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

---

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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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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