The interpretation of free energy as bit-erasure capacity

Our paper discussed in the previous blog post might prompt this question: Is there still a way to use Landauer’s principle to convert the free energy of a system to its bit erasure capacity? The answer is “yes”, which we can demonstrate with a simple argument.


Summary: The correct measure of bit-erasure capacity N for an isolated system is the negentropy, the difference between the system’s current entropy and the entropy it would have if allowed to thermalize with its current internal energy. The correct measure of erasure capacity for a constant-volume system with free access to a bath at constant temperature T is the Helmholtz free energy A (divided by kT, per Landauer’s principle), provided that the additive constant of the free energy is set such that the free energy vanishes when the system thermalizes to temperature T. That is,

    \[N = \frac{A}{kT} = \frac{U-U_0}{kT} - (S - S_0),\]

where U_0 and S_0 are the internal energy and entropy of the system if it were at temperature T. The system’s negentropy lower bounds this capacity, and this bound is saturated when U = U_0.


Traditionally, the Helmholtz free energy of a system is defined as \tilde{A} = U - kTS, where U and S are the internal energy and entropy of the system and T is the constant temperature of an external infinite bath with which the system can exchange energy.Here, there is a factor of Boltzmann’s constant k in front of TS because I am measuring the (absolute) entropy S in dimensionless bits rather than in units of energy per temperature. That way we can write things like N = S_0 - S.a   (I will suppress the “Helmholtz” modifier henceforth; when the system’s pressure rather than volume is constant, my conclusion below holds for the Gibbs free energy if the obvious modifications are made.)

However, even in the case of fixed bath temperature, we cannot naively use Landauer’s principle to divide the free energy by kT to get the erasure capacity.… [continue reading]

On computational aestivation

People often say to me “Jess, all this work you do on the foundations of quantum mechanics is fine as far as it goes, but it’s so conventional and safe. When are you finally going to do something unusual and take some career risks?” I’m now pleased to say I have a topic to bring up in such situations: the thermodynamic incentives of powerful civilizations in the far future who seek to perform massive computations. Anders Sandberg, Stuart Armstrong, and Milan M. Ćirković previously argued for a surprising connection between Landauer’s principle and the Fermi paradox, which Charles Bennett, Robin Hanson, and I have now critiqued. Our comment appeared today in the new issue of Foundations of Physics:

Comment on 'The aestivation hypothesis for resolving Fermi's paradox'
Charles H. Bennett, Robin Hanson, C. Jess Riedel
In their article [arXiv:1705.03394], 'That is not dead which can eternal lie: the aestivation hypothesis for resolving Fermi's paradox', Sandberg et al. try to explain the Fermi paradox (we see no aliens) by claiming that Landauer's principle implies that a civilization can in principle perform far more (~1030 times more) irreversible logical operations (e.g., error-correcting bit erasures) if it conserves its resources until the distant future when the cosmic background temperature is very low. So perhaps aliens are out there, but quietly waiting. Sandberg et al. implicitly assume, however, that computer-generated entropy can only be disposed of by transferring it to the cosmological background. In fact, while this assumption may apply in the distant future, our universe today contains vast reservoirs and other physical systems in non-maximal entropy states, and computer-generated entropy can be transferred to them at the adiabatic conversion rate of one bit of negentropy to erase one bit of error.
[continue reading]

FAQ about experimental quantum Darwinism

I am briefly stirring from my blog-hibernationThis blog will resume at full force sometime in the future, but not just yet.a   to present a collection of frequently asked questions about experiments seeking to investigate quantum Darwinism (QD). Most of the questions were asked by (or evolved from questions asked by) Phillip Ball while we corresponded regarding his recent article “Quantum Darwinism, an Idea to Explain Objective Reality, Passes First Tests” for Quanta magazine, which I recommend you check out.


Who is trying see quantum Darwinism in experiments?

I am aware of two papers out of a group from Arizona State in 2010 (here and here) and three papers from separate groups last year (arXiv: 1803.01913, 1808.07388, 1809.10456). I haven’t looked at them all carefully so I can’t vouch for them, but I think the more recent papers would be the closest thing to a “test” of QD.

What are the experiments doing to put QD the test?

These teams construct a kind of “synthetic environment” from just a few qubits, and then interrogate them to discover the information that they contain about the quantum system to which they are coupled.

What do you think of experimental tests of QD in general?

Considered as a strictly mathematical phenomenon, QD is the dynamical creation of certain kinds of correlations between certain systems and their environments under certain conditions. These experiments directly confirm that, if such conditions are created, the expected correlations are obtained.

The experiments are, unfortunately, not likely to offer many insight or opportunities for surprise; the result can be predicted with very high confidence long in advance.… [continue reading]

Branches as hidden nodes in a neural net

I had been vaguely aware that there was an important connection between tensor network representations of quantum many-body states (e.g., matrix product states) and artificial neural nets, but it didn’t really click together until I saw Roger Melko’s nice talk on Friday about his recent paper with Torlai et al.:There is a title card about “resurgence” from Francesco Di Renzo’s talk at the beginning of the talk you can ignore. This is just a mistake in KITP’s video system.a  





[Download MP4]   [Other options]

In particular, he sketched the essential equivalence between matrix product states (MPS) and restricted Boltzmann machinesThis is discussed in detail by Chen et al. See also good intuition and a helpful physicist-statistician dictionary from Lin and Tegmark.b   (RBM) before showing how he and collaborators could train an efficient RBM representations of the states of the transverse-field Ising and XXZ models with a small number of local measurements from the true state.

As you’ve heard me belabor ad nauseum, I think identifying and defining branches is the key outstanding task inhibiting progress in resolving the measurement problem. I had already been thinking of branches as a sort of “global” tensor in an MPS, i.e., there would be a single index (bond) that would label the branches and serve to efficiently encode a pure state with long-range entanglement due to the amplification that defines a physical measurement process. (More generally, you can imagine branching events with effects that haven’t propagated outside of some region, such as the light-cone or Lieb-Robinson bound, and you might even make a hand-wavy connection to entanglement renormalization.) But I had little experience with constructing MPSs, and finding efficient representations always seemed like an ad-hoc process yielding non-unique results.… [continue reading]

Models of decoherence and branching

[This is akin to a living review, which will hopefully improve from time to time. Last edited 2017-11-26.]

This post will collect some models of decoherence and branching. We don’t have a rigorous definition of branches yet but I crudely define models of branching to be models of decoherenceI take decoherence to mean a model with dynamics taking the form U \approx \sum_i \vert S_i\rangle\langle S_i |\otimes U^{\mathcal{E}}_i for some tensor decomposition \mathcal{H} = \mathcal{S} \otimes \mathcal{E}, where \{\vert S_i\rangle\} is an (approximately) stable orthonormal basis independent of initial state, and where \mathrm{Tr}[ U^{\mathcal{E}}_i \rho^{\mathcal{E} \dagger}_0 U^{\mathcal{E}}_j ] \approx 0 for times t \gtrsim t_D and i \neq j, where \rho^{\mathcal{E}}_0 is the initial state of \mathcal{E} and t_D is some characteristic time scale.a   which additionally feature some combination of amplification, irreversibility, redundant records, and/or outcomes with an intuitive macroscopic interpretation. I have the following desiderata for models, which tend to be in tension with computational tractability:

  • physically realistic
  • symmetric (e.g., translationally)
  • no ad-hoc system-environment distinction
  • Ehrenfest evolution along classical phase-space trajectories (at least on Lyapunov timescales)

Regarding that last one: we would like to recover “classical behavior” in the sense of classical Hamiltonian flow, which (presumably) means continuous degrees of freedom.In principle you could have discrete degrees of freedom that limit, as \hbar\to 0, to some sort of discrete classical systems, but most people find this unsatisfying.b   Branching only becomes unambiguous in some large-N limit, so it seems satisfying models are necessarily messy and difficult to numerically simulate.… [continue reading]

Comments on Weingarten’s preferred branch

A senior colleague asked me for thoughts on this paper describing a single-preferred-branch flavor of quantum mechanics, and I thought I’d copy them here. Tl;dr: I did not find an important new idea in it, but this paper nicely illustrates the appeal of Finkelstein’s partial-trace decoherence and the ambiguity inherent in connecting a many-worlds wavefunction to our direct observations.


We propose a method for finding an initial state vector which by ordinary Hamiltonian time evolution follows a single branch of many-worlds quantum mechanics. The resulting deterministic system appears to exhibit random behavior as a result of the successive emergence over time of information present in the initial state but not previously observed.

We start by assuming that a precise wavefunction branch structure has been specified. The idea, basically, is to randomly draw a branch at late times according to the Born probability, then to evolve it backwards in time to the beginning of the universe and take that as your initial condition. The main motivating observation is that, if we assume that all branch splittings are defined by a projective decomposition of some subsystem (‘the system’) which is recorded faithfully elsewhere (‘the environment’), then the lone preferred branch — time-evolving by itself — is an eigenstate of each of the projectors defining the splits. In a sense, Weingarten lays claim to ordered consistency [arxiv:gr-qc/9607073] by assuming partial-trace decoherenceNote on terminology: What Finkelstein called “partial-trace decoherence” is really a specialized form of consistency (i.e., a mathematical criterion for sets of consistent histories) that captures some, but not all, of the properties of the physical and dynamical process of decoherence.[continue reading]

Symmetries and solutions

Here is an underemphasized way to frame the relationship between trajectories and symmetries (in the sense of Noether’s theorem)You can find this presentation in “A short review on Noether’s theorems, gauge symmetries and boundary terms” by Máximo Bañados and Ignacio A. Reyes (H/t Godfrey Miller).a  . Consider the space of all possible trajectories q(t) for a system, a real-valued Lagrangian functional L[q(t)] on that space, the “directions” \delta q(t) at each point, and the corresponding functional gradient \delta L[q(t)]/\delta q(t) in each direction. Classical solutions are exactly those trajectories q(t) such that the Lagrangian L[q(t)] is stationary for perturbations in any direction \delta q(t), and continuous symmetries are exactly those directions \delta q(t) such that the Lagrangian L[q(t)] is stationary for any trajectory q(t). That is,

(1)   \begin{align*} q(t) \mathrm{\,is\, a\,}\mathbf{solution}\quad \qquad &\Leftrightarrow \qquad \frac{\delta L[q(t)]}{\delta q(t)} = 0 \,\,\,\, \forall \delta q(t)\\ \delta q(t) \mathrm{\,is\, a\,}\mathbf{symmetry} \qquad &\Leftrightarrow \qquad \frac{\delta L[q(t)]}{\delta q(t)} = 0 \,\,\,\, \forall q(t). \end{align*}

There are many subtleties obscured in this cartoon presentation, like the fact that a symmetry \delta q(t), being a tangent direction on the manifold of trajectories, can vary with the tangent point q(t) it is attached to (as for rotational symmetries). If you’ve never spent a long afternoon with a good book on the calculus of variations, I recommend it.

Footnotes

(↵ returns to text)

  1. You can find this presentation in “A short review on Noether’s theorems, gauge symmetries and boundary terms” by Máximo Bañados and Ignacio A. Reyes (H/t Godfrey Miller).
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How to think about Quantum Mechanics—Part 7: Quantum chaos and linear evolution

[Other parts in this series: 1,2,3,4,5,6,7.]

You’re taking a vacation to Granada to enjoy a Spanish ski resort in the Sierra Nevada mountains. But as your plane is coming in for a landing, you look out the window and realize the airport is on a small tropical island. Confused, you ask the flight attendant what’s wrong. “Oh”, she says, looking at your ticket, “you’re trying to get to Granada, but you’re on the plane to Grenada in the Caribbean Sea.” A wave of distress comes over your face, but she reassures you: “Don’t worry, Granada isn’t that far from here. The Hamming distance is only 1!”.

After you’ve recovered from that side-splitting humor, let’s dissect the frog. What’s the basis of the joke? The flight attendant is conflating two different metrics: the geographic distance and the Hamming distance. The distances are completely distinct, as two named locations can be very nearby in one and very far apart in the other.

Now let’s hear another joke from renowned physicist Chris Jarzynski:

The linear Schrödinger equation, however, does not give rise to the sort of nonlinear, chaotic dynamics responsible for ergodicity and mixing in classical many-body systems. This suggests that new concepts are needed to understand thermalization in isolated quantum systems. – C. Jarzynski, “Diverse phenomena, common themes” [PDF]

Ha! Get it? This joke is so good it’s been told by S. Wimberger“Since quantum mechanics is the more fundamental theory we can ask ourselves if there is chaotic motion in quantum systems as well.[continue reading]