Down memory lane: Quantum Computing

I spent about 2.5 years working on variational quantum algorithms for noisy intermediate-scale quantum (NISQ) devices [1]. The question was straightforward: can we do anything useful with these noisy, small-scale, "universal" quantum devices? Here useful was typically to mean faster, but could also mean solve far too complex problems for classical quantum chemistry.

The short answer I came to was: not in any way that clearly demonstrated a speedup over well-tuned classical solvers. The longer answer: you can get them to run, get numbers back, even match the literature, but there's no clear, reproducible speed-up. Most of the effort is in engineering the system to produce anything coherent, not in pushing computational frontiers. I'm not sure if this is a good thing or a bad thing.

So what is the state now? Are there any clear signs of utility for variational quantum algorithms? Are there any new quantum algorithms for chemistry or physics that require error correction but have proven advantage over classical computers? My guess is the answer is no not really, but I'm not sure and for now don't have the time to read up and investigate.

What I saw with Universal Quantum Computing in the NISQ Era

Universal quantum computing here means we have a qubit, quantum information analog of digital bits, and we can do arbitrary single-qubit rotations and controlled two-qubit gates to produce logic operations that put qubits into superposition and can entangle them. We can also compose them into circuits that approximate any unitary evolution.

For famous algorithms like Shor's or Grover's, the path to usefulness is clear if you had fault-tolerant quantum computing with sufficient qubits. But in the NISQ setting, VQA [2] or QAOA are the only viable options. There might be some new class of NISQ-like algorithms, I'm not sure, but it's probably still safe to say VQA and QAOA are dominant.

Lets say that you moved beyond the NISQ era, which may well be happening, and thus the noise and decoherence are under control, there's still the ansatz problem, that is how do you pick a circuit structure that can efficiently represent the solution state in the first place?

The Ansatz Problem

An ansatz is a parameterized (quantum circuit) guess for the form of your quantum state. In VQAs, it's the fixed sequence of gates you tune with a classical optimizer. In phase estimation (PEA) or Hamiltonian simulation, it's often the state-preparation step for your quantum algorithm.

The difficulty is balancing expressivity and feasibility:

• Too shallow: can't represent the physics; optimizer converges to the wrong state.
• Too deep: hardware noise kills you in NISQ; in fault-tolerant hardware, depth inflates T-gate and qubit costs.
• Too generic: risks barren plateaus [3].
• Too problem-specific: works only on one Hamiltonian.

In PEA, the ansatz problem just shifts to the state-preparation step. You might nail the controlled-unitary and inverse QFT, but if you can't efficiently prepare the eigenstate, you'll likely get garbage.

So what did I work on?

Most of the creativity in the research I did comes all from my colleague/co-author, I was mostly involved in domain application, instrumentation, and analysis. There were two works [4-5] but the one I'll highlight is probably the least interesting: we developed a hybrid quantum–classical eigensolver without variation or parametric gates[4]. The idea is to project the problem Hamiltonian into a smaller subspace, measured term-by-term with short circuits, and diagonalized classically.

This allowed us to:

• Extract ground and excited states for small molecules (BeH₂, LiH).
• Validate against exact diagonalization.
• Run on the quantum hardware at the time (i.e., IBM devices).

It avoided long, problem-tailored ansatz circuits, but the choice of basis in the subspace projection is still a hidden ansatz.

A Self-Critique of Our Hybrid Eigensolver Work

We avoided variational loops and deep, problem-specific ansätze: no parameterized circuits, no barren plateau optimizers. The issue, though, was

1. We dodged the ansatz problem: The reduced-space basis is still an ansatz, thus performance depends on making a smart choice. We didn't quantify sensitivity.
2. Hardware vs. simulation gap unexplored IBM runs matched noiseless simulations, but was this due to noise resilience, shallow circuits, or luck?
3. Thin classical comparisons We used exact diagonalization due to simplicty of the chemical system and basis. Real claims require benchmarks vs. DMRG [6], coupled-cluster, etc.

Why NISQ VQAs Struggle

There has been a lot of work on this in the last few years and I'm not fully up to date but this is what I gather:

• Noise vs. depth: deeper means more decoherence.
• Barren plateaus: gradients vanish exponentially with qubits.
• Optimizer instability: hardware drift, shot noise, optimizer quirks.
• Classical competition: tensor networks, DMRG [6] often scale better [7].
• Ansatz rigidity: wrong ansatz wastes all gates and shots.

Summary of My Position

I'm not a seasoned quantum algorithm researcher, but from my limited research, I see NISQ-era of VQAs as mainly useful for benchmarking, with their progress limited by the challenge of designing effective ansätze. In quantum chemistry, there is little convincing evidence so far that VQAs offer practical advantages¹. Digital quantum simulation is highly flexible but comes with significant resource costs. Analog simulation, which directly emulates physical systems, is already useful in certain specialized areas but will always be a niche. Looking ahead to fault-tolerant quantum computing, real breakthroughs may be possible, but efficient state preparation will remain a central obstacle.

Footnotes

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1. Maybe this has changed but I wager probably not and the paper by Lee et al. [7] is a good indicator. ↩

References

[1] J. Preskill, Quantum Computing in the NISQ era and beyond, arXiv (2018). [2] M. Cerezo et al., Variational Quantum Algorithms, arXiv (2021). [3] M. Larocca et al., Barren Plateaus in Variational Quantum Computing, arXiv (2024). [4] P. Jouzdani & S. Bringuier, Hybrid Quantum–Classical Eigensolver Without Variation or Parametric Gates, Quantum Reports 3, 8 (2021). DOI [5] P. Jouzdani, S. Bringuier, M. Kostuk, A method of determining molecular excited-states using quantum computation, MRS Advances 6 (2021) 558–563. DOI [6] S. R. White, Density matrix formulation for quantum renormalization groups, PRL 69, 2863 (1992). [7] S. Lee et al., Evaluating the evidence for exponential quantum advantage in ground-state quantum chemistry, Nat. Commun. 14, 1952 (2023). DOI

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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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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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Ramble: Science Shill & Semantic Bias

 While doing my regular reading-up on quantum computing, I came across a tweet that referenced a recent Joe Rogan podcast with Michio Kaku as well as the blog post on Not Even Wrong. I have never read any of Michio Kaku's books but I always thought he was really good at communicating popular physics, that's until I watch the JRE episode. Yikes! why did Micho Kaku talk so confidently about technical details he seems so unfamiliar with. His talking points on quantum computing seem off-base. It's not that he is completely wrong about the prospects of quantum computers it's just he makes such grandiose propositions that don't seem to have a clear line of thinking. For example, I'm going to paraphrase, he mentions using quantum computers as fact-checkers for generative LLMs, like ChatGPT. Okay, what are you specifically thinking about? Why is a quantum computer ideal for this or better than a classical computing approach? Is he referring to some kind of complexity problem related to watermarking AI outputs where only a quantum algorithm could find the solution in polynomial time? Without these kinds of additional details, it seems like he just is making things up! These little statements by Kaku were all throughout the episode and made him seem like he was just a  hype man for science, not a knowledgeable professor or researcher.  Prof. Woit over on Not Even Wrong basically lambasts Kaku about this behavior.

My bias with semantics in QM

The other thing Kaku kept saying during the JRE episode when referencing quantum mechanics was "parallel universes". For some reason this phrase has always bothered my ears; it's pretty silly I know and probably of little importance, or maybe I'm wrong here. I just don't like the word "parallel" mainly because I think it conjures up notions of communication or connections between "parallel" universes, but this would be misleading in my view. If I want to take a simple example to explore why I think this way, then take the Bell quantum state of two qubits:

\begin{equation*}|\Psi\rangle = \frac{1}{\sqrt{2}}\left(\vert00\rangle +|11\rangle\right)\end{equation*}

What about the two states $\vert 00\rangle$ and $\vert 11\rangle$ are "parallel"? Nothing to me! For one, the inner product between the two basis states is zero because they are orthogonal: $\langle 00|11 \rangle = \langle 11|00\rangle = 0$, so nothing "parallel" about them there. Yes, you may say well each qubit is in a superposition state and so they have a "parallel" configuration, but my premise is this is misleading because in my view it's all about the configurations provided by Hilbert space. So you say, "Well then what do you propose? Or why does this even matter it's just semantics!", for which I say: your right, it probably doesn't! but I think popular science communicators may be providing mental imagery to the broader public that is fanciful. Someone should do a survey of none STEM individuals and see what they say comes to mind when the words "parallel universes" is used. This goes back to my concerns at the beginning of this post.

Example of a lenticular image. All images exist in the same physical space but depending on what angle you view it from you see different information. We can think of each version of Sayin as a view of a basis state but no information is exchanged nor are they "parallel"

What I'm trying to say with my ramble is that we should use words that better articulate what we interpret as our best theory to describe. My personal opinion is that the word lenticular, coming from lenticular imaging and which is a process used to create the perception of multiple images on the same print (see figure above). At different viewing angles, you only see one image but viewing the print at a set of angles will show superimposed images. This is what I think the quantum state is like. Nothing is parallel only that certain "views" show specific information; here a view is a stand-in for one of the basis states in the quantum state/wavefunction. You could argue that no one is familiar with the word lenticular, and you would be right, but we can use more common descriptors like "tilt-card". One thing that could also cause trip-ups is that a lot of people confuse the lenticular prints for holographic ones, they are different. In holography, the light field is what is captured whereas in lenticular prints it's the use of interlacing and the lenses.

I should probably provide a conclusive statement on what I want to replace "parallel universes" with. Let me mention that "many worlds" is much better and I definitely like "branches of the quantum state". However, I would probably wager for something like the Tilt-Card Universe. Nothing parallel about the name, hopefully elicits thoughts of multiple "views", and cements the concept that all exist at once.

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Book Review: Quantum Entanglement, Jed Brody

I just finished reading through this short monograph on quantum entanglement. The approach taken by the author is to provide what quantum entanglement is through conceptual examples. There are no wave functions or quantum states discussed in this book. At first, the reader is introduced to two very important concepts in the philosophy of physics; realism and locality. In realism, the assumption is that any physical objects have properties regardless of whether another object with agency (i.e., a person) is observing that object. A typical example of this concept is the following questions:

Does a falling tree in the forest make a sound when no one is listening? 

Realism says yes, it does. In the case of the tree, it has a center of mass that gives the tree some gravitational potential energy that upon falling is converted to kinetic energy and then generates sound waves in the air once it hits to ground. The tree had mass, potential energy, and kinetic energy which according to realism exist objectively. The opposing view is that it was the sound wave came into existence because an agent was listening. This seems absurd and it is in classical physics, but not necessarily in quantum physics.

Locality refers to the fact that observing, measuring, or disturbing objects in a region of a space does not affect other objects at arbitrary distances in that space. Here, I'm using space in an abstract sense not necessarily a Euclidean 3D space. I do note that in the book the discussion of locality is with regard to distances in 3D Euclidean geometry but I think I'm correct in that locality applies to non-Euclidean 3D spaces and this would be the more general statement. Locality is a pretty important concept in physics and is one of the reasons we got the famous EPR paper from Einstein. 

After the book presents these two concepts it gradually moves into the concept of hidden variables, that is properties of objects that can change aspects of the object when observed, but yet the variables themselves are never observed. Hidden variables satisfy realism. Much of the subsequent chapters present examples that lead to the famous Bell inequality which arises due to correlations in probabilities. The bell inequality needs to be satisfied for a theory to contain locality, if it is violated the theory is non-local. As it turns out, at least to our ability to experimental test the theory of quantum mechanics, it is a non-local theory without hidden variables because. All experiments that have been conducted to date violate Bell's inequality and suggest that correlations are instantaneous within the quantum mechanics framework. It should be noted that you could have a quantum hidden variable theory (i.e. Bohemian mechanics) that is non-local which would describe experimental results, but I guess the argument against this is why introduce hidden variable theory if a non-hidden variable theory doesn't provide any additional clarity other than satisfying realism.

It is pretty well documented, or at least we are made to think, that Einstein had serious issues with the non-local (dubbed "spooky action at a distance") behavior of quantum theory as well as the mainstream interpretations not satisfying a realism philosophical perspective. More specifically, the Copenhagen interpretation posits that the wavefunction/quantum state is more of a mathematical tool and is not necessarily a physical object since it only provides a way to extract probabilities of observable properties.

Going back to the book, chapters 3 and 4 provide different and simple experimental setups that look at probabilities and their correlations to arrive at Bell's inequality. The author then reminds the reader that quantum mechanics violates this inequality. Chapters 1-4 are written in a direct and comprehendible manner, but the truth is, I find it actually easier to understand the Bell inequality and violations of it by actually following the simple Linear algebra of quantum theory.  Trying to think through all the words describing the setup and outcomes can become burdensome. Given  the current focus on quantum computing, there are a lot of good books that go through the same results using simple linear algebra. I think it would have been easy to introduce most readers interested in this book to the basics of a qubit, Hilbert space, and corresponding operations, which could help readers understand these concepts more easily.

Chapter 5 goes through the potential inconsistencies of quantum mechanics with special relativity. Personally, I found this chapter was not delivered in the most impactful way, but it addresses the original concerns of physicists.  The final concluding sentence that indicates everything is okay in the end is:

"... the linkage between entangled particles conveys neither mass nor messages"

The author ends the book with a chapter regarding realism and its validity.  I think this is the best section of the book. The author gives their thinking on the topic of local realism by stating:

"The only fact that's (almost) certain is local realism cannot account for measured results"

Thus, local realism is a dead concept in the author's eyes As frustrating as it feels, I would agree with the author. The remainder of the chapter deals with interpretations of quantum theory from philosophical perspectives and you get a nice concrete quadrant table to decide what path to take, namely:

Find falsehoods in assumptions	Abandon locality & realism
Abandon locality & keep realism	Abandon realism & keep locality

I'm not going to go through and explain each of these because I want to leave some excitement, but I think this is the most interesting part of the book. 

I recommend reading this book if you're going to be studying quantum mechanics in any way because it will help with some of the philosophical thinking behind the theory. The reading is extremely accessible to any background and it is very short making for a good weekend read. Here's the book:

MIT Press Store

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The Framework of Quantum Theory

This post is my rewording of the postulates of quantum mechanics/theory. I'm trying to do it first without using mathematical notation; we will see how good I do. Then I'll add the math.

Many texts have slightly different ways of presenting the postulates of quantum theory and it usually depends on which domain the framers are coming from. I particularly like the quantum computing perspective because its abstract from any physical system and is strictly discussed from the mathematics. References are provided at the end of the post.

What is meant by quantum theory?

To answer this question I would like to first  understand what is meant by theory, I'll use the basic definition:

An uncertain belief or a system of ideas intended to explain something, usually using a general and abstract framework.

You may ask how does this differs from a law. Well to be honest, the difference between a "law" and "theory" seems mostly semantics, but usually a law is based strictly on factual observations, whereas in by the definition above a theory is a hypothesis or concept used to explain observations. 

So how does quantum theory get formed, well we invoke a "system of ideas" in the form of postulates. A postulate is defined as:

Assume something to be true (i.e. axiom) or factual-based for use as a foundation for reasoning.

Therefore what is done in quantum theory is to put forth postulates that act as the foundation of the theory and are applied to every system we are interested in describing within the theory. Now listing  out what the pioneers of quantum theory took as the postulates.

Postulates of quantum theory 

Depending on which text you reference there are may be a different number of postulates (5-7) presented, here I'll present 5; sometimes they are in a different order.

Below I will first state the postulates in general language and then reiterate them in a stronger mathematical sense.

1. A physical system can be completely described by a quantum state existing in a vector space (i.e. Hilbert space). The information extracted from the quantum state about the physical system is given in terms of probability.
2. To interpret a classical physical quantity (e.g. momentum) associated with a quantum state, a corresponding operator exist that acts upon the quantum state.
3. The realization of observing an operator, that is measurement of a physical system described by a quantum state acted upon by an operator, gives a measurable outcome (real-valued). In addition, the quantum state collapses to a specific point in vector space corresponding to the measured value.
4. The average value measured of a physical quantity associated with the quantum state is provided by the probabilistic expected value (i.e., expectation) of the operator.
5. The deterministic time evolution of the quantum state is given by the Schrödinger equation.

The framework is not too crazy, you just have to buy into the fact that there exists an abstract object called the quantum state (p-1) that evolves according to the Schrodinger equation (p-5), and any information in the classical physical sense is obtained through operators that can be measured. 

One side note, we probably don't say " the laws of quantum mechanics" because as mentioned laws are derived from factual observables, and in the case of the quantum state, we actually never observe it directly. Remember postulates 1-3, tell us we only see real-valued outcomes with some probability. As a matter of fact, we don't even know if the quantum state corresponds to an epistemic (i.e. knowledge we have) or ontologic (i.e., the nature of reality) perspective. Stated more succinctly, does the quantum state correspond to information about reality or reality itself.

A more mathematical form

Here I'll restate the postulates but with a more mathematical signature, but first I'm going to define the core concept in quantum theory, the quantum state:

$|\Psi\rangle$ is the quantum state which is a vector in Hilbert space — a finite or infinite dimensional space that contains inner products ­— and the amplitudes of each component of the quantum state vector  are complex valued. The specific notation, the Greek letter $\Psi$ sandwiched between a vertical line and right angle, indicates that this is a vector in Hilbert space. However, in many physics and chemistry oriented problems the quantum state is represented in a position basis, i.e., the amplitudes of the quantum state vector depend on position, and therefore we can define a continuous function $\Psi(\bf{r}) = \langle \bf{r} | \Psi \rangle$ ubiquitously dubbed the "wave function". The term $\langle \bf{r} |$ is the complement quantum state in position basis, i.e. the  quantum state is written in terms of each unique position value given by the amplitude and orthogonal vector in Hilbert space.

and now the postulates with some mathematical notation: 

1. A physical system is described by a quantum state, $|\Psi\rangle$,  that exist in $\mathcal{H}$ space who's elements are $\mathbb{C}^N$ and can be written in terms of linear combinations of elements in $\mathcal{H}$, for example, $|\Psi\rangle = \sum_i c_i |\psi_i\rangle$. The quantum state carriers information about the physical system which is extracted with reference to probability, given by $|\Psi|^2 = \langle \Psi | \Psi \rangle $. 
2. To interpret a classical physical quantity associated with a quantum state, $|\Psi\rangle$, a corresponding, $\hat{\mathcal{O}}$, exist that acts upon the quantum state linearly, $\hat{\mathcal{O}}|\Psi\rangle$. The operator is "moving" the quantum state around in Hilbert space.
3. The realization of observing an operator acting on a quantum state is framed in terms of positive operator-valued measure (POVM). If we have the operator, $\hat{O}$, we say that the operator can be defined as a sum of projector operators, $\hat{O}=\sum_i \Phi_i$, where the projector operators act on a subspace and are orthogonal. It then follows that the quantum state acted upon by each projector "collapses" to the state $|\Psi^{\ast}\rangle=\frac{\Phi_i |\Psi\rangle}{\sqrt{\langle \Psi | \Phi_i | \Psi \rangle}}$  with probability $\langle \Psi | \Phi_i | \Psi \rangle$. Keep in mind that what we measure physically are the eigenvalues with outcomes given by the probability.
4. The probabilistic expected measured value for an operator, $\hat{O}$ given a quantum state $\Psi$, will be given by the expectation value, $\langle O \rangle = \langle \Psi| \hat{O} | \Psi \rangle$.
5. The governing evolution of the quantum state is given by the time-dependent Schrödinger equation, $i \hbar \frac{d}{dt} |\Psi(t)\rangle = \hat{H}|\Psi(t)\rangle$, where $\hat{H}$ is the Hamiltonian operator. The time-dependent Schrödinger equation is due to the fact that the time evolution of the quantum state must be unitary (i.e. preserves inner products) and is given by $\hat{U}(t) = e^{-i  \hat{H} t / \hbar}$.. 

Finally I'll conclude with a quote from Paul Dirac, a interesting physicist with tremendous mathematical acumen.

“God is a mathematician of a very high order and He used advanced mathematics in constructing the universe.” — Paul A. M. Dirac

 

References

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Not so hidden: A clearly written popular quantum physics book by a masterful communicator

$^\dagger$My Commentary

First of all, I am a very big fan of Sean Carroll and in general many of the fantastic science and technology communicators in the 21st century. As a society we are in debt to their efforts and time in aiding in understanding the fascinating and intriguing universe we live in. What I enjoy about Sean Carroll's approach to communicating physics concepts and topics is his unmatched clarity in delivery and pace of speech. If you've every listened to his podcast (Mindscape) or others that he has been a guest on, you know what I'm talking about. His most recent popular physics book titled "Something Deeply Hidden: Quantum Worlds and the Emergence of Spacetime" continues his trend of excellence in science communication.

In this book, Sean focuses on the foundations of quantum physics and how we ( the community scientist) have become complaisant with the "shut-up and calculate" mentality when dealing with the quantum realm. The issues stems from the early pioneers of quantum mechanics who could not make sense of the predictions from the mathematics and the experimental observations, namely, that the quantum object/information we call the wavefunction, does not manifest as described by the math when measured in the lab. Lets throw some math into the mix just to make things clear. In nonrelativistic quantum mechanics we have an equation which describes the relationship between the time evolution of the quantum state (i.e. wavefunction) to the evolution of energy content of a system given that quantum state; this is the what we are told the infamous Schrödinger equation:

$$ i\hbar \frac{\partial} {\partial t} | \Psi \rangle= \hat{H} | \Psi \rangle$$

The quantum state function, $\Psi$ is called the wavefunction because in many cases it has a functional form that reassembles wave-like characteristics. This wavefunction differs from our intuitive notion of classical waves in that the amplitude is a complex number.  Sean Carroll's argument is that the wavefunction is all information needed to describe a quantum system. Furthermore, he argues that we take this equation at face value for what it tells us, namely, that given a quantum wavefunction we can describe its evolution deterministically. This is a key statement, because in popular science you may hear that quantum mechanics is a probabilistic theory, that is only true  if we are concerned with knowing additional information about the system using the wavefunction. For example, if we want to know the position, momentum, or energy then we can only speak in terms of probabilistic outcomes of those observable. But the wavefunction always evolves deterministically via the Schrödinger equation. For example if we want to know the probability expectation for observing  or measuring the momentum of a wavefunction describing a particle, we would write something down like:

$$ \langle \Psi | \hat{p} | \Psi \rangle $$

this equation provides us with a mathematical result about the momentum representation of the wavefunction  in a probablistic manner. It is  beyond the scope of my intent for this blog but the reason we can't say that the observed momentum of the wavefunction  $\Psi$ is exact, is related to the fact that the wavefunction is a superposition of equally valid solutions in what is known as Hilbert space. 

Now the main focus of the book is on an alternative understanding for the measurement catastrophy in quantum physics, that is to say, when experiments are performed on quantum systems we don't get the entire probability distribution of the wavefunction for an observable/expectation as an output. What we get is a single data point from that sampling space. If we conduct enough experiments, then of course we recover the distribution. But why don't we get the entire wavefunction probability when we measure? The historical and mainstream thought on this is that something happened by which when an observer (e.g., the human eye or a digital sensor) measured the quantum state/wavefunction so that it "collapased" into a single value. Now you say "What do mean? What forced the function to collapse?", yes this is indeed a strange phenomena. The Schrödinger equation nor any other mathematical interpretation tells us anything about a wavefunction "collapse". For many years I never really thought about this, but more recently it really is bothersome that a quantum state just "collapses" to a single value as if something forced it for which we don't know anything about. Albert Einstein's thought on this was that the forced collapse is due to local hidden variables; things we are unable to identify as being part of the system and are locally causal.

Now Sean Carroll's approach is more epistemic, we know we have this quantum mathematical object called the wavefunction and every thing in the universe can be described by it, so what happens when the quantum system I am describing interacts with an observer who is also treated as being a quantum system. The outcome is that we get  parallel quantum states that are very much deterministic and in existence, but having different probabilistic outcomes (I think this is how I understand it?). In other words, the act of quantum systems interacting produces many outcomes that in a sense occur in parallel worlds, hence the many worlds. To be clear we don't need to think of the same physical space being occupied, but rather that in some abstract representation of many isolated outcomes have occurred with validity. This approach goes by the Everettian or Many Worlds interpretation.

The main tenants of the the many worlds argument are 1.) we should not try and interpret the meaning of the Schrodinger equation but just follow the mathematics as providing what is real and 2.) don't select which systems behave quantum mechanically, assume every physical object in the universe can be described by a quantum state.

Although I very much appreciate Sean's insight and excellent introduction to this foundation quantum physics vantage point, my own human bias doesn't want to agree. Its not that I don't think its a valid understanding of the outcomes of the Schrodinger equation, but more that it leaves me wondering about the other "branches" of the wavefunction. For example, can one roll-back time to traverse a new branch? This should be possible since the time evolution operator in the Schrödinger equation is a unitary operator.

There are two things I should mention, 1.) I'm not a quantum physicist by training so my understanding of the topic could have gone wary, 2.) I haven't finished the book.

$^{\dagger}$ I haven't finished the book yet so this is a partial commentary.

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Simple Example of Entangled States & My Thoughts

Entanglement

In the famous EPR paper, the primary argument is related to the bizarre non-local consequences of entangled quantum states. Einstein's main objection is related to the non-local (spatial) characteristic of entangled states. He suggested that quantum mechanics must be an incomplete theory and argued that our experience in nature appears local. He therefore concluded that hidden variables or additional degrees-of-freedom most likely exist and are not captured by current quantum theory formulations. Unfortunately, well maybe fortunately, Einstein appears to be incorrect about this as exemplified by the work of John S. Bell. The main outcome of Bell's work is that non-local theories can exist without the need for hidden variables and thus entangled quantum states are not restricted by spatial locality. I won't dive to deep into the foundations of quantum theory, but rather just look at a simple mathematical argument for why entangled states exist.

Simple Example

Let me first start with describing a quantum state (i.e. wavefunction) with basis vectors$^*$:

$$|0\rangle = \begin{bmatrix}
1 \\
0
\end{bmatrix} ,\;
|1\rangle = \begin{bmatrix}
0 \\
1
\end{bmatrix}
$$

This is a orthogonal basis set and corresponds to a Hilbert space (abstract function vector space) of $2^n$, where $n$ is the number of particles or objects. Each particle or object can be represented by a linear combination vectors in the Hilbert space. The basis corresponds to that used to describe quantum bits or qubits. 

The basis states can be used to form product basis states, which are given by the tensor product, such that for a system of two objects the  Hilbert space  is $2^2 = 4$ and can be written as:

$$
|00\rangle =\begin{bmatrix}
1 \\
0 \\
0 \\
0 \end{bmatrix} ,\;
|01\rangle =\begin{bmatrix}
0 \\
1 \\
0 \\
0 \end{bmatrix} ,\;
|10\rangle =\begin{bmatrix}
0 \\
0 \\
1 \\
0 \end{bmatrix} ,\;
|11\rangle =\begin{bmatrix}
0 \\
0 \\
0 \\
1 \end{bmatrix} \;,
$$

where the state of a particle/object is represented by a linear superposition of the product basis states, each having a complex amplitude (can be real if the imaginary part is zero). Now we can do something interesting, what if we take the product basis just written above and construct a  potential wavefunction with the following features:

$$ |\Psi\rangle = \frac{1}{\sqrt{2}} \left( |00\rangle + |11\rangle\right).$$

Since we know from that the states $|00\rangle$ and $|11\rangle$ can also be written as product states, as shown above, lets do the following:

$$|\psi_1\rangle = \alpha_1 |0\rangle + \beta_1|1\rangle $$
$$|\psi_2\rangle = \alpha_2 |0\rangle + \beta_2|1\rangle. $$

Now from these definitions lets take the tensor product:

\begin{align}
|\psi\rangle &= |\psi_1\rangle \otimes |\psi_2\rangle \\
 &= \alpha_{1} \alpha_{2} |0\rangle|0\rangle + \alpha_1 \beta_2 |0\rangle|1\rangle + \beta_1 \alpha_2 |1\rangle|0\rangle + \beta_1 \beta_2 |1\rangle|1\rangle \\
 &= \alpha_{1} \alpha_{2} |00\rangle + \alpha_1 \beta_2 |01\rangle + \beta_1 \alpha_2 |10\rangle + \beta_1 \beta_2 |11\rangle .\\
\end{align}

So we now have the given state $|\Psi\rangle$ and the product state $|\psi\rangle$, and if we compare the terms we immediately observe that:

$$\alpha_1 \alpha_2 = \frac{1}{\sqrt{2}} \; \text{and} \;  \beta_1 \beta_2 = \frac{1}{\sqrt{2}}$$

However, since for an orthonormal basis we must have that, $||\Psi\rangle|^2 = 1$ , it then has to be such that:

$$\alpha_1 \beta_2 = 0 \; \text{and} \; \beta_1 \alpha_2 = 0.$$

This would be a contradiction though because its is not feasible  given $\alpha_1 \alpha_2 = \frac{1}{\sqrt{2}}$ and $\beta_1 \beta_2 = \frac{1}{\sqrt{2}}$. We therefore say that quantum state,

$$ |\Psi\rangle = \frac{1}{\sqrt{2}} \left( |00\rangle + |11\rangle\right),$$

represents a quantum entangled state, and cannot be written in terms of product states. Moreover, the determination or expectation of a single particle/object in this entangled state immediately tells us about the state of the other particle/object. For example, if the expectation value of particle 1 is $0.5$ in $|0\rangle$ or $|1\rangle$ then we will know with unity the state of particle 2 is. 

What does it mean

The feature of entanglement is exclusive to quantum mechanics. One way I try to think of things is that quantum states can be composed of two types, those that exist due to the "combination" of quantum states (i.e., product states) and states which manifest as unique quantum solutions which are indecomposable into any other state or description. This fact, of entanglement is one of the great mysteries in quantum physics but its existence is enabling fantastic technologies.

There are two quotes from Niels Bohr's that I think fit well with this post, the first is:

"Einstein, stop telling God what to do [with his dice]!"

-Niels Bohr's, A response to Einstein's assertion that "God doesn't play dice".

This was in response to Einstein's dislike for many of the unexplained conundrums of quantum mechanics. The second quote:

"Anyone who is not shocked by quantum theory has not understood it."

-Niels Bohr's, The Philosophical Writings of Niels Bohr (1987),

Which captures the unexpected and possibly bizarre way of thinking one needs to succumb to in order to appreciate the predictive power of quantum theory (i.e. mathematics and interpretations). Despite this lack of comfort, quantum theories have made extremely accurate predictions and have been validated numerous times through meticulously controlled experiments.

I personally still find the foundations of quantum physics to be nebulous, but this is probably due to my own fallibility. To me its is unsatisfying that we do not not know the true meaning of the wavefunction or more specifically what is the meaning of a Universal wavefunction? Maybe its to complex that we will never know. Then there are questions about why entanglement exist, is it necessary to be consistent with physics as a whole (i.e., General Relativity)? Have we dismissed other understandings to quickly? Does the dendritic many-worlds interpretation originally proposed by Hugh Everett describe reality? What about revisiting non-local hidden variable theories such as Bohmian mechanics (also known as pilot-wave theory and De Broglie-Bohm theory)? How does non-locality make sense, is our notion of space being innate to the universe incorrect? Does space emerge from something else?

All of these question intrigue me, however, I have only scratched the surface  and look forward to learning more about research focused on the foundations of quantum physics. There is a considerable learning curve and start-up time  for me since my formal training is not in theoretical physics, but that won't stop me.

$^*$ The basis vectors are represented using the bra―ket notation pioneered by Paul Dirac. This is a very useful notation but may be unfamiliar to materials science people given that our solid state physics education, to my knowledge, never goes over this because we typically always deal with quantum states (i.e., wavefunctions) in a position or wave-vector basis, for example, $\psi_i(x) = A_i e^{\alpha_i \left(x-x_o\right)^2}$ or $\psi_{k}(x) = \frac{1}{\sqrt{V}} e^{ik\cdot x}$. Therefore the more broad and useful bra―ket notation goes unused.

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Discussion on Quantum States & Corresponding Electrons

Possible Conceptual Issues

My intent of for this blog post was to explore pedagogical approaches to explaining solutions (i.e.  wavefunctions) for the time-independent Schrodinger equation and how electrons are associated with those solutions. In its present form, the blog is not polished and may have erroneous statements, so keep in mind the blog just follows my train of thought. I apologize in advance for confusion or mistakes. For clarity I will try to avoid any gobbledygook (i.e., unnecessary nomenclature).

Quantum States$^\dagger$: A Train of Thought

The first thing to discuss is that we are interested in explaining the motion and interactions of our – humans that is – notion of particles that make up matter. The main confusion is that our intuition/experience tells us to think of the world as particles, however, the whole premise of quantum mechanics is that the motion and interactions cannot be described by classical point particles but rather by "wave-like"$^\mp$ states which are solutions to quantized operators/transformations. It behooves us then to concede that with our current mathematical tools (i.e. equations of quantum mechanics) we are told that the material world, at its smallest scales, is in  "wave-like" states. What does "wave-like" states even mean? It means that the mathematics used to treat the motion and interactions of what we call particles$^\ddagger$ within the theory of quantum mechanics (QM) looks similar to the mathematics used to describe the motion of waves. The difference from the classical picture of waves arises in the nature of the quantum mechanical solutions, that is, the "wave-like" states and their amplitudes exist in a complex number space.

The essence in interpreting the physical meaning of the "wave-like" state solutions, is an additional layer of struggle and falls within the philosophical interpretation of quantum physics. For example what is the meaning of the "wave-like" state if when I interact with the physical world I always observe or measure particles — this is what experiments show us. This bewildering problem still exist even though it has been nearly 100 years since the conception of QM. So why don't we know the meaning? I'm not sure. Its not to say that many great minds haven't proposed insightful ideas, but that there is no clear victor. The most widely accepted and used approach is commonly referred to as the Copenhagen interpretation, which was primarily pushed by Niels Bohr. The main premise is that it is only the magnitude of the complex numbered "wave-like" state that we can assign meaning to as a kind of probability about our [human] intuition of where particles could be. Then there are the many-worlds theory initially proposed by Hugh Everett, pilot-wave theory of David Bohm and De Broglie, and the spontaneous wavefunction collapse theory (GRW). It seems to me that the complexity of trying to explain the philosophical implications of the quantum "wave-like" state is something to admire given that it eludes some of most brilliant of minds. The general theme in physics seems to focus on QM (i.e., wavefunction) predictive description of reality not the inference of its existential meaning.

Okay, lets shift back and think about some examples where waves describe the behavior of a classical physical system. The two most simplest to start with are a vibrating violin string  and drum membrane. In both cases a medium (the string and drum membrane materials) acts as a field — a property given at a point in space, here the property is the elasticity of the material. It turns out that because the string and drum head are pinned down at the edges, only certain characteristic vibrations can exist, these are called modes or in mathematics eigenstates (eigen in German means self/characteristic). The characteristic vibrations are a state that the system (i.e., string or drum head) can exist in, but they do not necessarily always exist, it depends on how the system is excited (i.e. plucked or banged). In many instances the overall vibration of the string or drum head is a combination of characteristic vibrations, an analogy one could think of is how we combine the colors red and green to get yellow. We can combine characteristic vibrations to get a new vibration, this is commonly called linear superposition or combination. 

So now that we've discussed a little about waves, lets revisit the "wave-like" aspect of quantum mechanics. We start by stating that the total motion and interactions of a quantum system is described by a "wave-like" state or wavefunction, $$\Psi\left(\chi_1,\chi_2,\chi_3,\cdots,\chi_N\right)$$,

where $\chi_i$ is a single state function that will depend on position, time, and the type/number of particles.  When the time dependence is removed from the description of the system, that is the system is time-independent, it described by the equation:

$$\hat{H}|\Psi(\chi_i,\cdots)\rangle=E|\Psi(\chi_i,\cdots)\rangle $$

This is the Schrödinger time-independent non-relativistic wave equation which is an eigenvalue problem. Lets describe each term in the equation in  more details. The term $\hat{H}$ is called the Hamiltonian operator named after William Rowan Hamiltonian and describes the dynamics, interactions, and energy components of the system. In Newtonian mechanics we describe a system in terms of the acting forces, in Hamiltonian's representation we use momentum and energy. For a quantum system we can write a generic electronic Hamiltonian as,

\begin{align}
H &= K.E. + P.E. \\
&=\frac{-\hbar}{2m_e}\sum_{i}^{N}\nabla_{i}^2+\frac{1}{2}\sum_{i,j}^N V(\mathbf{r}_i,\mathbf{r}_j)
\end{align}.

The first term representing kinetic energy (momentum) and potential energy (field interactions). In this representation the K.E. and P.E. are said to be operators — I tend to think of these as being equivalently equal to transformations —  that are "quantized" in their representation. These operators act on the "wave-like" state, $\Psi$, of a system producing a representation of the system in terms of K.E. and P.E. At this point the representation is a linear combination of characteristic "wave-like" states that the system could be in.

But what happens if I ask the question "What energy is associated with a particle in a given state and where can that particle be in physical space?", more specifically what can I know about the particle that makes sense or has meaning to me, the human. See at this point we just know a spectrum of energies associated with the "wave-like" solution states, which doesn't have a clear comprehensible meaning to us/me yet. It is at this junction where the "reality" of outcomes of Q.M. falls in the the realm of philosophical physics, or in other words, subjected to interpretation. In quantum physics pedagogy the stance is we need to make an expectation or guess on what the outcome is to be. To do this we say we have to project/collapse/measure the "wave-like" states onto a known meaningful platform, a set of states, to get a understanding of where a particle might be. It turns out that this projection is best interpreted as information about the probability or probability of transition of  "wave-like" state; that is with respect to the state its in and the state I know. This is typically what is understood as the expectation value or measurement collapse argument (also more commonly called the Copenhagen interpretation), which is necessary for us (humans) to relate the math or experimental results of quantum mechanics to our notion of particles. To write this out we would have:

$$ <\Psi|H|\Psi> = E <\Psi|\Psi> $$

I use this equation to try and understand the measurement outcome in quantum experiments, that is, the act of observing in an experiment is akin to enforcing a projection onto a known state/solution.

So lets recap, we mentioned that quantum systems can be treated mathematically similar to wave dynamics. We said that these quantum systems have characteristic "wave-like" solutions, e.g., eigenvalues and eigenvectors. Then an equation relating the operation on or transformation of the quantum "wave-like" state resulted in the same state multiplied by the eigenenergies, but we said it only tells us about the energy spectrum and the states the quantum system could be in. Finally, we tried to understand this by projecting or collapsing the "wave-like" solutions onto known solutions (I forgot to mention these could initial states we used prior to applying any operators or transformations).

All this discussion has so far abstracted the "wave-like" states from actual matter that occupies the states. This is akin to our string or drum which we known has vibrating solutions regardless of if its actually been plucked or banged. The next step is to understand how our notion of particles or matter is associated with the "wave-like" states. In other words, how particles or matter can occupy those solutions. It turns out that electrons, which are the basic building blocks for how we (humans that is) experience chemistry and materials, have very specific rules for how they assume a given state. Electrons are indistinguishable particles, meaning they cannot be individual tracked or resolved,  but they have a unique property associated with the "wave-like" state that requires them to take on a specific intrinsic angular momentum which is called spin$^\star$. This spin characteristic gives rise to "wave-like" states such that only a single electron is allowed to acquire a give "wave-like" state of specified spin. In QM this is called the Pauli exclusion principal for fermions which was postulated by Wolfgang Pauli based on spin-statistics theorem. It wasn't till Paul Dirac's work with relativistic QM of an electron that "spin" was deduced to be a consequence of the inclusion of Einstein's special relativity, at least this is my understanding of the outcome of the Dirac Equation.

So now let me try and give analogy to re-explain this, not sure it will work out.

The Race Track (Quantum) Architect

We start our conceptual  understanding of a quantum system which describes electrons in an environment of atomic nuclei.

A quantum architect is told that they will need to design a race track. The race track has several constraints in how it can be built. The first piece of information the architect is given is where the race track will be built. This will dictate the shape, number of bends and slopes of the race track. The race track terrain is our analogy for the potential energy surface created by the combinatoric interactions among the atomic nuclei and electrons.

We then tell the quantum architect how many lanes/tracks are needed to accommodate the drivers, these are our analogy for the "wave-like" states an electron can be in. There are additional pieces of information we need to tell our architect about this race track, that is, the drivers of this race track have unique characteristics. The first is that they all look identical and drive the same car, so we cant tell them apart, therefore we never know who is really in each lane. The next unique characteristic is that each driver has a preferred direction (spin) of driving, forward or reverse, and preferred lane (spatial location). The third characteristic is that each driver dislikes all other drivers, so they try to avoid each other at all costs and never use the same lane, with one exception, that is when two drivers who drive in opposite directions use the same lane/track they both usually perform better — we will take this to mean lower energy — but no lane can ever accommodate drivers who are moving in the same direction.

So lets break down our analogy. The number of lanes corresponds to the number of "wave-like" state solutions we can have which depends on the type of system we are interested in; here the system is our race track configuration based on the number of drivers, terrain, etc. The fact that all the drivers look identical and drive the same car is a characteristic of electrons, they are indistinguishable particles. The preferred driving direction of each driver corresponds to a innate property of electrons (more broadly fermions) dubbed "spin", where its value corresponds to a positive (up) or negative (down) sign due to its behavior in magnetic fields. The fact that the drivers dislike each other but can have special configurations in the lanes corresponds to the Coulomb and exchange forces, where the latter is due to spin and the Pauli exclusion principal.

With all this information the architect is ready to build our race track, but, we through in an additional caveat, that we need him/her to build it in such a way that the number and shape of the lanes can change dynamically. At this point he/her is annoyed, but says their company is so skilled that they can dynamically adjust the state of the race-track to accommodate any number of drivers. So now drivers can be added or removed from the rack track simply by the architects ability to add or remove lanes.

Okay, so we have described the quantum state of electrons using an analogy.  I'm not sure how good this was but I will continually revisit it to improve. No quote for this blog given it is not final.

$^\dagger$ Please comment if you find any explanation of analogy incorrect. This blog post was a result in trying to explain to my wife, who is not trained in a STEM field, the physics that gives rise to the chemical and material world around us. What inspired this was my reading of books discussing self-learning approach dubbed the Feynman technique, pioneered by the famous physicist Richard Feynman. The idea is you take a topic or idea that your familiar with but not sure your level of understanding and work through the topic/idea (e.g. Quantum States) as if you were teaching it to someone else. The purpose is to help identify areas where you may lack understanding or capability (i.e. mathematics). Since I'm always trying to improve my level of knowledge in quantum mechanics, this blog post was my attempt to do so.

$^\mp$ I have chosen to use the quoted term "wave-like" instead of the more common and probably accurate term wavefunction. My reason for doing so is at this level of discussion I don't think any additional clarity is gained by using the word wavefunction, but "wave-like" in my opinion, conveys that the wavefunction will have a functional form that is reminiscent of a wave or wave packet. Anyhow, I will use the two interchangeably throughout this blog.

$^\ddagger$ My understanding is that the leading view in quantum (field) physics is that particles are a manifestation of excitations in quantum fields. In other words, it is quantum fields that exist not particles, and when we speak of particles, say an electron, we are just talking about quantized excitation in an electron field.

$^\star$ The origin of the term spin is unfortunate because it forces a physical picture of the electron which isn't necessarily true in the framework of QM. It comes from the initial view that an electron is not a structure-less point object in space, but that it has some diameter/volume  associated with it and spins about its own axis (as the earth does). To my knowledge the diameter of a electron has never been observed or measured. As mentioned in the main text, the spin is a result of special relativity which adds chirality (handedness) to the spatial part of the wavefunction, also called a spinor.

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