing a quantum computer halfway through its calculation, when every qubit in its memory is in a superposition of states from your perspective, and play around with these transformations to find the perspective of every qubit in that moment. If the qubit interacted with another such that it became perfectly correlated with it, you will always find that from its perspective, the qubit it is correlated with is not in a superposition of states.
The whole point of a measuring apparatus is to correlate itself with what it is measuring, so naturally, relative to a measuring apparatus, a system will always be in an eigenstate predicted by the Born rule. This is not because it is a measuring apparatus, however, but this occurs even for individual particles.
These perspective transformations are not arbitrary. You can empirically verify these perspective transformations work in real life just by using a person as the basis of the perspective you are translating into. You could carry out much more complex experiments than the Wigner’s friend scenario where particles are constantly placed into superposition of states and then measurements are made on them, and then new superposition of states are created based on those measurement outcomes.
If you had very large sample sizes, you would get a probability distribution of the eigenstates at all points of measurement in the experiment, and you could compare it to the perspective transformations someone outside the experiment would make, and verify they match.
Hence, this is not really even an “interpretation,” but what the mathematics outright says if you take it at face value: if you start with the premise that both the Schrodinger equation and Born rule are fundamental laws of nature, that neither are problematic, that there is no need for adding anything to the theory like a definition of measurement, so on and so forth.
After laying it out, it becomes clear that this is all an obvious conclusion of the principle of complementarity. If I measure a particle’s position, its momentum is now a superposition of states from my perspective. If you then measure its momentum, your memory state (what you believe you saw) must be correlated with the particle’s momentum (what you actually saw).
If you are statistically correlated with a superposition of states, well, that must itself be a superposition of states, i.e., it’s an entangled superposition of states. But, obviously, from your perspective, you wouldn’t perceive that, you would perceive an eigenstate with probabilities given by the Born rule. And that is exactly what a perspective transformation on ρ accomplishes: it gives you the probabilistic eigenstate, the possible eigenstates weighted by their probabilities, for what the other person would perceive.
