In Everettian quantum mechanics, what does branching look like if the universe is in a non-unitarily evolving mixed state?

Question

In Cuffaro and Hartmann's (2021) Open Systems View, the authors suggest, "The problem is that if the physical universe is not represented as a unitarily evolving state vector, then it becomes difficult to conceive of it in terms of a multiverse of effectively independently evolving worlds."

Why is this the case? Can we not get branching if the universe is in a non-unitarily evolving mixed state represented by a density operator?

For those sceptical of non-unitary evolution in EQM, both Wallace (Emergent Multiverse 2012 §10.5) and Carroll (Reality as a Vector in Hilbert Space 2021) consider this to be a possibility.

Answer 1

One of the main points of the Many-Worlds Interpretation (MWI) is that measurements are not special, they are merely an entangling interaction between the system being measured and the measurement apparatus. Because measurements are not special, there is nothing nonunitary. In fact, I can tell you what the measurement unitary is, both in toy models and for real physical experiments.

I should also point out that there isn't any active "branching" in MWI. What I mean is that the branches are already there: It's just a question of basis. To see this, suppose you prepare two qubits in the Bell state $\left| {\rm Bell}\right\rangle = \left( \left| 0 0 \right\rangle + \left| 1 1 \right\rangle \right)/\sqrt{2}$ (in the $Z$ basis) and give one qubit to Alice and the other to Bob.

If Alice measures $Z$ on her qubit, she does so using a measurement apparatus initially in the "default" state $\left|0 \right\rangle$ (this is merely a convention). After measuring her qubit, the MWI state is now $\left| {\rm Bell}\right\rangle \to \left( \left| 0 0 0\right\rangle + \left| 1 1 1 \right\rangle \right)/\sqrt{2}$ where the third qubit reflects the state of the measurement apparatus (0 if no photon detected / "default", 1 if photon detected / "click"). If Bob also measures $Z$, the state becomes $\left| {\rm Bell}\right\rangle \to \left( \left| 0 0 0 0 \right\rangle + \left| 1 1 1 1 \right\rangle \right)/\sqrt{2}$.

As you can see, the state of the physical system (the two qubits originally in the Bell state) is completely unchanged. We've merely entangled the measurement devices to that state in the same basis. This is how measurements work. It's also easy to check that this continues to hold no matter what Alice and Bob measure: As long as we write the pre-measurement state in the basis of the operators we intend to measure, every measurement will look like what I wrote above. This shows that nothing drastic happens upon measurement, nor does anything nonlocal happen.

Importantly, the main premise of MWI is that you never consider open systems, but instead consider the smallest effectively closed system. You then realize that regular old quantum mechanics (QM) without weird features (like nonlocality / wavefunction collapse) explain things that used to seem weird (like measurements), if you recognize that the measurement apparatus is part of the system. So the lesson is that QM works just fine for closed systems.

MWI is the realization that (i) measurements aren't weird, (ii) if you identify a truly closed system, all QM is just unitary time evolution, and (iii) the apparent randomness of measurement outcomes is an artifact of our being internal to a larger closed system.

I don't know why anyone would think or write about MWI for open systems. MWI is not a theory but an interpretation. It's a way of explaining how something we originally thought to be weird and "different" (measurement) is actually just another example of regular QM. Prior to MWI, people wondered if observers, consciousness, or even humans were essential. But quantum describes the universe, which does not need to contain humans (and didn't always).

If anything, the lesson of MWI is that you should either (1) not do quantum mechanics on open systems, but instead identify the smallest system that can be treated as closed and do unitary QM there or (2) forfeit the right to complain if some aspect of QM seems to stop working / seems different. Considering an "open system" is literally just an approximation to considering a larger closed system (where MWI is fine), in which we ignore the details of a large part of the system, which we call the "environment".

The Copenhagen picture with all of its weirdness is still useful for calculations. The machinery of open quantum systems is even more practically useful. But neither of these makes sense if you're trying to address fundamental questions about the universe or quantum mechanics, since the universe is fundamentally closed and QM can always be represented unitarily (if you've accounted for all degrees of freedom). The whole point of MWI is that the weird / bothersome aspects of quantum mechanics are artifacts of considering open systems.

So to me it's not sensible / enlightening to consider MWI with open systems. And that's because there is no place for open systems (nor nonunitary channels) in MWI, because those are just approximations of regular QM for some closed system that we were too lazy to identify. Also, it's not altogether surprising (nor is it interesting) that if you try to use the concepts of MWI on an open system, they may stop making sense. Finally, as I said before, there is no "active" branching in MWI: The branches are always there.

Answer 2

The debate about the interpretation of quantum theory is about what is happening in reality to produce the results of quantum mechanical experiments. Non-MWI interpretations either deny that there is such a thing as objective reality, e.g. - the Copenhagen and statistical interpretations, or favour modifying quantum theory to eliminate the multiple versions that arise from quantum theory without collapse.

In his paper on the relative state interpretation, Everett wrote:

This paper proposes to regard pure wave mechanics (Process 2 only) as a complete theory. It postulates that a wave function that obeys a linear wave equation everywhere and at all times supplies a complete mathematical model for every isolated physical system without exception. It further postulates that every system that is subject to external observation can be regarded as part of a larger isolated system.

Everett writes a paragraph about branching:

We thus arrive at the following picture: Throughout all of a sequence of observation processes there is only one physical system representing the observer, yet there is no single unique state of the observer (which follows from the representations of interacting systems). Nevertheless, there is a representation in terms of a superposition, each element of which contains a definite observer state and a corresponding system state. Thus with each succeeding observation (or interaction), the observer state “branches” into a number of different states. Each branch represents a different outcome of the measurement and the corresponding eigenstate for the object-system state. All branches exist simultaneously in the superposition after any given sequence of observations.

In a footnote, Everett writes more about branches:

The whole issue of the transition from “possible” to “actual” is taken care of in the theory in a very simple way—there is no such transition, nor is such a transition necessary for the theory to be in accord with our experience. From the viewpoint of the theory all elements of a superposition (all “branches”) are “actual,” none any more “real” than the rest. It is unnecessary to suppose that all but one are somehow destroyed, since all the separate elements of a superposition individually obey the wave equation with complete indifference to the presence or absence (“actuality” or not) of any other elements. This total lack of effect of one branch on another also implies that no observer will ever be aware of any “splitting” process.

So according to Everett's paper, the independence of branches depends on the linearity of the equations of motion.

As a historical matter, the Everett interpretation was formulated in terms of open systems being incorporated in isolated systems evolving according to the Schrodinger equation, hence the apparent inconsistency with an open sytems view of quantum theory.

More recently, branching has been discussed as being an emergent result of quantum theory that depends on decoherence:

http://philsci-archive.pitt.edu/8888/1/Wallace_chapter_in_Oxford_Handbook.pdf

https://arxiv.org/abs/quant-ph/0104033

This view of quantum theory may be compatible with non-unitary and non-linear variants of quantum theory that don't eliminate other branches.

There have been some papers that claim that under some circumstances quantum systems would undergo non-linear evolution, e.g. - near closed timelike curves. This model has been criticised because it assumes standard quantum measurement theory still works in the context of a non-linear equation of motion:

https://arxiv.org/abs/0908.3023

See also:

https://arxiv.org/abs/2205.02797