# How deterministic are large open quantum systems (e.g. with humans)?

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I hope to formulate the question, "How much quantum indeterminism is present in human-scale phenomena?" or "How much does the wavefunction of a human branch over time?" without delving into interpretational questions.

Consider some large system modeled as an open quantum system -- say, a person in a room, where the walls of the room interact in a boring way with some environment.  Begin with a pure initial state describing some comprehensible configuration. (Maybe the person is sitting down.)  Generically, the system will be in a highly mixed state after some time.  Both normal human experience and the study of decoherence suggest that this state will be a mixture of orthogonal pure states that describe classical-like configurations.  Call these configurations branches.

How much does a pure state of the system branch over human time scales?  There will soon be many (many) orthogonal branches with distinct microscopic details.  But to what extent will probabilities be spread over macroscopically (and noticeably) different branches?

The answer seems impossible to quantify by calculating anything directly.  Simple reasoning already demonstrates the answer must depend significantly on the initial state.  If the initial state describes a person in deep sleep, then after a few seconds there will be no high-probability branches in which the person is awake.  But if the person has an alarm clock triggered by nuclear decay, there may well be 50/50 branching in which the person is awake or asleep.  (Note that you couldn't get such branching with a regular alarm clock.)

The decay-triggered alarm clock demonstrates that the presence/absence of noticeably different branches depends on what's in the room.  What if two people are talking in a sparse room (with no decay-triggered alarms)?  Will there be different branches of non-negligible probability in which the people say different things?  And after how long?

Although I'm sympathetic to objections, I believe these questions are somewhat well-formulated within most conventional accounts of quantum mechanics.  That is, most physicists believe it is hypothetically possible to model a large system in this way -- perhaps using non-relativistic quantum mechanics or the Standard Model -- and to calculate branches, where the probabilities of these branches correspond to observation.

In response to the helpful comments, let me now pose a more general and careful version of this question.

I start with a widespread assumption: that it's hypothetically possible to perform some quantum simulation of a real-life macroscopic system, perhaps at the atomic level or quantum field-theoretic level. I'm assuming the simulation yields (perhaps probabilistic) predictions about macroscopic behavior and that these predictions match observation.  The initial condition for the simulation should somehow represent a classical state, as well-specified as possible.  I haven't been specific about what sort of "quantum simulation" this is -- whether we start with a pure/mixed state, whether we use an open system, etc.  But if the assumptions seem plausible, fill in the details however you like.  Make it a simple macroscopic system. I'm deliberately asking about a hypothetical simulation/model (rather than "the actual system") to separate this question from interpretational issues, if possible.

Now here's the question: Are my assumptions okay?  If so, what happens to the probability distributions for macroscopic observables as you run the simulation in time?  Does the simulation predict a single classical configuration?  (I don't think so.)  Does it predict some distribution over different classical configurations?  (What does the distribution look like?)  And what's the evidence?

edited Oct 12, 2015

The crux of your question appears to be, how to quantify "amount of quantum indeterminism" or "amount of branching" for *anything*. A human being is an excessively complicated place to start for such questions.

Sure Mitchell, but he explicitly says that human is just an example.  The important point is that it's a macroscopic system, so one can take a thermodynamic limit.  I'm sure he'd be happy with answers that only address macroscopic non-hu

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On human time scales, objects of human size can never be assumed to be in a pure state.

Indeed, environmental decoherence immediately (on human time scales) transforms a pure state into a state well described by a density matrix of the form $e^{-S/k}$ where $S$ has the form of a 1-particle operator in the asymptotic free Fock space plus an integral over the energy density. This state can be well-described by nonequilibrium thermodynamics, and is the reason why the latter describes everyday phenomena.

On the other hand, nonequilibrium thermodynamics is in general chaotic, hence for practical purposes as nondeterministic as quantum mechanics, unless the system is sufficently close to equilibrium.

answered Oct 8, 2015 by (15,608 points)

@dranard: I do not have a reference. But branching in the Everett sense is strictly speaking a property of wave functions, hence of pure states only. To give the concept of branching a meaning in the case of mixtures one must therefore consider systems that interact strongly with a system in which a pure state has been prepared, so that one can say the mixture branches when the pure system branches. The latter is possible only for very tiny systems - in practice with either one continuous degree of freedom or a few discrete degrees of freedom.

For describing how nonequilibrium thermodynamics captures the macroscopically accessible part of quantum systems see, e.g., books and papers by Roger Balian.

@GiulioPrisco: But your chain of finite beings must stop because of the limited size of the universe accessible to any being. Thus there is a barrier in principle.

Concerning "cleaner": The notion of fundamental randomness is on logical grounds intrinsically meaningless. I.e., one cannot in principle give an executable operational definition of its meaning. I therefore believe that quantum randomness is not fundamental but a consequence of a not yet found highly chaotic deterministic description. (I don't count Bohmian mechanics as such adescription, since it is not applicable to relativistic quantum fields.)

@ArnoldNeumaier re "I therefore believe that quantum randomness is not fundamental but a consequence of a not yet found highly chaotic deterministic description." - very interesting, thanks. Is this the "Thermal Interpretation" mentioned on your website? What should I read first?

re "The notion of fundamental randomness is on logical grounds intrinsically meaningless. I.e., one cannot in principle give an executable operational definition of its meaning." - How about I flip a coin next time I have to make a decision? (OK flipping a coin is chaotically deterministic, but I am sure you see what I mean). What's intrinsically meaningless in the notion of fundamental randomness?

@ArnoldNeumaier - by the way Arnold, your website is great. Thanks for curating it.

@GiulioPrisco: What's intrinsically meaningless in the notion of fundamental randomness?

This is off-topic here, but if you formulate an appropriate question about fundamental randomness in a separate thread, I'd argue it there.

My thermal interpretation is not directly related to determinism; it is only a first step towards a possible deterministic description. To understand it, you could start with my slides ''Optical models for quantum mechanics'' linked to from my profile page

@ArnoldNeumaier - Thanks for the link. I will think of how to formulate an appropriate question about fundamental randopmess and open a new thread.

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Your question is central to my research interests, in the sense that completing that research would necessarily let me give you a precise, unambiguous answer.  So I can only give you an imprecise, hand-wavy one.  I'll write down the punchline, then work backwards.

Punchline: The instantaneous rate of branching, as measured in entropy/time (e.g., bits/s), is given by the sum of all positive Lyapunov exponents for all non-thermalized degrees of freedom.

Most of the vagueness in this claim comes from defining/identifying degree of freedom that have thermalized, and dealing with cases of partial/incomplete thermalization; these problems exists classically.

Elaboration: Your question postulates that the macroscopic system starts in a quantum state corresponding to some comprehensible classical configuration, i.e., the system is initially in a quantum state whose Wigner function is localized around some classical point in phase space.  The Lyapunov exponents (units: inverse time) are a set of local quantities, each associated with a particular orthogonal direction in phase space.  They give the rate at which local trajectories diverge, and they (and their associated directions) vary from point to point in phase space.

Lyapunov exponents are defined by the linearized dynamics around a point, and therefore they are constant on scales smaller than the third derivative of the potential.  (Perfectly linear dynamics are governed by a quadradic Hamiltonian and hence a vanishing third derivative.)  So if your Wigner function for the relevant degree of freedom is confined to a region smaller than this scale, it has a single well-defined set of Lyapunov exponents.

On the other hand, the Wigner function for degrees of freedom that are completely thermalized is confined only by the submanifold associated with values of conserved quantities like the energy; within this submanifold, the Wigner function is spread over scales larger than the linearization neighborhood and hence is not quasi-classical.

As mentioned above, I don't know how to think about degree of freedom which are neither fully thermalized nor confined within the linear neighborhoods.

Argument: We want to associate (a) the rate at which nearby classical trajectories diverge with (b) the production of quantum entanglement entropy.  The close relationship between these two has been shown in a bunch of toy models.   For instance, see the many nice cites in the introduction of

Asplund and Berenstein, "Entanglement entropy converges to classical entropy around periodic orbits", (2015).  arXiv:1503.04857.

especially this paper by my former advisor

Zurek and Paz,"Quantum chaos: a decoherent definition", Physica D 83, 300 (1995). arXiv:quant-ph/9502029.

The very crude picture is as follows. An initially pure quantum state with Wigner function localized around a classical point in phase space will spread to much larger phase-space scales at a rate given the Lyapunov exponent.  The couplings between systems and environments are smooth functions of the phase space coordinates (i.e., environments monitor/measure some combination of the system's x's and p's, but not arbitrary superpositions thereof), and the decoherence rate between two values of a coordinate is an increasing function of the difference. Once the Wigner function is spread over a sufficient distance in phase space, it will start to decohere into an incoherent mixture of branches, each of which are localized in phase space. See, for instance, Fig. 1 in Zurek & Paz:

Hence, the rate of trajectory divergence gives the rate of branching.

With regard to this:

But if the person has an alarm clock triggered by nuclear decay, there may well be 50/50 branching in which the person is awake or asleep.  (Note that you couldn't get such branching with a regular alarm clock.)

It's true you couldn't get such branching with a highly-reliable deterministic alarm clock, but you could dispense with the nuclear decay by measuring any macroscopic chaotic degree of freedom on a timescales longer than the associated Lyapunov time constant.  In particular, measuring thermal fluctuations of just about anything should be sufficient.

One more thing (rather controversial): The reason this questions was so hard to even formulate is two fold:

1. No one has a good definition of what a branch, nor how to extract predictions for macroscopic observations directly from a general unitarily evolving wavefunction of the universe.  (My preferred formulation of this is Kent's set selection problem in the consistent histories framework.)
2. Branching is intimately connected to the process of thermalization.  Although some recent progress in non-equilibrium thermodynamics has been made for systems near equilibrium (especially Crooks Fluctuation Theorem and related work), folks are still very confused about the process of thermalization even classically.  See, for instance, the amazingly open question of deriving Fourier's law from microscopic first principles, a very special case!
answered Oct 19, 2015 by (220 points)
edited Oct 19, 2015

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