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Moyal bracket with manifest (affine) symplectic covariance

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Moyal’s equation for a Wigner function W of a quantum system with (Wigner-transformed) Hamiltonian H is \partial_t W = \{ H,W \}_\hbar where the Moyal bracket is a binary operator on the space of functions over phase space. Unfortunately, it is often written down mysteriously as

(1)   \begin{align*} \{ A,B \}_\hbar = \frac{2}{\hbar} A \sin \left( \frac{\hbar}{2} \left(\overleftarrow{\partial}_x\overrightarrow{\partial}_p - \overleftarrow{\partial}_p\overrightarrow{\partial}_x \right) \right) B, \end{align*}

where the arrows over partial derivatives tell you which way they act, i.e., C (\overleftarrow{\partial}_x \overrightarrow{\partial}_p ) D = (\partial_x C)(\partial_p D). This only becomes slightly less weird when you use the equivalent formula \{ A,B \}_\hbar = (A \star B - B\star A)/(i\hbar), where “\star” is the Moyal star product given by

(2)   \begin{align*} A \star B = A \exp \left( \frac{i\hbar}{2} \left(\overleftarrow{\partial}_x\overrightarrow{\partial}_p - \overleftarrow{\partial}_p\overrightarrow{\partial}_x \right) \right) B. \end{align*}

The star product has the crucial feature that \widehat{A \star B} = \widehat{A}\widehat{B}, where we use a hat to denote the Weyl transform (i.e., the inverse of the Wigner transform taking density matrices to Wigner functions), which takes a scalar function over phase-space to an operator over our Hilbert space. The star product also has some nice integral representations, which can be found in books like Curtright, Fairlie, & ZachosThe complete 88-page PDF is here.a  , but none of them help me understand the Moyal equation.

A key problem is that both of these expressions are neglecting the (affine) symplectic symmetry of phase space and the dynamical equations. Although I wouldn’t call it beautiful, we can re-write the star product as

(3)   \begin{align*} A \star B = A \exp \left( \frac{i\hbar}{2} \overleftarrow{\partial}_a\overrightarrow{\partial}^a \right) B. \end{align*}

where a=x,p is a symplectic index using the Einstein summation convention, and where symplectic indices are raised and lowered using the symplectic form just as for Weyl spinors: v_a = \epsilon_{ab}v^b and w^a = \epsilon^{ab}w_b, where \epsilon is the antisymmetric symplectic form with \epsilon^{xp} = +1 = \epsilon_{px}, and where upper (lower) indices denote symplectic vectors (co-vectors).

With this, we can expand the Moyal equation as

    \begin{align*} \{ H,W \}_\hbar &= \frac{2}{\hbar} H \sin \left( \frac{\hbar}{2} \overleftarrow{\partial}_a\overrightarrow{\partial}^a \right) W \\ &= \sum_{n=0}^\infty \frac{(-\hbar^2/4)^n}{(2n+1)!} \left(\partial_{a_1}\cdots \partial_{a_{2n+1}} H\right)\left(\partial^{a_1}\cdots \partial^{a_{2n+1}} W\right) \\ &= (\partial_a H)(\partial^a W) - \frac{\hbar^2}{24}(\partial_a\partial_b\partial_c H)(\partial^a \partial^b \partial^c W) \\ &\qquad \qquad + \frac{\hbar^4}{80640}(\partial_a\partial_b\partial_c \partial_d \partial_e H)(\partial^a \partial^b \partial^c \partial^d \partial^e W) - \cdots \end{align*}

where we can see in hideous explicitness that it’s a series in the even powers of \hbar and the odd derivates of the Hamiltonian H and the Wigner function W.… [continue reading]

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Quantum computing timelines

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[Jaime Sevilla is a Computer Science PhD Student at the University of Aberdeen. In this guest post, he describes our recent forecasting work on quantum computing. – Jess Riedel]

In Short: We attempt to forecast when quantum computers will be able to crack the common cryptographic scheme RSA2048, and develop a model that predicts less than 5% confidence that this capability will be reached before 2039. A preprint is available at arXiv:2009.05045.

Advanced quantum computing comes with some new applications as well as a few risks, most notably threatening the foundations of modern online security.

In light of the recent experimental crossing of the “quantum supremacy” milestone, it is of great interest to estimate when devices capable of attacking typical encrypted communication will be constructed, and whether the development of communication protocols that are secure against quantum computers is progressing at an adequate pace.  

Beyond its intrinsic interest, quantum computing is also fertile ground for quantified forecasting. Exercises on forecasting technological progress have generally been sparse — with some notable exceptions — but it is of great importance: technological progress dictates a large part of human progress.

To date, most systematic predictions about development timelines for quantum computing have been based on expert surveys, in part because quantitative data about realistic architectures has been limited to a small number of idiosyncratic prototypes. However, in the last few years the number of device has been rapidly increasing and it is now possible to squint through the fog of research and make some tentative extrapolations. We emphasize that our quantitative model should be considered to at most augment, not replace, expert predictions.  Indeed, as we discuss in our preprint, this early data is noisy, and we necessarily must make strong assumptions to say anything concrete.… [continue reading]

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Comments on “Longtermist Institutional Reform” by John & MacAskill

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Tyler John & William MacAskill have recently released a preprint of their paper “Longtermist Institutional Reform” [PDF]. The paper is set to appear in an EA-motivated collection “The Long View” (working title), from Natalie Cargill and Effective Giving.

Here is the abstract:

There is a vast number of people who will live in the centuries and millennia to come. In all probability, future generations will outnumber us by thousands or millions to one; of all the people who we might affect with our actions, the overwhelming majority are yet to come. In the aggregate, their interests matter enormously. So anything we can do to steer the future of civilization onto a better trajectory, making the world a better place for those generations who are still to come, is of tremendous moral importance. Political science tells us that the practices of most governments are at stark odds with longtermism. In addition to the ordinary causes of human short-termism, which are substantial, politics brings unique challenges of coordination, polarization, short-term institutional incentives, and more. Despite the relatively grim picture of political time horizons offered by political science, the problems of political short-termism are neither necessary nor inevitable. In principle, the State could serve as a powerful tool for positively shaping the long-term future. In this chapter, we make some suggestions about how we should best undertake this project. We begin by explaining the root causes of political short-termism. Then, we propose and defend four institutional reforms that we think would be promising ways to increase the time horizons of governments: 1) government research institutions and archivists; 2) posterity impact assessments; 3) futures assemblies; and 4) legislative houses for future generations.

… [continue reading]
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Living bibliography for the problem of defining wavefunction branches

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[Last updated: Nov 27, 2021.]

This post is (a seed of) a bibliography covering the primordial research area that goes by some of the following names:

Although the way this problem tends to be formalized varies with context, I don’t think we have confidence in any of the formalizations. The different versions are very tightly related, so that a solution in one context is likely give, or at least strongly point toward, solutions for the others.

As a time-saving device, I will mostly just quote a few paragraphs from existing papers that review the literature, along with the relevant part of their list of references. Currently I am drawing on four papers: Carroll & Singh [arXiv:2005.12938]; Riedel, Zurek, & Zwolak [arXiv:1312.0331]; Weingarten [arXiv:2105.04545]; and Kent [arXiv:1311.0249].

I hope to update this from time to time, and perhaps turn it into a proper review article of its own one day. If you have a recommendation for this bibliography (either a single citation, or a paper I should quote), please do let me know.

Carroll & Singh

From “Quantum Mereology: Factorizing Hilbert Space into Subsystems with Quasi-Classical Dynamics” [arXiv:2005.12938]:

While this question has not frequently been addressed in the literature on quantum foundations and emergence of classicality, a few works have highlighted its importance and made attempts to understand it better.

… [continue reading]
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How to think about Quantum Mechanics—Part 8: The quantum-classical limit as music

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[Other parts in this series: 1,2,3,4,5,6,7,8.]

On microscopic scales, sound is air pressure f(t) fluctuating in time t. Taking the Fourier transform of f(t) gives the frequency distribution \hat{f}(\omega), but in an eternal way, applying to the entire time interval for t\in [-\infty,\infty].

Yet on macroscopic scales, sound is described as having a frequency distribution as a function of time, i.e., a note has both a pitch and a duration. There are many formalisms for describing this (e.g., wavelets), but a well-known limitation is that the frequency \omega of a note is only well-defined up to an uncertainty that is inversely proportional to its duration \Delta t.

At the mathematical level, a given wavefunction \psi(x) is almost exactly analogous: macroscopically a particle seems to have a well-defined position and momentum, but microscopically there is only the wavefunction \psi. The mapping of the analogyI am of course not the first to emphasize this analogy. For instance, while writing this post I found “Uncertainty principles in Fourier analysis” by de Bruijn (via Folland’s book), who calls the Wigner function of an audio signal f(t) the “musical score” of f.a   is \{t,\omega,f\} \to \{x,p,\psi\}. Wavefunctions can of course be complex, but we can restrict ourself to a real-valued wavefunction without any trouble; we are not worrying about the dynamics of wavefunctions, so you can pretend the Hamiltonian vanishes if you like.

In order to get the acoustic analog of Planck’s constant \hbar, it helps to imagine going back to a time when the pitch of a note was measured with a unit that did not have a known connection to absolute frequency, i.e.,… [continue reading]

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A checkable Lindbladian condition

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Summary

Physicists often define a Lindbladian superoperator as one whose action on an operator B can be written as

(1)   \begin{align*} \mathcal{L}[B] = -i [H,B] + \sum_i \left[ L_i B L_i^\dagger - \frac{1}{2}\left(L_i^\dagger L_i B + B L_i^\dagger L_i\right)\right], \end{align*}

for some operator H with positive anti-Hermitian part, H-H^\dagger \ge 0, and some set of operators \{L^{(i)}\}. But how does one efficiently check if a given superoperator is Lindbladian? In this post I give an “elementary” proof of a less well-known characterization of Lindbladians:

A superoperator \mathcal{L} generates completely positive dynamics e^{t\mathcal{L}}, and hence is Lindbladian, if and only if \mathcal{P} \mathcal{L}^{\mathrm{PT}} \mathcal{P} \ge 0, i.e.,

    \[\mathrm{Tr}\left[B^\dagger (\mathcal{P} \mathcal{L}^{\mathrm{PT}} \mathcal{P})[B]\right] \ge 0\]

for all B. Here “\mathrm{PT}” denotes a partial transpose, \mathcal{P} = \mathcal{I} - \mathcal{I}^{\mathrm{PT}}/N = \mathcal{P}^2 is the “superprojector” that removes an operator’s trace, \mathcal{I} is the identity superoperator, and N is the dimension of the space upon which the operators act.

Thus, we can efficiently check if an arbitrary superoperator \mathcal{L} is Lindbladian by diagonalizing \mathcal{P}\mathcal{L}^{\mathrm{PT}} \mathcal{P} and seeing if all the eigenvalues are positive.

A quick note on terminology

The terms superoperator, completely positive (CP), trace preserving (TP), and Lindbladian are defined below in Appendix A in case you aren’t already familiar with them.

Confusingly, the standard practice is to say a superoperator \mathcal{S} is “positive” when it is positivity preserving: B \ge 0 \Rightarrow \mathcal{S}[B]\ge 0. This condition is logically independent from the property of a superoperator being “positive” in the traditional sense of being a positive operator, i.e., \langle B,\mathcal{S}[B]\rangle \ge 0 for all operators (matrices) B, where

    \[\langle B,C\rangle \equiv \mathrm{Tr}[B^\dagger C] = \sum_{n=1}^N \sum_{n'=1}^N B^\dagger_{nn'} C_{n'n}\]

is the Hilbert-Schmidt inner product on the space of N\times N matrices. We will refer frequently to this latter condition, so for clarity we call it op-positivity, and denote it with the traditional notation \mathcal{S}\ge 0.

Intro

It is reasonably well known by physicists that Lindbladian superoperators, Eq. (1), generate CP time evolution of density matrices, i.e., e^{t\mathcal{L}}[\rho] = \sum_{b=0}^\infty t^b \mathcal{L}^b[\rho]/b! is completely positive when t\ge 0 and \mathcal{L} satisfies Eq.… [continue reading]

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Review of “Lifecycle Investing”

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Summary

In this post I review the 2010 book “Lifecycle Investing” by Ian Ayres and Barry Nalebuff. (Amazon link here; no commission received.) They argue that a large subset of investors should adopt a (currently) unconventional strategy: One’s future retirement contributions should effectively be treated as bonds in one’s retirement portfolio that cannot be efficiently sold; therefore, early in life one should balance these low-volatility assets by gaining exposure to volatile high-return equities that will generically exceed 100% of one’s liquid retirement assets, necessitating some form of borrowing.

“Lifecycle Investing” was recommended to me by a friend who said the book “is extremely worth reading…like learning about index funds for the first time…Like worth paying 1% of your lifetime income to read if that was needed to get access to the ideas…potentially a lot more”. Ayres and Nalebuff lived up to this recommendation. Eventually, I expect the basic ideas, which are simple, to become so widespread and obvious that it will be hard to remember that it required an insight.

In part, what makes the main argument so compelling is that (as shown in the next section), it is closely related to an elegant explanation for something we all knew to be true — you should increase the bond-stock ratio of your portfolio as you get older — yet previously had bad justifications for. It also gives new actionable, non-obvious, and potentially very important advice (buy equities on margin when young) that is appropriately tempered by real-world frictions. And, most importantly, it means I personally feel less bad about already being nearly 100% in stocks when I picked up the book.

My main concerns, which are shared by other reviewers and which are only partially addressed by the authors, are:

  • Future income streams might be more like stocks than bonds for the large majority of people.
… [continue reading]
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COVID Watch and privacy

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[Tina White is a friend of mine and co-founder of COVID Watch, a promising app for improving contact tracing for the coronavirus while preserving privacy. I commissioned Tom Higgins to write this post in order to bring attention to this important project and put it in context of related efforts. -Jess Riedel]

Countries around the world have been developing mobile phone apps to alert people to potential exposure to COVID-19. There are two main mechanism used:

  1. Monitoring a user’s location, comparing it to an external (typically, government) source of information about infections, and notifying the user if they are entering, or previously entered, a high-risk area.
  2. Detecting when two users come in close proximity to each other and then, if one user later reports to have been infected, notifying the second user and/or the government.

The first mechanism generally uses the phone’s location data, which is largely inferred from GPS.In urban areas, GPS is rather inaccurate, and is importantly augmented with location information inferred from WiFi signal strength maps.a   The second method can also be accomplished with GPS, by simply measuring the distance between users, but it can instead be accomplished with phone-to-phone bluetooth connectionsA precursor to smartphone-based contact tracing can be found in the FluPhone app, which was developed in the University of Cambridge Computer Laboratory in 2011. (BBC Coverage.) Contact tracing was provided over bluetooth and cases of the flu were voluntarily reported by users so that those with whom they had come into contact would be alerted. Despite media coverage, less than one percent of Cambridge residents downloaded the app, whether due to a lack of concern over the flu or concerns over privacy.… [continue reading]