# Does time emerge from entanglement?

I stumbled upon the hypothesis that time emerges from quantum mechanics as long as the observer is part of the system. That is, the evolution of an entangled state can serve as a clock for an observer within the system. An outside observer would not see a change at all and thus no time. The theory goes back to Page and Wooters, there also was a recent experiment seemingly confirming this view.

To me, this seems a very interesting ansatz, but I have never heard of it before. It does not seem to be mainstream, why is this? Maybe because we cannot imagine an observer outside our universe who would be able to confirm the theory? Or are there deeper contradictions? Does the experiment actually confirm anything?

[1] Page and Wootters, Phys. Rev. D 27, 2885 (1983).

[2] Moreva et al., Phys. Rev. A 89, 052122 (2014).

• Hi, would be a mischaracterization of your question to change the title to something along the lines of "What is the status of Page-Wootters mechanism in contemporary physics?"? The current title couples to numerous distinct approaches to quantum gravity whereas the body of your question nicely focuses on one specific construction of time in quantum mechanics.
– ACat
Jan 24 at 16:28
• @DvijD.C. You've set a bounty for this question - are you looking for an answer specifically about Page & Wootters, or for an answer addressing the title hypothesis in general? Jan 24 at 18:31
• @MitchellPorter Yeah, I am specifically looking for an answer about Page and Wootters but I am not sure if I can officially restrict the bounty to it without OP agreeing to edit the question. And the question is not different enough for me to ask it separately.
– ACat
Jan 25 at 2:26
• I was not aware of "numerous distict approaches" all arguing that time emerge from entanglement nor that there is such a thing as a "Page-Wooters mechanism". I agree that the body of the question seems a lot more specific than the title. So feel free to change the title to a more appropriate one (although my interest is rather general and I feel a good answer should address related ideas as well).
– dodi
Jan 25 at 8:53
• @dodi Thanks for the clarification. I won't be changing the title since I suppose it's a good practice to conserve the OP's intention behind their question -- all else being equal in some sense. I suppose "Page-Wootters mechanism" is, although not widespread, a fairly used phrase, see, for example, articles citing the 1983 paper: journals.aps.org/prd/cited-by/10.1103/PhysRevD.27.2885. As for the other approaches, there is, for example, loop quantum gravity where spacetime is quantum mechanical. There is also the amplituhedron program where spacetime is expected to be emergent, etc.
– ACat
Jan 25 at 9:34

The Page-Wootters (PW) approach to the problem of time is still alive and kicking today in quantum foundations and quantum information theory.

The main idea is that instead of using a label $$t$$ to denote the time, one introduces a clock system, with its own Hilbert space and Hamiltonian and rigs the total Hamiltonian of the clock+rest of the world such that the eigenfunctions of the total Hamiltonian is something of the form $$\int\mathrm{d}t\;|t\rangle_\mathrm{clock}\otimes|\psi(t)\rangle_\mathrm{world}$$ where $$|\psi(t)\rangle_\mathrm{world}$$ satisfies the time-dependent Schrödinger equation. The idea being that what you always do to know a property of a system at some time, you first measure at the clock (read the time) and then you measure the system.

The approach had fallen out of grace after Kuchar pointed out that there seemed no good way to get correct two-time probabilities out of it (things like the probability of finding the particle at $$x$$ at time $$t_2$$ given you had found it at $$y$$ at time $$t_1$$). This problem has recently been solved by unifying the PW approach with other relational approaches to quantisation.

PW is not "mainstream" in the sense that, to do laboratory physics, the normal approach (with $$t$$ being just a label) works perfectly so there is no use for PW. Secondly, note that the PW relies on an idea of a universal wavefunction that does not collapse upon measurement and, as I imagine you know, this idea is not accepted by everyone.

Finally, I doubt that there will be an experiment "confirming" or "disproving" PW any time soon. It is designed to reduce to standard QM in laboratory settings. However, PW or some related idea could help in building a theory of quantum gravity (there is no external time in GR, either). In that case, one would be more justified in thinking that this is the correct way of understanding time.

• Thanks! This is exactly the kind of answer I was hoping to get with the bounty :) I basically wanted to know if there are known fatal issues with the approach or if the approach is still considered a viable option. And this addresses it perfectly, so thanks again!
– ACat
Jan 30 at 11:31
• One additional question, I understand how this approach would be super useful for quantum gravity but I didn't quite get why/how it is useful in quantum information theory. I know there are quantum information-theoretic approaches to quantum gravity and so I can imagine it being useful in that vein. But is it also useful for the quantum computing adjacent flavor of quantum information theory?
– ACat
Jan 30 at 11:36
• I see it in following way: in 4D spacetime we assign an entropy scalar at each point. The direction of increasing entropy defines the arrow of time. In each 3D slice, perpendicular to time, an entanglement can occur. The entangled particles belong to the same 3D slice (slice which is perpendicular to time). The process of disentangling them (thus increasing entropy, thus moving forward in time to the next 3D slice) is also a loss of information. Hence time, entanglement, entropy and information are all affecting each other. Jan 31 at 12:07
• @DvijD.C. You're welcome! I don't know of any direct applications to quantum computing, but the second paper I cited relates PW to work in quantum reference frames (QRF), which is of interest to QI folk. See arxiv.org/abs/1712.07207 where we can see that entanglement and superposition are frame-dependent. In arxiv.org/abs/quant-ph/0610030, although not about quantum reference frames, you can find further motivation to as why reference frames are of interest for QI. Feb 2 at 22:10

I find this idea intriguing too! The idea that time emerges from entanglement is mainstream enough to have a 2014 article in Quanta Magazine on the topic, "Time’s Arrow Traced to Quantum Source". Here is an excerpt:

In 2009, the Bristol group’s proof resonated with quantum information theorists, opening up new uses for their techniques. It showed that as objects interact with their surroundings — as the particles in a cup of coffee collide with the air, for example — information about their properties “leaks out and becomes smeared over the entire environment,” Popescu explained. This local information loss causes the state of the coffee to stagnate even as the pure state of the entire room continues to evolve. Except for rare, random fluctuations, he said, “its state stops changing in time.”

Consequently, a tepid cup of coffee does not spontaneously warm up. In principle, as the pure state of the room evolves, the coffee could suddenly become unmixed from the air and enter a pure state of its own. But there are so many more mixed states than pure states available to the coffee that this practically never happens — one would have to outlive the universe to witness it. This statistical unlikelihood gives time’s arrow the appearance of irreversibility. “Essentially entanglement opens a very large space for you,” Popescu said. “It’s like you are at the park and you start next to the gate, far from equilibrium. Then you enter and you have this enormous place and you get lost in it. And you never come back to the gate.”

In the new story of the arrow of time, it is the loss of information through quantum entanglement, rather than a subjective lack of human knowledge, that drives a cup of coffee into equilibrium with the surrounding room.

The article credits Seth Lloyd with the idea that entanglement drives the arrow of time. No mention is made of Page or Wootters.

However, the Wikipedia entry for Quantum Entanglement mentions both Page and Wootters and Lloyd regarding the relationship between entanglement and time:

There have been suggestions to look at the concept of time as an emergent phenomenon that is a side effect of quantum entanglement.[51][52] In other words, time is an entanglement phenomenon, which places all equal clock readings (of correctly prepared clocks, or of any objects usable as clocks) into the same history. This was first fully theorized by Don Page and William Wootters in 1983.[53] The Wheeler–DeWitt equation that combines general relativity and quantum mechanics – by leaving out time altogether – was introduced in the 1960s and it was taken up again in 1983, when the theorists Don Page and William Wootters made a solution based on the quantum phenomenon of entanglement. Page and Wootters argued that entanglement can be used to measure time.[54]

...

Physicist Seth Lloyd says that quantum uncertainty gives rise to entanglement, the putative source of the arrow of time. According to Lloyd; "The arrow of time is an arrow of increasing correlations."[55] The approach to entanglement would be from the perspective of the causal arrow of time, with the assumption that the cause of the measurement of one particle determines the effect of the result of the other particle's measurement.

Unfortunately, I can't tell from such descriptions the degree of similarity between Page and Wootters theory and Lloyd's theory.

• All RQM ideas are quite similar. And none of those theories produced actual scheme of experiment which would allow to differentiate it from others. Jan 24 at 17:43

I have explored this topic myself in a couple of ArXiv papers The Concept of Entropic Time and Quantum Information and the Mind-Body Problem, so it interesting to know about the Page/Wooters theory and Quantum Magazine article, that I had not come across before. Thank you for that!

To my mind, the proper framework for this discussion is decoherence theory, which describes how information can be transferred from one system to another via entanglement. However, it is important to realise that entanglement is itself a reversible process - systems that have become entangled can become unentangled through the unitary evolution of the joint system.

What cannot be effectively reversed, from the subjective point of view of an observer, is information accrual. If information becomes erased through two systems becoming unentangled, then it is entirely forgotten and cannot be known about.

Our subjective perception of time must therefore involve stored information about 'past' events. An interesting aspect of this is that it is the logical relation between 'sets of stored facts' about the world that determines their causal (i.e. perceived temporal) relation and need not be related to the temporal parameter 't' representing time in the formal mathematical expressions for the unitary evolution of the joint systems.

I believe that both Zeh and Zurek have explored similar ideas in some depth (these guys are the heavyweights of decoherence theory) so this view of things is gaining some ground.

I don't think this works. "Entropy and time", "Time and QM" and so on were discussed long enough so we can firmly say that additional theoretical progress could only result from more experimental basis. And this article of Page & Wooters does not add up to anything. Sadly but science has no additional input on Relativistic Quantum Mechanics (RQM) outside of Gordon-Dirac equations and similar things. I will formulate it like this, try to make an experiment to check that version of RQM which you got, and I will do it for you. Good Luck!