Quantum State, Non-Locality, and Interpretation

As someone coming from a materials science and engineering background, I've always approached quantum mechanics with a pragmatic mindset: it's a tool to make useful predictions. My guess is most in applied disciplines have a smiliar perspective. But sometimes I find myself asking detailed questions on the language used to describe the theory itself. I mean what does non-local actually refer to? When you try to find out just plain boring explanations I think your meet with assumptions that both reader and author are on the same standing about the meaning of words. This poor assumption is where I think, or at least I found for me, the confusion surrounding QM stems from sloppy language and mismatched conceptual frameworks, particularly around the nature of the quantum state space (i.e., wave function) and the term non-locality. I'm going to write about how I've come to understand the perspectives on the foundations of quantum mechanics.

Quantum State/Wave Function

I will use the quantum state and wave function interchangeably, but this is ironic because I've just lambasted about the lack of clarity in the language used to describe QM. The specifics are the quantum state is a complex-valued vector in a Hilbert space, while the wave function is the projection² of this quantum state onto a physical space(i.e., our 3D space and 1D time) basis of position eigenstates.

The Core Question: What Is the Quantum State?

At the heart of QM is the quantum state/wave function ($\Psi$), a complex-valued mathematical or physical object, depending on your view, that is used to compute probabilities of outcomes of observables, thats really all its utility! The fundamental nature of the quantum state is where things become difficult to wrap your head around (for a review see [2]). The nature of the quantum state, $\Psi$ has two "beliefs"¹, which are:

1. It is an ontological object, meaning it is something real, part of the physical world we observe and exist in.
2. Or it is an epistemological object meaning it is a representation of our knowledge or information about the world that we inhabit.

The view one takes fundamentally determines how one interprets the foundations of quantum behavior: entanglement and "non-local" (more on this next) correlations. But from a "shut-up and calculate" [1] perspective, it doesn't matter.

Non-Locality

In QM, the word "non-locality" is thrown around often, but it's crucial to disentangle (😆) its usage in physical reality versus mathematical contexts. Non-locality is an underpinning concept of the mathematical formalism of QM; it refers to a property of a mathematical space (not our physical space and time) that has a structure allowing for correlations (i.e., entanglement) which appear coordinated over spatial separations without a direct mediating causal signal limited by the speed of light. This 'acausal' nature of correlations in the mathematical formalism is distinct from, and does not enable violation of the principles of causality in physical spacetime as described by special relativity. How can this be? What such space has that feature? Turns out in information theory, the space of all possible distributions of probability over a finite set of outcomes has such a property.

Digression: Bell's Theorem

Bell's theorem shows that no theory of local hidden³ variables can reproduce all the predictions of QM [3]. John S. Bell formulated this as an mathematical inequality, which if violated, rules out the possibility of combining locality AND hidden variables.

So far all experiments have verified violation of Bell's inequality, and thus we can have a local theory with hidden variables or a non-local theory without hidden variables. The former (i.e., local hidden variables) is not consistent with any observations and thus is not considered a valid theory. This means we must give up either locality or hidden variables (or both). Various interpretations of QM make different choices: Copenhagen and QBism interpretations maintain locality by giving up hidden variables, while Bohmian mechanics maintains hidden variables by embracing non-locality. The theorem doesn't force us to conclude QM is inherently non-local, its more that it depends on your interpretational framework.

Physical Space (Spacetime)

In physical spacetime, which has a continuous geometric structure defined by coordinates, metrics, and causality, locality is built-in. Things happen through the continuity of the space and time. Events occur at points in space and time. Forces and interactions are local as they propagate through space over time and cannot influence distant points instantaneously without violating causality. Putting it more basic, point A cannot go to point B without going through the points in between.

This geometric structure ensures that:

• Causality is respected.
• Influence is constrained by light cones.
• Distances and continuity matter.

Hilbert Space (Quantum State Space)

In contrast, the quantum state lives in Hilbert space, which is a mathematical geometric space, again its mathematical not the physical space that we are familiar with. It has a structure:

• Inner product
• Norms
• Angles

This allows for some very important and convenient mathematical results. But again, it is not directly apart (embedded?) of spacetime, and therefore:

• Does not need to have spatial locality, i.e., the distance between two points can be disjointed in this space.
• Does not need to obey causal propagation in this geometric space.
• Can encode non-local correlations purely as part of its informational structure

I think this is a big reason people often get confused. More over we are usually taught from the position basis projection² perspective of the quantum state onto physical space so it makes this decoupling something we have to backtrack on to understand. Physics PhDs and mathmeticians probably have no issues here, but others I think might (I know I did!). The key point is Hilbert space is geometric, but not physically local in our familiar spacetime. The language used in our courses/textbooks tends to blend physical and mathematical notions carelessly.

Two Views of the Quantum State

Ontological Interpretation

If you treat the Quantum State as a real object in the physical world, then you're saying there exists an object -- the quantum state -- that is part of everything. This assumption then leads to:

• The Quantum State can be non-local in that it encodes information (i.e., entanglement), even if in a projected position-basis that information is spatially separated systems.
• It evolves unitarily even across space-like separations.
• Measurement collapse (i.e., observables/observational outcomes) is a physical process, albeit a non-unitary one.

This leads to ideas like Many-Worlds and Bohmian mechanics [4]. It demands that we expand our notion of physical reality beyond just local spacetime. Space and time are still very much local, but the quantum state is not.

Epistemological Interpretation

If one views the quantum state as just a tool that encodes information (as in Copenhagen or QBism [5]), then it's not a real object. It reflects an observer's state of belief about outcomes.

• Measurement collapse isn't physical, it's a Bayesian update (i.e., we got more info from observing).
• Entanglement reflects only non-local information, not non-local causes.
• No signal travels faster than light—only correlated expectations are updated.

This is why the quantum state can have non-local structure without violating causality. It's not describing the world directly as it is but describing what we know (or can infer) about it.

In this view, quantum mechanics becomes an information theory, with the quantum state as a formal device for encoding and updating expectations.

• This "information" doesn't have to live in physical space.
• It can exist in mathematical abstract spaces (e.g., Hilbert space), where correlations span across what would be distant points in spacetime.
• There's no requirement for continuity, geometry, or causality in this mathematical space.

This aligns with modern perspectives from quantum information theory, where entanglement is a resource, and non-locality is an informational, not physical, phenomenon.

Thoughts on how to communicate this

First, its okay to be verbose and expressive when explaining. We can say things like:

• Our reality of physical geometry means space and time is always local because it is continuous, if we say otherwise then causality is violated and we've got a problem because we don't observe this nor does general relativity tell us this!
• Mathematical geometries which refers to abstract concepts like Hilbert space have a structure that has similar properties of physical geometry (i.e., a metric⁴), but does not have to maintain locality. Things can be far apart in this geometric space and still be connected by the metric (i.e., entanglement).

Making these distinctions should help fade the confusion with these conceptual aspects. I could be wrong, but I think it helps.

The quantum state, depending on your interpretation, is either a very weird, non-local physical object or an informational object living in a non-local, non-physical space. Either way, the math is consistent and making predictions occurs without conflict. The confusion is in the language we use to describe such perspectives, well at least its confusing for me.

Footnotes

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1. Yes, seems like a strange word to use, surely I can't mean ones evidence lacking perspective and feelings about something. But this is exactly what I mean. I think whether you view the quantum state as an ontological or epistemological object is a question of your prior. ↩

2. Hilbert space states can be projected onto physical space through a choice of basis—commonly the position basis $( |x\rangle )$. The Hamiltonian operator, which encodes dynamics and interactions, is expressed in this basis to produce differential equations in space and time (e.g., the Schrödinger equation). This projection allows abstract quantum states to manifest as real-space amplitudes and enables direct modeling of physical interactions in quantum field theory and many-body systems. ↩↩

3. Hidden variables are variables that are not directly observable, but are used to explain the results of experiments, they are not part of the quantum state. ↩

4. A metric is a function that defines the distance between any two points in a space. Its a "ruler" that can be adjusted to work correctly in the space of interest. ↩

References

[1] N.D. Mermin, Could Feynman Have Said This?, Physics Today 57 (2004) 10–11. https://doi.org/10.1063/1.1768652.

[2] Leifer, M. S. (2014). Is the quantum state real? An extended review of $\psi$-ontology theorems. Quanta, 3(1), 67-155. https://doi.org/10.12743/quanta.v3i1.22.

[3] Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V., & Wehner, S. (2014). Bell nonlocality. Reviews of Modern Physics, 86(2), 419–478. https://doi.org/10.1103/RevModPhys.86.419.

[4] Maudlin, T. (2011). Quantum Non-Locality and Relativity: Metaphysical Intimations of Modern Physics (3rd ed.). Wiley-Blackwell. URL.

[5] Fuchs, C. A., Mermin, N. D., & Schack, R. (2014). An introduction to QBism with an application to the locality of quantum mechanics. American Journal of Physics, 82(8), 749-754. https://doi.org/10.1119/1.4874855.

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Entropy: Thinking Beyond Disorder

Every so often, I casually refer to terms like entropy or free energy and forget to really think about what the fundamental meaning of those words are. So, with this post, I aim to focus on the meanings of these terms (i.e., without involving math), particularly entropy.

Entropy: More Than Just Disorder

Entropy is often described in terms of 'disorder' or 'randomness', but these terms, in my opinion, can be misleading and confusing. When we consider epistemology of a system, that is the study of what we can know about a system, entropy is meaningful only from a statistical perspective. This is because, scientifically, we understand that everyday matter¹ is made up of microscopic constituents (i.e., atoms or molecules), yet it's impossible² to know all details about these constituents. Therefore, we rely on a statistical representation of their collective behavior. Entropy is fundamentally about the number of ways a system composed of matter (or information) can be arranged at the microscopic level while still appearing the same at the macroscopic level. It measures the diversity of microstates corresponding to a given macrostate.

As a result, I think it's best not to conceive 'disorder' in the traditional sense, where objects are randomly spread out in a chaotic fashion, but rather as the numerous microstates a system can adopt while maintaining familiar macroscopic properties, such as temperature or pressure. Thus, entropy reflects a lack of specific information³ about individual microstates of a system, while still enabling knowledge of macroscopic properties.

The 'randomness' in entropy relates to the probabilistic nature of microstates. At the microscopic level, the exact configuration of particles in a system follows the laws of probability. This aspect of randomness is key to understanding why macroscopic properties of systems emerge as averages over many microstates. Simply put, we can never truly know what microstate the system is in, only the expectation value of a given microstate while considering all other microstates.

Nature's Tendency To Maximize

What we empirically observe in nature, is that matter at the microscopic and mesoscopic scales, tends towards the largest possible configurational space of microscopic states. This tendency to maximize the number of accessible microstates drives the natural progression towards states of higher entropy and thermodynamic equilibrium. Recall that entropy relates the microstates to the macrostate, and thus nature aims to maximize the number of microstates corresponding to an equilibrium macrostate.

To recap, entropy, often encapsulated in terms like 'disorder' and 'randomness', is more easily thought of as a nuanced measure of the probabilistic distribution of microstates in a system. Understanding entropy in this way helps better appreciate the intricate relation between microscopic thermodynamics (i.e., statistical mechanics) and macroscopic thermodynamics.

Bonus: Free Energy

Free Energy is another term worth digesting. It's best understood as the energy freely available in a system to do work, hence the name free energy. This concept is intertwined with the energy content of a system and its capacity to perform work. Under the laws of thermodynamics, free energy represents the principle that, while energy cannot be created or destroyed, it can be transferred or transformed. Thus, free energy is a conceptual tool that helps us understand how energy moves within a system and its surroundings.

Footnotes

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1. A famous thought experiment that attempts to circumvent the idea of not being able to determine what every constituent particle is doing is Maxwell's Demon. It was used as a way to violate the second law of thermodynamics for an adiabatic system, $\Delta S \ge 0$. ↩

2. Our best scientific theories, confirmed by experiment, indicate the fundamental constituents of matter are more elementary particles than atoms and molecules. However, for the most part, our interactions with the physical world at the smallest scales can be well described by thinking in terms of electrons, atoms, molecules, and the like. ↩

3. In information theory, entropy is used to quantify details about information. So that rather than the microscopic states of matter, one thinks about how information can be represented. This was one of the seminal results put forth by Claude Shannon and I believe inspired by John von Neumann. ↩

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Stranded on a Quantum Branch

I'm going to try to cover an important concept that was put forth in quantum mechanics during the 1970's, which is quantum decoherence. In short decoherence is the process that provides a way to discuss the transition of a system from one described using quantum theory to one which can be explained using classical theories. It basically allows us (i.e. the scientist) to say something about why electrons in an atom are best described quantum mechanically yet a ping-pong ball is not¹. One thing that has piqued my interest in this topic again is a the whiteboard session Sean Carroll and Timothy Nguyen had on Timothy's podcast The Cartesian Cafe. The focus was on many-worlds interpretation, but what stood out to me was how it seemed many-worlds viewpoint intrinsically includes the idea of decoherence. After some reading about the history it seems this is indeed the origin.

Comment

It may be the case that I conceptual get things wrong about many-worlds and decoherence. My goal is to try and use this post as a recitation of my understanding from the video.

What is Decoherence?

Starting with a system in a quantum superposition state, which can be written as a linear combination:

$$|\Psi \rangle = a|\psi_1 \rangle + b|\psi_2 \rangle + c|\psi_3 \rangle + ...$$

Here, $|\psi_1 \rangle$, $|\psi_2 \rangle$, $|\psi_3 \rangle$, etc., represent the different states in the superposition, and $a$, $b$, $c$, etc., are the complex coefficients determining the probability amplitude of each state. In many-worlds its the probability of being on a "branch" and in traditional views its the probability of the collapsed² wavefunction in that state.

Now, suppose this system interacts with the environment, represented by the state $|E\rangle$. Before the interaction, the total system plus environment can be described as the product state $|\Psi ⟩ \otimes |E \rangle$. However, due to the interaction, the system and environment become entangled:

$$|\Psi ⟩ \otimes |E \rangle \rightarrow \alpha|\psi_1 \rangle |E_1 \rangle + \beta|\psi_2 \rangle |E_1 \rangle + \gamma|\psi_3 \rangle |E_1 \rangle + ... + \zeta_{ij}|\psi_i\rangle|E_j\rangle $$

The coefficients $\alpha$, $\beta$, $\gamma$, and are now also associated with specific environmental states, e.g., $|E_1 \rangle$, and are not separable. The system and environment have become correlated, i.e., entangled, through the Schrödinger equation.

Decoherence in the Many-Worlds Interpretation

The Many-Worlds Interpretation of quantum mechanics interprets decoherence in a unique way. From my understanding, each term in the superposition after decoherence can be viewed as a separate "world". Moreover, the observer also becomes entangled with the system and environment:

$$\begin{align*}|\Psi \rangle \otimes |E \rangle \otimes |\text{O} \rangle \rightarrow& \aleph\,|\psi_1 \rangle |E_1 \rangle |\text{O}_1 \rangle + \beth\,|\psi_2 \rangle |E_1 \rangle |\text{O}_1 \rangle +\\ &\daleth\,|\psi_3\rangle |E_1\rangle |\text{O}_1⟩ + ... + \Omega_{ijk} |\psi_i\rangle |E_j\rangle |\text{O}_k\rangle\end{align*}$$

Here, $|\text{O}_1 \rangle$, etc., represent the state of the observer who has become entangled with the system to be in the state $|\psi_1 \rangle$, $|\psi_2 \rangle$, etc. Keep in mind the Schrodinger equation is what evolves $|\Psi⟩ \otimes |E\rangle \otimes |\text{O} \rangle$. This leads to a superposition of all possible outcomes, each corresponding to a different branch in the many-worlds interpretation. It's not entirely clear to me if I'm getting the representation of the environment and observer right.

So what does this represent? Each component would be a branch where the observer is in their respective world, it appears as though the system has collapsed into a definite state. However, from an "God's eye" perspective, all outcomes have occurred, and all observer states exist, each in their respective branch, 🤯.

Question

There is one thing that I'm still very confused about. It could be that I'm getting the details above wrong, however, what is the size of the Hilbert space for $|E\rangle$ and $|O\rangle$? Because if it is large then most certainly there will be branches where things are very strange. Phrasing it another way, if the probability amplitudes for the branches are heavily distributed to say, states $|\psi_i\rangle |E_1\rangle |\text{O}_1\rangle$, then we may confidently say all outcomes/branches will appear to have identical states for the observer and environment. But what if this is not the case? What if there is an observer state where all electrons in the observer's body corresponds to un-bonded atoms, would there be a branch where the observer is a dissociated mess of nothing consistent with a conscious human? Admittedly, even if this is indeed true, the chance that observer is on this branch can be taken to have such a miniscule probability amplitude as to be irrelevant, however, its not zero! To me this is somewhat reminiscent to Schrodinger's cat.

The Role of the Hamiltonian in Decoherence

How does entanglement and decoherence come about? In this context, the Hamiltonian is an essential element. It is the operator that corresponds to the total energy of the system, and it dictates the time-evolution of the system via the Schrödinger equation, that is, how the state of the system changes over time. When a system interacts with its environment, the Hamiltonian leads to an entangled state, which causes the state, $|\Psi\rangle$, of the system to become decoherent.

Trying to Summarize Many-Worlds

In many-worlds, each potential quantum state within the superposition corresponds to a separate "world". What the traditionalist call a quantum measurement is viewed as a branching where all possible outcomes materialize in some Hilbert space³. Decoherence, in this perspective, is the mechanism (via the Schrödinger equation). Once decoherence has occurred, the various outcomes can no longer interfere with each other. It's this process that effectively 'splits' these worlds.

The entanglement caused by the interactions between a quantum system, observer, and surrounding environment, as governed by the Schrödinger equation, leads to decoherence. This process is at the core of the transition from quantum to classical behavior, as well as the basis of the many-worlds interpretation of quantum mechanics. It's an ongoing area of study and a fascinating frontier in our quest to explain reality.

References

[1] Timothy Nguyen, Sean Carroll, The Many Worlds Interpretation & Emergent Spacetime, The Cartesian Cafe podcast. Accessed July 12, 2023. https://www.youtube.com/watch?v=LGtimjuA5gA.

Footnotes

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1. My understanding is that a ping-pong ball has all the degrees of freedom entangled with it's environment, while an atom does not, although it can once the state of an observer comes into play. An important distinction is that things become entangled with the environment, not only interact, because in principal it is possible to have interacting systems who's states are just products of the individual states, i.e., product states. This latter scenario would mean that it would be possible reverse things to get back the individual quantum states. This is like doing the double slit experiment in reverse such that you reconstruct the wavefunctions of the electron and the observer before they interact, which never happens especially if you taken the viewpoint of wavefunction collapse. ↩

2. This is whats referred to as the measurement problem, because if we take the expectation value for some observable/operator like position operator, $|\psi(x)|^2 = \langle \psi |\hat{x}| \psi \rangle$, this only tells us the probability of measuring something at $x$. The problem is that this is not considered a physical process (i.e., the Schrodinger equation), so we need mechanism to localize to a specific state in the superposition of $|\psi\rangle$. Mathematically we can do this with projection-valued measures which introduces a way to describe measurement. The important point to keep in mind this is not part of the postulates of quantum mechanics and therefore is more like a addendum. Many-worlds has this baked via the perspective of branching. ↩

3. The more fundamental question that arises for me is, is Hilbert space "The Universe"? Is it what is underlying reality? Basically is Hilbert space the fabric of the universe? ↩

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