Paternity leave: reflections

Paternity leave: reflections 19 April 2018

[I wrote most of this post two years ago, at the end of my six months' paternity leave with K      . Here I am, nearly two years later, finally polishing it up and posting it towards the end of my six months' paternity leave with E      and K      ! I've left my thoughts from two years ago as I wrote them then. Maybe in another two years' time I'll post a follow-up reflecting on what was different about the experience the second time around.]

When K      was still at the theory stage, S      and I spent inordinate amounts of time discussing how we wanted to organise things once she arrived. One of the few things we spent almost no time discussing was what to do about parental leave. It just seemed obvious that we'd split it 50/50, and each take 6 months.It didn't occur to us we were doing anything unusual. If we thought about it at all, I guess it just seemed fair to give each of us an equal share of time with K      , and also to tryto share any career impact it might have.

Once I started my six months leave, I didn't need to look up the statistics to quickly realise how few men were taking any parental leave at all. It was a rare baby singing, music, swimming or soft-play class where I wasn't the only man present. I never found that an issue. To their credit, all the mothers I met made me feel as accepted and welcome as any other parent of a young baby, and I made some really good friends. There's honour among thieves…and solidarity amongst sleep-deprived parents of any gender!

So how do I feel about it having done it? Two things struck me particularly forcefully over the six months.

I didn't understand how hard it was until I was doing it myself

I wasn't /

LaTeX: Cleveref package

LaTeX: Cleveref package 27 March 2018 The Cleveref package (also available from CTAN) does clever things with cross-references:

  • automatic formatting of cross-references based on the type of object referred to (chapter, section, equation, theorem, etc.);
  • full control and customisation of the reference format;
  • cross-references and page references to multiple items;
  • automatic (optional) sorting and compression of multiple cross-references or page references;
  • optional output of a sed script that can strip out Cleveref commands and replace them with standard LaTeX, allowing Cleveref to be used e.g. in articles sent to journals or collaborators that don't (yet!) support Cleveref.

A number of other LaTeX packages provide similar features. Some are venerable enough to be documented in The LaTeX Companion, or even included in the standard LaTeX distribution itself. But all those that I've come across provide only a subset of the Cleveref features. (See the Introduction to the package documentation for some comparisons.)


Full documentation is available in a variety of formats, for download and viewing online. (Since the package is in DocTeX format, the source for the documentation is the same as the package itself.)

  • PDF file
  • Postscript file (gzipped)

Download and Installation

The latest release of the Cleveref package is version 0.21.4 (released March 2018).

Note that this is a pre-release version, which is a not currently available on CTAN, and should not (yet) be considered fully stable. Please report any issues or bugs that you encounter, and check back here periodically for updates. Note that the version number is only finally fixed once the version is uploaded to CTAN. Pre-release versions may be updated without any change to the version number. Compare package dates to see if you hav

Truths about proofs and groups

Truths about proofs and groups 9 February 2018

A while back, a PhD student in our group asked me whether the sum over all elements of a stabilizer is a projector. If you know what this means, you're probably (a) a quantum information theorist (in which case, stop reading here) and (b) already know the answer and how to prove it (unlike me, who'd forgotten both).

The answer is no doubt well-known to anyone who works on quantum stabilizer codes, and we could have just googled for the result. It seemed like a nice, self-contained mathematical question, though. So rather than googling, we tried to figure it out for ourselves at the blackboard.

If you just want to see the simple final answer, skip to the end. But then you'll miss all the fun and the main point of this post. The way we came up with the solution makes for a nice toy example of the convoluted, messy and inelegant process by which mathematical results are really proven. Before they get polished up into the simple, elegant, pristine proofs "from The Book" that are all you ever get to see in textbooks and research papers. The unspoken (or at least unpublished) reality is that elegant proofs invariably emerge after following numerous blind alleys, unjustified intuitive leaps, and inelegant, round-the-houses arguments. All of which get simplified away in time for publication. (Or maybe that's just my proofs!)

Instead of just explaining the elegant final answer, I'm going to explain the inelegant process we went

Emacs: Data structure packages

Emacs: Data structure packages 16 August 2017 These packages provide basic (and not so basic) data structures. They are all relatively stable, though bug-fixes and new features are added occasionally. (Latest update: August 2017).

In recent versions of Emacs (>=24.1), you can install all the non-obsolete packages from within Emacs itself, via GNU ELPA. Use M-x list-packages and take it from there. This is the preferred installation method. (Occasionally, the ELPA version might lag slightly behind the latest version available here.)

  • Git repository:
  • heap.el (version 0.5)
  • queue.el (version 0.2)
  • avl-tree.el (already included in Emacs >=24.1)
  • tNFA.el (version 0.1.1; requires queue.el)
  • tstree.el (obsolete! Use trie.el instead)
  • trie.el (version 0.4; requires everything except dict-tree.el)
  • dict-tree.el (version 0.14; requires everything else)


The functions these packages provide are well documented using Emacs' built-in documentation features. Brief descriptions of the data structures follow:


A heap is a form of efficient self-sorting tree. In particular, the root node is guaranteed to be the highest-ranked entry in the tree. (The comparison function used for ranking the data can, of course, be freely defined). They are often used as priority queues, for scheduling tasks in order of importance, and for implementing efficient sorting algorithms (such as heap-sort).

  • heap.el (version 0.5)


A queue can be used both as a first-in last-out and as a first-in first-out stack, i.e. elements can be added to and removed from the front or back of the queue. (This library is an updated re-implementation of the old Elib queue library.)

  • queue.el (version 0.2)

Tagged Non-deterministic Finite state Automata

Features of modern regexp implementations, including Emacs', mean

Advanced Quantum Information Theory course

Advanced Quantum Information Theory course 23 June 2017 Lectured from 2017 as an optional advanced theory course for the UCL quantum CDT.

Based on a course I originally designed and lectured from 2013 to 2015 as a Part III Mathematics course at the University of Cambridge.

Lecture notes

  • Notation and terminology
  • Bibliography

Section 1: Hamiltonian Complexity

  • Lecture 1: Whistle-stop introduction to computability and complexity theory
  • Lectures 2: Local Hamiltonians
  • Lectures 3: Kitaev's Theorem
  • Lecture 4: Local clock construction

Section 2: Lieb-Robinson techniques

  • Lecture 5: Many-body quantum physics introduction
  • Lecture 6: Lieb-Robinson bounds
  • Lectures 7-8: Exponential decay of correlations

Problem sheets

  • Examples sheet 1: Complexity theory; Hamiltonian complexity
  • Examples sheet 2: Kitaev's theorem; Many-body physics; Lieb-Robinson bounds

Course description

Quantum information theory is neither wholly physics (though it's mostly about quantum mechanics), nor wholly mathematics (though it mostly proves rigorous mathematical results), nor wholly computer science (though it's mostly about storing, processing, or transmitting information). Over the last two decades, it has developed into a rich mathematical theory of information in quantum mechanical systems, that draws on all three of these disciplines. More recently, this has been turned on its head: quantum information is beginning to be used to attack deep problems in physics, computer science, and mathematics.

The aim of this course is to select one or two advanced topics in quantum information theory, close to the cutting edge of research, and cover them in some depth and rigour.

This time around, I will focus on quantum information in many-body systems. What do these two topics have to do with each other? Quantum computation aims to engineer complex many-body systems to process information in ways that would not be possible classically. Many-body physics aims to understand the complex behaviour of nat

Dear Diane

Dear Diane 27 January 2017

[Letter sent to my local MP, Diane Abbott. Do the same! Keep up the pressure on MPs to represent your views.]

Dear Diane,

When I first moved to Hackney, I was proud to tell people I had you as my MP. As one of the few voices on the Labour backbenches consistently voting according to conscience, defying the party whip when it was at odds with your principles and your constituents' interests, you stood out from the crowd of political apparatchiks toting the party line. On issues ranging from the Iraq war, to defending the NHS from privatisation, to resisting the incoming tide of government mass surveillance, your voting record aligned even more closely with my views than the overall Labour party line. Though I've been a lifelong Labour voter, I was even happier to be a Diane Abbot voter.

You campaigned for remaining in the EU. Your constituents voted overwhelmingly remain, the joint-second highest remain vote (with Lambeth) after Gibraltar. You know that opposing a hard Brexit and fighting to keep the UK in the common market is in the best interests of your constituents, not just economically but also socially. You know that, for 40 years, the strongest bulwark against dismantling of social protections, civil liberties, and workers' rights in the UK has been European legislation. You know that fighting to retain as much of that as possible is fighting to prevent Theresa May's race to the very bottom in pandering to Trump, legitimising corporate tax evasion, liberating corporations to exploit employees, and dismantling and privatising the services that provide a safety net to so many in the UK.

Now you are in the unenviable position of being in the shadow cabinet of a party imposing a three-line whip to vote against your own views, and against the overwhelming majority view of your own constituents. If you abandon your principles and independent-mindedness now, just because you sit in the shadow cabinet, will you be able to live with your own capi

Quantum Computation and Complexity course

Quantum Computation and Complexity course 15 July 2016 Lectured at the 2016 Autrans summer school on Stochastic Methods in Quantum Mechanics. The notes are adapted from the first half my Advanced Quantum Information Theory course, with additional material on the basics of computation and complexity theory.

Lecture notes

  • Lectures 1-2: Computation and Complexity
  • Lecture 3: Local Hamiltonians
  • Lectures 3-4: Kitaev's Theorem
  • Lecture 4: Local clock construction

Recommended reading

The Arora-Barak book gives an excellent, modern treatment of the theory of computation and complexity, going far beyond what's covered in this short course. The proof of Kitaev's theorem closely follows the original from the Kitaev-Schen-Vyalyi book. The other references are review papers on Hamiltonian complexity, which may also be of interest.

  • Arora and Barak, "Complexity Theory: A Modern Approach, Cambridge University Press
  • Kitaev, A., Shen, A., and Vyalyi M. "Classical and Quantum Computation", American Mathematical Society
  • Aharonov, D. and Naveh, T. "Quantum NP - a Survey"
  • Gharibian, S., Huang, Y. and Landau, Z. "Hamiltonian Complexity"

The following is a selective and incomplete list of links to the arXiv versions of papers that proved key results in Hamiltonian Complexity post-Kitaev. (These are the papers I mentioned in the brief survey at the very end of the lecture course.)

QMA-completeness with stronger locality conditions, and related results

  • J. Kempe and O. Regev, "3-local Hamiltonian is QMA-complete" (2003)

Proves QMA-completeness of the k-local Hamiltonian problem for \(k=3\).

  • J. Kempe, A. Kitaev and O. Regev, "The Complexity of the Local Hamiltonian Problem" (2004)

Proves QMA-completeness of the k-local Hamiltonian problem for \(k=2\). Introduces the perturbation gadget technique.

  • R. Oliveira, B. Terhal, "The complexity of quantum spin systems on a two-dimensional square lattice" (2007)

Proves QMA-completeness of the k-local Hamiltonian problem for nearest-neighbo

Matrix Product States and PEPS

Matrix Product States and PEPS 14 July 2016 Notes from David Perez-Garcia's lecture course on Matrix Product States and PEPS at the 2016 Autrans summer school on Stochastic Methods in Quantum Mechanics.

The slides are courtesy of David. The lecture notes are my handwritten notes from the whiteboard section of David's lectures. All content by David; all mistakes by me!

Lecture notes

  • MPS motivation (slides)
  • MPS lecture notes (handwritten)
  • PEPS and topological order (slides)

(The slides are copyright © 2016 David Perez-Garcia, with all rights reserved. The handwritten notes are copyright © 2016 Toby Cubitt, and are licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.)

Paternity leave: statistics

Lies, damn lies, and…

A couple of months ago, the statistic that only 1% of men had taken up shared parental leave was splashed all over the British media. (Shared Parental Leave was introduced in the UK in 2015, and essentially allows parents to share 12 months of leave however they like. Taking it consecutively, simultaneously, alternating blocks of leave between both parents, or a mixture of the above are all permitted.)

My Family Care, the company that carried out the survey on which this statistic was based, apparetly asked Human Resources directors at 200 businesses what percentage of men in their company had taken shared parental leave in the year since it was introduced. But, as Radio's 4's excellent More or Less programme pointed out, they forgot to ask what percentage of those men were actually elligible for parental leave in the first place! Most of them won't have had children at all in the last year. Some of them won't even have any children!! Ooops.

Clearly, the fact that 1% of all men (fathers or otherwise) have taken up shared parental leave tells us next to nothing about the take up of shared parental leave. We can try to extract from this a very crude estimate of the percentage of eligible new fathers taking it up, using average birth rate figures. We definitely shouldn't be doing this, for all kinds of reasons. For one thing, applying the average birth rate to the demographic of employed men is so dubious it's almost certainly plain wrong. The birth rate amongst employed men will differ substantially from the overall birth rate per head of population - the latter figure includes retired men, for example. (This will skew the estimate far more than the points My Family Care later raised: that adoptive parents and same-sex parents are also eligible, and some fathers won't have worked long enough to qualify.) More heinous still, we're ignoring all the uncertainties in the data. With such small samples and small percentages,

Paternity leave: reactions

Paternity leave: reactions 27 May 2016 I've been on paternity leave since January, taking care of K      full time. The reactions I've had from people when they discover I'm on paternity leave for half a year have been entirely positive. But some of the comments I've had in response have been interesting. I've collected the ones that stuck in my mind, together with my thoughts on them.

(If you know me personally, and think you recognise something you've said, you don't! These aren't direct quotes. I've paraphrased things that have been said to me multiple times by many different people.)

"I wish I could have taken paternity leave, but I can't really do that in my job."

My first instinct is to reply: actually, you can do this, in any job! You have a legal entitlement to shared parental leave in the UK, so if you want to take it your employer can't refuse. (Even the rules on qualifying periods are surprisingly lenient.)

Of course, this isn't what they mean. I think what people mean by this comment is some combination of:

  1. "My employer would find it difficult to get someone to cover my job for six months";
  2. "Taking six months off might negatively impact my career";
  3. "The culture in my workplace wouldn't be supportive";
  4. "No one has ever taken paternity leave before in my company";
  5. "I'm subconciously using my job as a reason to avoid seriously contemplating taking paternity leave".

It's revealing (but not very surprising) that this comment has only ever been said to me by men, never by women. Yet at least 1. and 2. apply just as much to women taking maternity leave. (No doubt 3. too in some workplaces.)

I'm sure the people who've said this to me didn't intend it, and wouldn't defend it. But implicit in what they said is the view that it's acceptable or inevitable that women will take a career hit when they have children. Whereas for men this is reason enough not to take paternity leave.

It shouldn't be acceptable or inevitable, for either gender.[fn::To be

Decoupling Method in Quantum Shannon Theory

Decoupling Method in Quantum Shannon Theory 19 October 2015 Originally lectured in 2015 as part of the quantum information theory masters course for the UCL quantum CDT.

Lecture Notes

  • Decoupling Method

Recommended reading

Much of the material covered here (and more!) was originally proven in the Mother of All Protocols paper by Abeyesinghe, Devetak, Hayden and Winter.

These notes largely follow Section 10.9 of Preskill's wonderful lecture notes, with a (very) few modifications and additions.

Emacs: Miscelaneous packages

Emacs: Miscelaneous packages 12 June 2015 These packages provide miscelaneous features I needed at some point. So I coded them. Currently, they're all to do with displaying useful information in the mode line.

show-point-mode displays the current value of the point in the mode line. I primarily find it useful when debugging Elisp code that uses overlays and markers.

wc-mode displays output similar to the Unix wc command in the mode line, i.e. the character count, word count and line count for the current buffer. (I primarily find this Useful when writing grant applications with character or word limits. Though I'm sure it's useful for other more productive activities, too…)

  • show-point-mode.el (version 0.3)
  • wc-mode.el (version 0.3)

Matlab code

Matlab code 28 December 2013 I've collected here various functions, routines, and other bits of Matlab, Octave and Mathematica code organized by topic, that might save someone, somewhere, from re-inventing the wheel. Some of them are so simple it would probably be quicker to re-code them than find this page, but since you're here anyway…

Comments within the code should be enough to figure out what they do and how to use them (try help <function> from within Matlab or Octave). No guarantee they work as advertised, but I use them myself so I do correct bugs when I come across them. The Matlab code should run under both Octave and Matlab.

All the Matlab, Octave and Mathematica code linked from this page is released under the GPL license, version 2 or later.

If you make use of this code in your research, consider including a citation to this web page in any resulting publication. Not only is it fair to give credit when you've made use of other people's work, it is important for scientific reproducability to document any code you used to help produce your results. Also, if you found this code useful, then others probably will too! Citing this page helps others find it.

I leave it to your judgement whether you feel your results made sufficient use of this code to warrant a citation; I do not insist on it. But consider whether using this code has been as helpful to your research as the least useful paper you are citing. (Typically, this sets the bar very low!). If so, you should probably include a citation to this web page.

Quantum Information Package

For convenience, all the Matlab/Octave functions related to quantum mechanics and quantum information theory are also available in a single bundle, as well as individually (below).

  • Quantinf package (version 0.5.1)

Linear algebra

Partial trace:\(TrX(\rho \text{ or } \psi, sys, dim)\)

Emacs: Undo Tree package

Emacs: Undo Tree package 8 August 2013 Emacs has a powerful undo system. Unlike the standard undo/redo system in most software, it allows you to recover any past state of a buffer (whereas the standard undo/redo system can lose past states as soon as you redo). However, this power comes at a price: many people find Emacs' undo system confusing and difficult to use, spawning a number of packages that replace it with the less powerful but more intuitive undo/redo system. (See the Emacs Wiki.)

Both the loss of data with standard undo/redo, and the confusion of Emacs' undo, stem from trying to treat undo history as a linear sequence of changes. It's not. The undo-tree-mode provided by this package replaces Emacs' undo system with a system that treats undo history as what it is: a branching tree of changes. This simple idea allows the more intuitive behaviour of the standard undo/redo system to be combined with the power of never losing any history. An added side bonus is that undo history can in some cases be stored more efficiently, allowing more changes to accumulate before Emacs starts discarding history.

It gets better. You don't have to imagine the undo tree, because undo-tree-mode includes an undo-tree visualizer which draws it for you, and lets you browse around the undo history.

The only downside to this more advanced yet simpler undo system is that it was inspired by Vim. But, after all, most successful religions steal the best ideas from their competitors!


Details of the undo-tree-mode commands and key bindings can be found in the Commentary section at the top of the undo-tree.el file itself, along with an extended explanation (with diagrams!) of the differences between the undo-tree-mode, standard Emacs' undo, and undo/redo systems.

The commands themselves are all documented using the standard, built-in Emacs documentation features. Customization options can be found under the undo-tree customization group.

Download and Installation

In recent v

Emacs: Auto-Overlays package

Emacs: Auto-Overlays package 22 February 2013 The Auto-Overlays package allows you to define overlays that are created (and updated and destroyed) automatically when text in a buffer matches a regular expression.

Various classes of automatic overlay are provided, to make it easy to define matches for different text regions: words, lines, regions enclosed by start and end tags, or regions enclosed by delimiters. You can also define your own custom classes.

The overlays are updated just before any buffer modification. The built in overlay classes only update as much as is necessary to ensure that overlays covering the point are consistent. Therefore the overlays at the point are guaranteed to be correct before any buffer modification takes place there, but updating the overlays is fast and usually causes no noticeable delay.

Download and Installation

In recent versions of Emacs (>=24.1), you can install the Auto-Overlays package from within Emacs itself, via GNU ELPA. Use M-x list-packages and take it from there. This is the preferred installation method. (Occasionally, the ELPA version might lag slightly behind the latest version available here.)

The current release of the Auto-Overlays package is version 0.10.8 (released February 2013). It's relatively stable, though bug fixes and new features are added occasionally.

If you want to live on the bleeding edge, the latest "development" version of the Auto-Overlays package is hosted in the same git repository as the Predictive Completion package.

  • Git repository:
  • Current version (0.10.8)

Note that the git repository URL is a git repository, not a web-site. You cannot view it in a web browser. To grab the latest development version, clone the repository using something like: #+BEGINEXAMPLE git clone http://www.d

Emacs: Completion User Interface package

Emacs: Completion User Interface package 22 February 2013 The Completion User Interface package is a library that implements user-interfaces for in-buffer completion.

Typically, in packages providing some kind of text completion, a large amount of code deals with providing the user interface rather than finding good completions. The goal of Completion-UI is to be the swiss-army knife of in-buffer completion user-interfaces; a library which any completion package can use to provide an in-buffer completion user-interface, thereby freeing completion package writers to concentrate on the task of finding the completions in the first place.

In fact, Completion-UI is even better than a swiss-army knife, because it's also extensible: it's easyto add new completion user-interfaces and hook them into Completion-UI. The new interface will then be available to any completion package that uses the Completion-UI library, without needing to make any changes to that package.

Various standard completion user-interfaces and commands are provided "out of the box". These can be separately enabled, disabled and tweaked by the Emacs user via the usual bewildering array of customization variables. The UIs provided with the package are:

Dynamic completion
Provisionally inserts the best completion candidate in the buffer, highlighting the completed portion.
Completion hotkeys
Single-key selection of a completion.
Cycle through completion candidates.
"Traditional" expansion to longest common substring.
Echo area
Display a list of completion candidates in the echo-area.
Display a list of completion candidates in a tool-tip located below the point.
Pop-up tip
Display a list of completion candidates in a pseudo tool-tip located below the point, using the <code>popup.el</code> library.Displays faster and looks better than real tooltips.
Pop-up frame
Allow completion candidates to be selected from a pop-up frame displayed below the point.
Completion menu
Allow completion candidates to be selected from a drop-down menu located below the point.

Emacs: Predictive Completion package

Emacs: Predictive Completion package 22 February 2013 The Emacs Predictive Completion package adds a new minor-mode to the GNU Emacs editor. When enabled, predictive mode exploits the redundancy inherent in languages in order to complete words you are typing before you've finished typing them (somewhat like the IntelliSense feature in some IDEs). It is highly customisable, and works happily alongside other Emacs major modes. See the documentation for more details.

Predictive mode only works under GNU Emacs, not under XEmacs. It may be possible to get it to work under XEmacs with a modicum of work. (At the very least, the overlay compatibility package would be required.) If you're interested in attempting this, then I'm happy to answer questions, but I have no plans to do it myself.

Download and Installation

The current release of the Predictive Completion package is version 0.24 (released February 2013). It's still under active development, so don't forget to check back here for updates every so often. If you want to live on the bleeding edge, the latest "development" version of the Predictive package is hosted in a git repository.

  • Git repository:
  • Current version (0.24)

Note that the git repository URL is a git repository, not a web-site. You cannot view it in a web browser. To grab the latest development version, clone the repository using something like:

git clone

Whether you downloaded the package or cloned the git repository, you need to perform some further steps in order to install the Predictive Completion package on your system. See the Obtaining and Installing section of the manual and the INSTALL file included in the package for installation instructions.


Full documentation is available in a variety of formats,

LaTeX: Quantum package

LaTeX: Quantum package 2 October 2012 The Quantum package defines a number of commands and short-hands useful when writing about quantum mechanics, and quantum information theory in particular.

There is no separate documentation; read the package source to find out what commands it provides.

  • Quantum package

LaTeX: Authord package

LaTeX: Authord package 27 July 2012 Gives a complete solution to the problem of precedence in scientific pubication, in a way that Don Knuth would surely approve of.

  • Authord package
  • Authord documentation

Quantum Mechanics for Mathematicians course

Quantum Mechanics for Mathematicians course 4 November 2011 Lectured in 2011 as the first section of a "Mathematics for Quantum Information" masters course given in the mathematics faculty of the Universidad Complutense de Madrid.

Lecture Notes

  • Section 0: Dirac notation;
    Section 1: The postulates of quantum mechanics
    (lecture 1)
  • Section 2: Combining quantum systems: tensor products
    (lecture 2)
  • Section 3: Non-locality and Bell inequalities
    (lecture 3)
  • Section 4: Ensembles and density operators;
    Section 5: Taking quantum systems apart: reduced states and the partial trace;
    Section 6: A brief introduction to entropy
    (lecture 4)

Recommended books

I have closely followed Chapter 2 of Nielsen and Chuang (which is by now the standard textbook on quantum information theory), with some additional mathematical content, and a more careful proof of the CHSH inequality.

The other books listed below may also be of interest:

  • "Quantum Computation and Quantum Information", Nielsen & Chuang Chapter 2 gives a concise but excellent introduction to quantum mechanics, more suitable for quantum information theory than most quantum mechanics textbooks. I have closely followed this chapter, but I have given additional mathematical results and proofs when desirable. Bell inequalities are also covered in this chapter, but this is not the focus of the book and the proof they give obscures some of the subtleties.
  • "An Introduction to Quantum Theory", Hannabuss A nice and more mathematically oriented quantum mechanics textbook, but still contains a lot more physics than covered in this course.
  • "Quantum Theory: Concepts and Methods", Asher Peres A delightful text book that contains a good treatement of the Bell experiment and much more.
  • "Bell Inequalities and Entanglement", Werner and Wolf arXiv:quant-ph/0107093 This review article gives a careful and rigorous discussion of Bell inequalities.
  • "Speakable and Unspeakable in Quantum Mechanics", John Bell A collection of insightful essays and papers by John Bell (of Bell inequality fame).
  • "Feynman Lectures vol. 3", Feynman, Leighton, Sands

Quantum Mechanics course

Quantum Mechanics course 11 May 2010 Lectured from 2007 to 2010 as the second part of the 3rd year mathematics undergraduate "Quantum Mechanics" course at the University of Bristol.

Lecture Notes

  • Section 1: Angular Momentum and Spin
    (lectures 1 and 2)
  • Section 2: Representations of Angular Momentum
    (lectures 3 to 5)
  • Section 3: Orbital Anglular Momentum
    (bonus lecture)
  • Section 4: Measurement
    (lecture 6)
  • Section 5: Multiple Particles and Tensor Products
    (lectures 7 and 8)
  • Section 6: Non-Locality and Bell Inequalities
    (lectures 9 and 10)

Problem Sheets

Problem sheets 7 and 8 correspond to my section of the course. I have removed the solution sheets, as the same problems may be used by future lecturers. If you want a copy of the solutions for purposes other than avoiding having a good go at the problems yourself, email me.

  • Problem Sheet 7
  • Problem Sheet 8

Recommended books

Angular momentum (lectures 1 to 5)

The main text book for this part of the course is the book by Hannabuss. But any good text book on quantum mechanics will cover this material. A sample of ones I like is listed below, but if you find one that presents the material in a way that you find easier, you should by all means make use of it.

  • "An Introduction to Quantum Theory", Hannabuss.
    The angular momentum section of the course closely follows chapter 8.
  • "Modern Quantum Mechanics", Sakurai
  • "Quantum Mechanics",Cohen-Tannoudji
  • "Group Theory in Physics", Cornwell.
    Chapter 12, Volume 2. For interest only; well beyond the level of the course.
  • "Feynman Lectures vol. 3", Feynman, Leighton, Sands.
    As an accompaniement to the other books, volume 3 of Feynman's famous lecture series contains a presentation of quantum mechanics with a different and somewhat less mathematical flavour, which some may find helpful or interesting.

Measurement, tensor products, non-locality, entanglement and Bell inequalities (lectures 5 to 10)

Why I use TMDA

Why I use TMDA 16 July 2005 Mine is a sad and familiar story. I was drowning in a deluge of spam (a.k.a. junk email), and it had become such a problem that email was fast becoming useless for me. Having to sort through and delete hundreds of spam emails per day was bad enough. Worse was the increasing frequency with which I was accidentally deleting legitimate email along with the spam.

There are various ways to fight this deluge of spam. The most common is to use a filter that tries to recognise and delete the spam (or, more usually, move it to a spam box for later perusal). This is quite effective. A small amount of spam will not be recognised as such (false-negatives), and will end up in your inbox anyway. But the amount of spam will usually be cut down to a manageable amount, rendering email usable again.

The problem with the filtering approach is the false-positives: legitimate mail that gets mis-identified as spam. Even if it's moved to a spam box rather than deleted, when you're searching through hundreds of spam emails you're almost certain to miss the one or two legitimate mails hiding amongst them, and you'll delete them along with the spam. (At least, that's what I found myself doing.)

A second approach is to use techniques such as domain blocking, real-time blacklists, and other methods of blocking whole groups of addresses known to send spam. But this is really just a variation on filtering (filters usually take the sender's address into account when deciding whether an email is spam or not, as well as the body).

I didn't want to run the risk of someone sending me an email, me deleting it accidentally, and them never knowing that I didn't received it. So I chose to use a third approach: white-list plus challenge/response (plus a number of other features of the impressive TMDA system).

The first part of this system is a `white-list' of email addresses belonging to people I know, or have exchanged email with in the past. Any email from an address on the white-l

Classical mechanics and electrodynamics

Classical mechanics and electrodynamics 14 May 2004 I have left up some of the material I prepared for classical mechanics and electrodynamics courses taught by Prof. Weise at the TUM (many, many years ago!) in case it's of use to someone.

Question Sheet Solutions

Given that the question sheets are substantially re-used in subsequent semesters, I've removed the worked solutions that were available here, to help you avoid the temptation to…ahem…short-cut the valuable learning process that struggling to solve the questions provides. (Believe it or not, the question sheets are not some obscure form of torture dreamed up by bitter and twisted physics professors).

If anyone involved in teaching the courses is interested in obtaining the solutions, drop me an email. I have scanned copies for about half the mechanics question sheets and all the electrodynamics question sheets.

Extra information sheets

  • Summation convention and \(\delta\)-functions

Recommended books

These books complement those recommended in the lectures, rather than replacing them. I recommend them as alternative references written in a less formal style, for when you're confused by the more formal approach (or just want to read more):

  • Feynman Lectures vols. 1 and 2
    Feynman, Leighton, Sands
  • Mathematical Methods for Physics and Engineering
    Riley, Hobson, Bence


I started making a list of physical and mathematical vocabulary in both English and German:

  • English/German mathematical vocabulary

If you have corrections or words you would like to see added, please email me.

Toby 'qubit' Cubitt

Who am I? (a brief Curriculum Vitae)

I'm a nationality-confused European, born and raised in Luxembourg but technically British.

I went to the European school in Luxembourg, graduating with the European Baccalaureate in 1998. From there, I hopped across the Channel to Churchill College, Cambridge, studying physics under the Natural Sciences Tripos at the University of Cambridge.

After graduating in 2002, I decided to see what the other end of Europe was like, and moved to the Max Planck Institute for Quantum Optics just outside Munich, Germany to do a PhD in quantum information theory under the wonderful Ignacio Cirac.

After finishing my PhD in 2006, I hopped back across the Channel, defecting from physics to maths en route (or maybe I'm just masquerading as a mathematician…), to the mathematics section of the quantum information theory group at the University of Bristol. I stayed there as a postdoc for four years, the latter two as a Leverhulme Early Career Fellow.

In 2010, my Leverhulme fellowship over, it was clearly time to live in a new country! So, still masquerading as a mathematician (or maybe I've really been converted now…), I moved to Spain with a Juan de la Cierva fellowship, joining the Mathematics and Quantum Information Theory group of old MPQ friend David Pérez-García, in the Departamento de Análisis (wow! just like a real mathematician!) within the Facultad de Ciencias Matemáticas at the Universidad Complutense de Madrid.

After two-and-a-bit fantastic years in Madrid, I was awarded a Royal Society University Research Fellowship. So, time to move back to the UK again, this time back to my alma mater, the University of Cambridge, where I was in the quantum information theory group based in the Department of Applied Mathematics and Theoretical Physics (thereby neatly sidestepping the need to resolve the physicist/mathematician ambiguity…).

Another two-and-a-bit years later, having worked in physics, applied maths and even pure math

Emailing me

Email address

I can be reached by email at, associated with this PGP key:

BB74 FB42 4C64 4CB7 3571  39AA A96F 4A67 4DC3 9B79

As of 2 August 2013, I transitioned from an old 1024-bit DSA key to this new 4096-bit RSA key. I will be signing all software releases with the new key. Please also use the new key for all correspondence. See the transition statement to certify the transition, and for more details.

Note that I use FLOSS spam-reduction software called TMDA to protect my addresses from junk-mail.

If you've never exchanged any email with me previously, you'll receive a message asking you to verify your email address. By simply replying to the message (literally just hit "Reply" then "Send"), your original message will be delivered. You'll only have to confirm your address once ever. All subsequent email from that address will be delivered directly.

Academic email

I don't generally use this email address for academic work. For academic-related email, you're better off using my university email address, which isn't protected by such stringent spam filters. It's easy enough to find my university email address online.

If you're an individual researcher, and you send academic-related email to this address, then as long you successfully navigate the anti-spam system your mail will still reach me and I'll read and respond to it. (Though I'll probably reply from my university email account.)

If you're an editor for a journal I've never previously published in, and you send a referee request to me at this address, then your request is very likely to get held in my anti-spam system and ignored. Even if it does get through, it will probably be silently ignored. I take a dim view of journals I have no previous association with sending referee requests to my personal email address, which they can only have found by googling my name, instead of having the common courtesy of looking up my pr


Publications You can also find all of my papers on the arXiv (which is sometimes more up-to-date than this list).

Published Papers

  1. Size-Driven Quantum Phase Transitions Johannes Bausch, Toby S. Cubitt, Angelo Lucia, David Perez-Garcia and Michael M. Wolf Proc. Natl. Acad. Sci. 115:1, p19–23 (2018) [18 pages] arXiv:1512.05687[quant-ph]
  2. The Complexity of Translationally-Invariant Spin Chains with Low Local Dimension Johannes Bausch, Toby Cubitt and Maris Ozols Annales Henri Poincaré, 18:11, p3449–3513 (2017) [63 pages] arXiv:1605.01718[quant-ph]
  3. Fundamental Limitations in the Purifications of Tensor Networks G. De las Cuevas, T. S. Cubitt, J.I. Cirac, M. M. Wolf and D. Perez-Garcia J. Math. Phys. 57, 071902 (2016) [8 pages] arXiv:1512.05709[quant-ph]
  4. The Complexity of Divisibility Johannes Bausch and Toby S. Cubitt J. Linear Alg. 504, p64–107 (2016) [50 pages] arXiv:1411.7380[math.PR]
  5. Complexity Classification of Local Hamiltonian Problems Toby Cubitt and Ashley Montanaro SIAM J. on Computing, 45:2, p268–316 (2016) [50 pages] arXiv:1311.3161[quant-ph]
  6. Simple Universal Models Capture all Classical Spin Physics Gemma de las Cuevas and Toby S. Cubitt Science, 351:6278, p1180-1183 (2016) [47 pages] arXiv:1406.5955[cond-mat.stat-mech]
  7. Area law for fixed points of rapidly mixing dissipative quantum systems F. G. S. L. Brandao, T. S. Cubitt, A. Lucia, S. Michalakis and D. Perez-Garcia J. Math. Phys. 56, 102202 (2015) [17 pages] arXiv:1505.02776[quant-ph]
  8. Undecidability of the Spectral Gap Toby S. Cubitt, David Perez-Garcia and Michael M. Wolf Nature, 528, p207–211, (2015) arXiv:1502.04135[quant-ph] (short version) arXiv:1502.04573[quant-ph] (full version, 143 pages)
  9. Quantum reverse hypercontractivity T. Cubitt, M. Kastoryano, A. Montanaro and K. Temme J. Math. Phys. 56, 102204 (2015) [14 pages] arXiv:1504.06143[quant-ph]
  10. Rapid Mixing and Stability of Quantum Dissipative Systems Toby S. Cubitt, Angelo Lucia, Spyridon Michalakis, and David Perez-Garcia Phys. Rev. A 91, 040302 (2015) arXiv:1409.7809[quant-ph]
  11. Unbounded Number of Channel Uses may be Required to Detect Quantum Capacity D. Elkouss, S. Strelchuck, W. Matthews, M. Ozols, D. Perez-Garcia and T. S. Cubitt Nature Communications 6, 7739 (2015) [11 pages] arXiv:1408.5115[quant-ph]
  12. Complexity Classification of Local Hamiltonian Problems Toby Cubitt and Ashley Montanaro IEEE 55th Annual Symposium on Foundations of Computer Science (FOCS), p120–129 (2014) arXiv:1311.3161[quant-ph]
  13. Bounds on Entanglement Assisted Source-Channel Coding via the Lovász Theta Number and its Variants Toby Cubitt, Laura Mancinska, David Roberson, Simone Severini, Dan Stahlke and Andreas Winter IEEE Trans. Inform. Theory 60, 7330 (2014) [15 pages] arXiv:1310.7120[quant-ph]
  14. Stability of local quantum dissipative systems Toby S. Cubitt, Angelo Lucia, Spyridon Michalakis, and David Perez-Garcia Commun. Math. Phys. 337, 1275 (2015) [38 pages] arXiv:1303.4744[quant-ph]
  15. Preparing Topological PEPS on a Quantum Computer M. Schwarz, K. Temme, F. Verstraete, D. Perez-Garcia and T. S. Cubitt Phys. Rev. A, 88, 032321 (2013) (Editors' suggestion) arXiv:1211.4050[quant-ph]
  16. Entanglement can Completely Defeat Quantum Noise Jianxin Chen, Toby S. Cubitt, Aram W. Harrow and Graeme Smith Phys. Rev. Lett. 107, 250504 (2011) (Editor's suggestion) arXiv:1109.0540[quant-ph] (highlighted in APS Physics article)
  17. Extracting Dynamical Equations from Experimental Data is NP-Hard Toby S. Cubitt, Jens Eisert and Michael M. Wolf Phys. Rev. Lett. 108, 120503 (2012) (Editor's suggestion) arXiv:1005.0005[quant-ph] (highlighted in Science NOW article and in APS Physics article)
  18. Zero-Error Channel Capacity and Simulation Assisted by Non-Local Correlations T. S. Cubitt, D. Leung, W. Matthews and A. Winter IEEE Trans. Inform. Theory 57:8, 5509–5523 (2011) [15 pages] arXiv:1003.3195[quant-ph]
  19. Super-duper-activation of the zero-error quantum capacity Jianxin Chen, Toby S. Cubitt, Aram W. Harrow and Graeme Smith IEEE International Symposium on Information Theory (ISIT), p2695–2697 (2010)
  20. An Extreme Form of Superactivation for Quantum Zero-Error Capacities Toby S. Cubitt and Graeme Smith IEEE Trans. Inform. Theory 58:3, 1953–1961 (2012) [9 pages] arXiv:0912.2737[quant-ph]
  21. Improving Zero-Error Classical Communication with Entanglement T. S. Cubitt, D. Leung, W. Matthews and A. Winter Phys. Rev. Lett. 104, 230503 (2010) arXiv:0911.5300[quant-ph]
  22. The Complexity of Relating Quantum Channels to Master Equations Toby S. Cubitt, Jens Eisert and Michael M. Wolf Commun. Math. Phys. 310, 383–417 (2012) [35 pages] arXiv:0908.2128[quant-ph]
  23. Superactivation of the Asymptotic Zero-Error Classical Capacity of a Quantum Channel Toby S. Cubitt, Jianxin Chen and Aram W. Harrow IEEE Trans. Inform. Theory 57:12, 8114–8126 (2011) [8 pages] arXiv:0906.2547[quant-ph]
  24. Non-Secret Correlations can be Used to Distribute Secrecy Joonwoo Bae, Toby S. Cubitt and Antonio Acín Phys. Rev. A 79, 032304 (2009) arXiv:0806.1606[quant-ph]
  25. The Structure of Degradable Quantum Channels Toby S. Cubitt, Mary Beth Ruskai and Graeme Smith J. Math. Phys. 49, 102104 (2008) [27 pages] arXiv:0802.1460[quant-ph]
  26. Counterexamples to Additivity of Minimum Output p-Rényi Entropy for p close to 0 Toby S. Cubitt, Aram W. Harrow, Debbie Leung, Ashley Montanaro and Andreas Winter Commun. Math. Phys. 284, 281–290 (2008) [9 pages] arXiv:0712.3628[quant-ph]
  27. Assessing non-Markovian Dynamics M. M. Wolf, J. Eisert, T. S. Cubitt and J.I. Cirac Phys. Rev. Lett. 101, 150402 (2008) arXiv:0711.3172[quant-ph]
  28. On the Dimension of Subspaces with Bounded Schmidt Rank Toby S. Cubitt, Ashley Montanaro and Andreas Winter J. Math. Phys. 49, 022107 (2008) arXiv:0706.0705[quant-ph]
  29. Engineering Correlation and Entanglement Dynamics in Spin Systems T. S. Cubitt and J.I. Cirac Phys. Rev. Lett. 100, 180406 (2008) arXiv:quant-ph/0701053
  30. Entanglement Flow in Multipartite Systems T. S. Cubitt, F. Verstraete and J.I. Cirac Phys. Rev. A 71, 052308 (2005) [12 pages] arXiv:quant-ph/0404179
  31. Separable States can be Used to Distribute Entanglement T. S. Cubitt, F. Verstraete, W. Dür, J.I. Cirac Phys. Rev. Lett. 91, 037902 (2003) arXiv:quant-ph/0302168 (highlighted in Science NOW article)


  1. Universal Quantum Hamiltonians Toby Cubitt, Ashley Montanaro and Stephen Piddock arXiv:1701.05182[quant-ph]
  2. Comment on "On the uncomputability of the spectral gap" Toby S. Cubitt, David Perez-Garcia and Michael M. Wolf arXiv:1603.00825[quant-ph]
  3. Universal Refocusing of Systematic Quantum Noise Imdad S. B. Sardharwalla, Toby S. Cubitt, Aram W. Harrow and Noah Linden arXiv:1602.07963[quant-ph]
  4. Undecidability of the Spectral Gap (full version) Toby S. Cubitt, David Perez-Garcia and Michael M. Wolf arXiv:1502.04573[quant-ph]
  5. An Information-Theoretic Proof of the Constructive Commutative Quantum Lovász Local Lemma M. Schwarz, T. S. Cubitt and Frank Verstraete arXiv:1311.6474[quant-ph]
  6. Are Problems in Quantum Information Theory (Un)decidable? Michael M. Wolf, Toby S. Cubitt and David Perez-Garcia arXiv:1111.5425[quant-ph]
  7. Entanglement in the Stabilizer Formalism David Fattal, Toby S. Cubitt, Yoshihisa Yamamoto, Sergey Bravyi and Isaac L. Chuang arXiv:quant-ph/0406168

Research interests

General interests

  • Quantum information theory
  • Many-body physics
  • Complexity theory
  • Hamiltonian complexity
  • Hamiltonian simulation
  • CP maps (a.k.a. quantum channels)
  • Entanglement theory
  • Probability theory
  • Algebraic geometry
  • Learning any other interesting new maths I come across…

That'll do for now.


You can find a (possibly not-quite-up-to-date) list of my publications on this web site with links to the papers, as well as the slides from some of my talks. For a more up-to-date list, try the arXiv.