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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