Home on EPHhttps://ericphanson.com/Recent content in Home on EPHHugo -- gohugo.ioen-usMon, 20 Aug 2018 17:35:28 -0400Is QSEP-CIRCUIT QMA-hard?https://ericphanson.com/work/presentation/is-qsep-circuit-qma-hard/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/presentation/is-qsep-circuit-qma-hard/“QSEP-CIRCUIT, as defined in a recent paper by Patrick Hayden, Kevin Milner, and Mark Wilde (http://arxiv.org/abs/1211.6120)in the computer science department here at McGill, is the problem of determining whether or not a quantum state given as a circuit is separable or not. The problem has been shown to be in QIP(2), and to be NP-hard and QSZK-hard, which lends credence to the idea that the problem is QMA-hard. The goal of this talk is to provide background for and explain in detail what QSEP-CIRCUIT and QMA are, explain why we care whether or not QSEP-CIRCUIT is QMA-hard, and hopefully showcase quantum information theory as an exciting and dynamic field.QSEP-STATE in QMIPnehttps://ericphanson.com/work/poster/qsep-state-in-qmipne/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/poster/qsep-state-in-qmipne/A poster on my work with Chris Bahr, under the supervision of Patrick Hayden, on a quantum complexity theory problem about entanglement, presented at McGill University for an undergraduate computer science poster session. Abstract: The promise problem QSEP-STATE asks if a quantum state described by a circuit is close to a separable (not entangled) state across a given cut, or not. A quantum multiprover protocol was found to give an upper bound for the computational complexity problem of QMIPne, which is the class of problems that can be solved by a computationally bounded quantum verifier exchanging messages with unentangled, computationally unbounded, quantum provers.Life on Phorcyshttps://ericphanson.com/work/misc/life-on-phorcys/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/misc/life-on-phorcys/2013 University Physics Competition submisssion. “In this article, we investigate the possible forms of life that may exist on planet Phorcys. Our approach is to determine key parameters describing the planet: temperature variation, ice cover, surface pressure and ocean presence and depth. We find that Phorcys is covered in a global ocean, with large ice caps and the possibility of an ice free equatorial region. The presence and size of this region is determined by a combination of the salinity of the ocean water, which affects the freezing point of water, and the day length of the planet, which affects how temperature varies with latitude.Non-local games and communication complexityhttps://ericphanson.com/work/notes/non-local-games-and-communication-complexity/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/notes/non-local-games-and-communication-complexity/“Synthesizing some of the literature on non-local games and communication complexity scenarios provides a deeper understanding of the fundamental differences between classical mechanics and quantum mechanics, as exemplified by the quantum violation of Bell inequalities, which hold in classical mechanics. Quantum mechanical protocols are shown to provide significant advantages in certain tasks over classical protocols, and could create the first loophole-free demonstration of quantum non-locality. Non-local games are a scenario where two players attempt to perform a task without communication; communication complexity scenarios generalize non-local games by allowing but attempting to minimize communication.Math 354 Honours Analysis 3 Noteshttps://ericphanson.com/work/notes/math-354-honours-analysis-3-notes/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/notes/math-354-honours-analysis-3-notes/Introduction to metric spaces, topological spaces.QSEP-State is in QMIPnehttps://ericphanson.com/work/presentation/qsep-state-is-in-qmipne/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/presentation/qsep-state-is-in-qmipne/“The promise problem QSEP-STATE asks if a quantum state described by a circuit is close to a separable(not entangled) state across a given cut, or not. A quantum multiprover protocol was found to give an upper bound for the computational complexity of QMIPne, which is the class of problems that can be solved by a computationally bounded quantum verifier exchanging messages with unentangled, computationally bounded, quantum provers.”The black hole information paradoxhttps://ericphanson.com/work/notes/the-black-hole-information-paradox/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/notes/the-black-hole-information-paradox/“The black hole information paradox concerns the intersection of quantum mechanics and general relativity,so its resolution could point the way towards a quantum theory of gravity. The paradox is a contradiction between our understanding of the limits of our physical theories and the consequences of them. With curvature on the scales of that in our solar system, current physical theories have been very successful experimentally; thus, there seems to exist a “solar system limit” in which the laws of physics reduce to those of quantum field theory.Landauer's Principlehttps://ericphanson.com/work/presentation/landauers-principle/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/presentation/landauers-principle/“Landauer’s principle states that the energy cost to erase one bit of information bythe action of a thermal reservoir at equilibrium temperature $T$ is always at least $kT$, and provides an interesting link between information theory and physics. My goal is to discuss Landauer’s principle generally, and for the case of a 2 level system coupled to a thermal reservoir modelled by an XY spin chain (or sequence of spin-1⁄2 particles), using the tools of quantum statistical mechanics.The existence and uniqueness of the Haar Measurehttps://ericphanson.com/work/notes/the-existence-and-uniqueness-of-the-haar-measure/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/notes/the-existence-and-uniqueness-of-the-haar-measure/“The Haar measure allows integration over topological groups,which has many applications; for example, the use of the Haar measure in proving the Fully Quantum Slepian Wolf (or state transfer) protocol fundamental to quantum information theory is briefly discussed in section VI. But first we build the definitions necessary to define the Haar measure, and state some fundamental theorems. Next, we show a left Haar measure exists on locally compact Hausdorff topological groups, and is unique (up to scale) on such groups which are also $\sigma$-compact.Landauer's Principle in Repeated Interation Systemshttps://ericphanson.com/work/paper/landauers-principle-in-repeated-interation-systems/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/paper/landauers-principle-in-repeated-interation-systems/“We study Landauer’s Principle for Repeated Interaction Systems (RIS) consisting ofa reference quantum system $\mathcal{S}$ in contact with a structured environment $\mathcal{E}$ made of a chain of independent quantum probes; $\mathcal{S}$ interacts with each probe, for a fixed duration, in sequence. We first adapt Landauer’s lower bound, which relates the energy variation of the environment $\mathcal{E}$ to a decrease of entropy of the system $\mathcal{S}$ during the evolution, to the peculiar discrete time dynamics of RIS.Landauer's Principle in RIShttps://ericphanson.com/work/presentation/landauers-principle-in-ris/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/presentation/landauers-principle-in-ris/Slides for a talk I gave to the Physics of Information lab at Waterloo, February 2016, about Landauer’s Principle in Repeated Interaction Systems. Based on the preprint arxiv/1510.00533.Combinatorics Noteshttps://ericphanson.com/work/notes/combinatorics-notes/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/notes/combinatorics-notes/Extremal combinatorics: Sperner systems, the Littlewood-Offord problem, intersecting hypergraphics, compression, Turan type problems, Ramsey theory, convexity, incidence problems, and algebraic methods. Contribute typo fixes here.Landauer's Principle in Repeated Interaction Systemshttps://ericphanson.com/work/essay/landauers-principle-in-repeated-interaction-systems/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/essay/landauers-principle-in-repeated-interaction-systems/My masters thesis; essentially, an expanded version of most of arXiv:1510.00533, in my own words. The abstract: “Landauer’s Principle states that there is a lower bound on the energy required to change the state of a small system from an initial state to a final state by interacting with a thermodynamic reservoir;of particular interest is when the bound is saturated and the minimal energy cost obtained for a given state transformation.Landauer’s Principle in Repeated Interaction Systemshttps://ericphanson.com/work/presentation/landauers-principle-in-repeated-interaction-systems/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/presentation/landauers-principle-in-repeated-interaction-systems/The slides for the talk I gave at the Autrans summer school Stochastic Methods in Quantum Mechanics on my work with Alain Joye, Yan Pautrat, and Renaud Raquepas on Landauer’s Principle in repeated interaction systems.Landauer's Principle in Repeated Interation Systemshttps://ericphanson.com/work/poster/landauers-principle-in-repeated-interation-systems/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/poster/landauers-principle-in-repeated-interation-systems/A poster on my work with Alain Joye, Yan Pautrat, and Renaud Raquepas on Landauer’s Principle in repeated interaction systems, presented at the one-day event CCIMI New Directions in the Mathematics of Information.Local continuity bounds for entropies of finite probability distributionshttps://ericphanson.com/work/poster/local-continuity-bounds-for-entropies-of-finite-probability-distributions/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/poster/local-continuity-bounds-for-entropies-of-finite-probability-distributions/A poster on my work with my supervisor Nilanjana Datta on local continuity bounds for the entropies of finite distributions, presented at the one-day event High Dimensional Mathematics. I gave a short “elevator pitch” to advertise the poster, the video recording of which is posted here.Local continuity bounds for entropies of finite probability distributionshttps://ericphanson.com/work/presentation/local-continuity-bounds-for-entropies-of-finite-probability-distributions/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/presentation/local-continuity-bounds-for-entropies-of-finite-probability-distributions/A presentation on my work with my supervisor Nilanjana Datta on local continuity bounds for the entropies of finite distributions, presented at the one-day event High Dimensional Mathematics. These are the slides I gave accompanying my short talk, the video of which is here.Landauer's Principle for Trajectories of Repeated Interaction Systemshttps://ericphanson.com/work/paper/landauers-principle-for-trajectories-of-repeated-interaction-systems/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/paper/landauers-principle-for-trajectories-of-repeated-interaction-systems/A refinement and generalization of our previous work on Landauer’s Principle in repeated interaction systems. We consider a two-time measurement protocol of the energy of the quantum probes, and recover a large deviations principle and a central limit theorem in the adiabatic limit of a repeated interaction system. Abstract: “We analyze Landauer’s principle for repeated interaction systems consisting of a reference quantum system $\mathcal{S}$ in contact with a environment $\mathcal{E}$ consisting of a chain of independent quantum probes.A universal construction of tight continuity bounds for entropieshttps://ericphanson.com/work/poster/a-universal-construction-of-tight-continuity-bounds-for-entropies/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/poster/a-universal-construction-of-tight-continuity-bounds-for-entropies/A poster on my work with my supervisor Nilanjana Datta on uniform continuity bounds for the single-partite entropies, presented at the conference Beyond IID 2017 in Singapore in August 2017, and the main conference of the thematic semester Analysis in Quantum Information Theory in Paris in December 2017.Maximum and minimum entropy states yielding local continuity boundshttps://ericphanson.com/work/paper/maximum-and-minimum-entropy-states-yielding-local-continuity-bounds/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/paper/maximum-and-minimum-entropy-states-yielding-local-continuity-bounds/We consider the geometry of the trace-ball of quantum states, find maximal and minimal states in a particular partial order called majorization, and use these states to construct local continuity bounds for quantum entropies. We also apply the theory of convex optimization to motivate the construction of the maximal state, and to find general optimality conditions for a particular class of functions subject to a trace-ball constraint. Abstract: “Given an arbitrary quantum state ($\sigma$), we obtain an explicit construction of a state $\rho^\varepsilon(\sigma)$ (resp.Operational Duality between Classical Source Coding with QSI and C-Q Channel Codinghttps://ericphanson.com/work/poster/operational-duality-between-classical-source-coding-with-qsi-and-c-q-channel-coding/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/poster/operational-duality-between-classical-source-coding-with-qsi-and-c-q-channel-coding/A poster on my work with Hao-Chung Cheng, my supervisor Nilanjana Datta, and Min-Hsiu Hsieh on operational duality of some quantum information-theoretic protocols, presented at the conference Beyond IID 2018 in Cambridge, UK in July 2018, and at Quantum Information Processing in Boulder, Colorado in January 2019.Quantum Information Theoryhttps://ericphanson.com/work/presentation/quantum-information-theory/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/presentation/quantum-information-theory/Winning entry for the 2017 CCIMI video contest. Video:
Many thanks to falcxne for allowing use of his songs “Said You Wouldn’t Change” (from the album Timbits III), “I Hate Homework (Interlude)” (from Timbits II) and “NYC (Interlude)” (from Timbits III); you can find these and more at https://falcxne.bandcamp.com/. And thanks to Phillipe Faist for creating the entropy zoo and allowing me to use it in the video. For more details on the continuity bounds mentioned, you can find a preprint here, which is joint work with my supervisor, Dr.Extrema in majorization order with applications to entropic continuity boundshttps://ericphanson.com/work/essay/extrema-in-majorization-order-with-applications-to-entropic-continuity-bounds/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/essay/extrema-in-majorization-order-with-applications-to-entropic-continuity-bounds/An essay submitted in the competition for a Smith-Knight or Rayleigh-Knight prize. This essay combines my work on uniform continuity bounds (arXiv:1707.04249) with that on local continuity bounds (arXiv:1706.02212), and unifies the notation. Abstract: “Majorization is a pre-order of vectors, giving a sense in which one vector can be said to be more disordered than another. Two given vectors, however, may be incomparable. This concept has been extendedto quantum mechanical states, to particular success in the theory of entanglement manipulation.When do we lose correlations under partial Markovian evolution?https://ericphanson.com/work/presentation/when-do-we-lose-correlations-under-partial-markovian-evolution/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/work/presentation/when-do-we-lose-correlations-under-partial-markovian-evolution/A brief presentation introducing partial Markovian evolution (classically), entanglement, and some of the results from arXiv:1902.08173, which was done in collaboration with Cambyse Rouzé and Daniel Stilck França. This talk was given at the March 2019 CCIMI retreat.When do we lose correlations under Markovian evolution?https://ericphanson.com/blog/2019/when-do-we-lose-correlations-under-markovian-evolution/Fri, 22 Mar 2019 00:00:00 +0000https://ericphanson.com/blog/2019/when-do-we-lose-correlations-under-markovian-evolution/⊕This post started as a talk for the CCIMI retreat, the slides of which are available here. But I added a lot of words to turn this into a blog post, so I encourage you to stay here instead!
Consider the following divided box with \(N=20\) particlesA discrete-time Ehrenfest model
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Toggle animation At each time step, exactly one particle jumps from one side of the box to the other.Another example of using type domain information in Juliahttps://ericphanson.com/blog/2019/another-example-of-using-type-domain-information-in-julia/Sat, 09 Mar 2019 00:00:00 +0000https://ericphanson.com/blog/2019/another-example-of-using-type-domain-information-in-julia/In a previous post, I discussed using type domain information to speed up generation of random density matrices with small dimension in Julia. There, we gave the Julia compiler knowledge of the dimension of the matrices at the time it generates code, instead of passing that dimension as a runtime variable, and saw significant runtime speedups as a consequence. This time, let’s push this further by giving the compiler a whole vector of numbers instead of a single integer.Carathéodory's theorem and the Holevo capacityhttps://ericphanson.com/blog/2018/caratheodorys-theorem-and-the-holevo-capacity/Tue, 18 Dec 2018 00:00:00 +0000https://ericphanson.com/blog/2018/caratheodorys-theorem-and-the-holevo-capacity/Let \(\Lambda: \mathcal{B}(\mathcal{H})\to \mathcal{B}(\mathcal{H})\) be a quantum channelcompletely positive and trace-preserving map
on a finite-dimensional Hilbert space \(\mathcal{H}\), and let \(d:= \dim \mathcal{H}\). The Holevo capacity of \(\Lambda\) is defined as \[ \chi(\Lambda) := \sup_{\{p_k, \rho_k\}} \left[ S\Big(\sum_k p_k \Lambda(\rho_k)\Big) - \sum_k p_k S( \Lambda(\rho_k)) \right] \] where supremum is over ensembles of quantum states \(\rho_k\) and probability distributions \(\{p_k\}\). The HSW theorem (Holevo 1998; Schumacher and Westmoreland 1997) states that the product state classical capacity of a quantum channel \(\Lambda\) is given by its Holevo capacity, thus providing the quantity with an operational meaning.Fast small random density matriceshttps://ericphanson.com/blog/2018/fast-small-random-density-matrices/Sun, 23 Sep 2018 20:07:42 +0000https://ericphanson.com/blog/2018/fast-small-random-density-matrices/Update (19 January 2019): I looked at this code again, and realized I had made a few basic mistakes, such as constructing a statically sized normally-distributed random matrix via SMatrix{d,d,Float64, d*d}(randn(Float64, d,d)) which constructs a random matrix (allocating memory dynamically) and then converts it to a static SMatrix, instead of randn(SMatrix{d,d,Float64}) which directly constructs an SMatrix without dynamic memory allocation. I also hadn’t written the function randsimplexpt in a very good way.Website rewritehttps://ericphanson.com/blog/2018/website-rewrite/Sat, 25 Aug 2018 19:26:42 -0400https://ericphanson.com/blog/2018/website-rewrite/I decided to redo my website using Hugo (before I was using Jekyll). Both are static site generators: you write in Markdown, a simple clean formatting language, and it generates HTML webpages following a consistent format. It’s great because the generation only has to occur once after the Markdown is written, and nothing active has to happen for each user of the website (unlike a so-called dynamic website). This reduces costs, increases reliability, and improves securitythe user needs very little access to the server hosting the webpages in a static siteLocally maximizing the Rényi entropieshttps://ericphanson.com/blog/2018/locally-maximizing-the-renyi-entropies/Sat, 25 Aug 2018 00:00:00 +0000https://ericphanson.com/blog/2018/locally-maximizing-the-renyi-entropies/As I was rewriting my website, I found some visualizations I had stored on my old website to show a collaborator, and I figured it was worth writing a little to have a more proper place to put them; hence this post 😊.
Probability distributions on three letters consist just of three non-negative numbers which add up to 1, which we can see as a vector in \(\mathbb{R}^3\). The set of all such distributions form a simplex, which looks like a 2D triangle laying in \(\mathbb{R}^3\):How to make an index in LaTeXhttps://ericphanson.com/blog/2018/how-to-make-an-index-in-latex/Wed, 20 Jun 2018 00:00:00 +0000https://ericphanson.com/blog/2018/how-to-make-an-index-in-latex/I’m sure there are many ways to make an index in LaTeX, but I was asked this question recently and thought I’d put my response here, which is based on what I did in my combinatorics notes.
In the preamble, make a command \defw (short for “define word”), use the package imakeidx, and call \makeindex:
\usepackage{imakeidx} \usepackage{xparse} \NewDocumentCommand{\defw}{m o}{% {\emph{#1}}% \IfNoValueTF{#2} {\index{#1}} {\index{#2}}% } \makeindex Generate the index at the end of the file with \printindex, like one does with a bibliography.arXiv-search (the sad goodbye-for-now post)https://ericphanson.com/blog/2018/arxiv-search/Mon, 14 May 2018 00:00:00 +0000https://ericphanson.com/blog/2018/arxiv-search/Update (26 January 10): The source code is now available under a MIT license at https://github.com/ericphanson/arxiv-search.
For a few months, I and a few others were working on a project we called “arxiv-search”, an attempt to search and sort all of the arxiv (~1 million papers). We were inspired by Andrej Karpathy’s arxiv sanity preserver which is an excellent tool for a limited set of papers (~50,000). Starting from that project, we ended up writing a new backend and frontend.A synchronized dance of eigenvalueshttps://ericphanson.com/blog/2017/a-synchronized-dance-of-eigenvalues/Tue, 26 Sep 2017 00:00:00 +0000https://ericphanson.com/blog/2017/a-synchronized-dance-of-eigenvalues/The motivation behind this post is to show some off some nice gifs. But I thought maybe they aren’t actually interesting without any context, so below I’ll try to explain what the gifs are about.
Brief mathematical introduction to Perron-Frobenius theory Consider the set of \(n\times n\) matrices \(M_n\), with complex entries.Perceptron demohttps://ericphanson.com/blog/2017/perceptron-demo/Wed, 19 Jul 2017 00:00:00 +0000https://ericphanson.com/blog/2017/perceptron-demo/A javascript demonstration of the perceptron algorithm, written for a reading group session. This was originally a separate webpage, but when I rewrote my website I decided to make it a blog post. This post can still be reached from /perceptron-demo, though.
Perceptron Perceptron is a very simple binary classification online learning algorithm, dating from the 1950s1. Such an algorithm tries to classify input (here, points in the plane) into one of two categories.The arrow of time in RIShttps://ericphanson.com/blog/2017/the-arrow-of-time-in-ris/Wed, 18 Jan 2017 21:09:03 +0000https://ericphanson.com/blog/2017/the-arrow-of-time-in-ris/Repeated interaction systems (RIS) Introduction to arrow of time A more careful description of the forward and backward processes Hypothesis testing on the arrow of time Connection to Landauer’s Principle I gave an informal talk today on the arrow of time in repeated interaction systems and I thought I’d write about it here.
Repeated interaction systems (RIS) For a recent review of RIS, see arxiv/1305.2472. We also introduce them in my and my coauthor’s work Landauer’s Principle in Repeated Interaction Systems.The traveling salesman and 10 lines of Pythonhttps://ericphanson.com/blog/2016/the-traveling-salesman-and-10-lines-of-python/Tue, 25 Oct 2016 00:00:00 +0000https://ericphanson.com/blog/2016/the-traveling-salesman-and-10-lines-of-python/Update (21 May 18): It turns out this post is one of the top hits on google for “python travelling salesmen”! That means a lot of people who want to solve the travelling salesmen problem in python end up here. While I tried to do a good job explaining a simple algorithm for this, it was for a challenge to make a progam in 10 lines of code or fewer. That constraint means it’s definitely not the best code around for using for a demanding application, and not the best for learning to write good, readable code.Setting up SublimeTexthttps://ericphanson.com/blog/2016/setting-up-sublimetext/Fri, 14 Oct 2016 00:00:00 +0000https://ericphanson.com/blog/2016/setting-up-sublimetext/I’m setting up a new laptop and figured it was a good chance to document setting up SublimeText for LaTeX, as quasi follow up to my recent post about live LaTeXing. However, installing LaTeX can be hard, and configuring the SublimeText package LaTeXTools to work with LaTeX has the potential to be hard (often just works, but if not, can be confusing), and I won’t write about those parts, because they are better documented elsewhere.Live notetaking with LaTeXhttps://ericphanson.com/blog/2016/live-notetaking-with-latex/Sat, 24 Sep 2016 00:00:00 +0000https://ericphanson.com/blog/2016/live-notetaking-with-latex/A friend suggested I write a guide about live notetaking in LaTeX, since it’s pretty useful and something I have a fair amount of experience with. I had been making attempts at live-LaTeXing course notes for a year or two before I was able to make it through a semester long class; the first course I fully LaTeX’d was Vojkan Jaksic’s excellent Analysis 3 (introduction to metric spaces and topology)PDF. Not my best work, but it did the job.Landauer's Principle and the balance equationhttps://ericphanson.com/blog/2016/landauers-principle-and-the-balance-equation/Mon, 29 Feb 2016 21:50:00 +0000https://ericphanson.com/blog/2016/landauers-principle-and-the-balance-equation/I’ve been working on my thesisEdit: this was my master’s thesis
over reading week, and I think I’ve finished my introduction to Landauer’s Principle. I ended up writing a pretty detailed derivation of the balance equation, and thus Landauer’s bound, so I thought it might be useful to post here.
Landauer's principle states that there is a minimal energetic cost for a state transformation \(\rho^\text{i}\to \rho^\text{f}\) on a system \(\mathcal{S}\) via the action of a thermal reservoir \(\mathcal{E}\) at temperature \((k_B\beta)^{-1}\)⊕\(k_B \approx 1.Cantor's set and functionhttps://ericphanson.com/blog/2016/cantors-set-and-function/Tue, 09 Feb 2016 00:00:00 +0000https://ericphanson.com/blog/2016/cantors-set-and-function/Cantor’s Set Construction Properties Cantor’s function I wrote these notes in February 2016 for an Analysis 2 tutorial when I was a teaching assistant at McGill, and always intended to put them here eventually; before August 2018 though, I hadn’t translated them to something web-friendly and only had posted a PDFThe web version has slightly improved wording in some parts.
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Cantor’s Set Cantor’s set is an interesting subset of \([0,1]\), with properties that help illuminate concepts in analysis.Completeness Ihttps://ericphanson.com/blog/2016/completeness-i/Fri, 01 Jan 2016 00:00:00 +0000https://ericphanson.com/blog/2016/completeness-i/Last semester, I helped a friend review McGill’s Analyis 3 course by trying to provide a better feel for completeness; this post will be a slightly edited version of that. Originally, I wanted to write about both completeness and compactness, and their connections, but I ended up only getting to completeness, and actually not everything I wanted to talk about. So I’ll call this Completeness I, leaving open the possibility for more of these in the future.New bloghttps://ericphanson.com/blog/2015/new-blog/Mon, 14 Dec 2015 00:00:00 +0000https://ericphanson.com/blog/2015/new-blog/I’ve made a new blog!
I started a “coffee blog” a month or so ago but only posted once (I think I’ll bring that post over here, too). I do like the idea though, and want to write about math and my research. So this should become a place for me to do that.
Part of my impetus is that journals and traditional publication methods don’t provide a pathway for discussing failed attempts.Abouthttps://ericphanson.com/about/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/about/I’m a PhD student in the Department of Applied Math and Theoretical Physics at Cambridge University, in my third year.
I’m interested in quantum statistical mechanics and quantum information theory; see here for my recent work.
I also made a video introducing quantum information theory (though it’s mostly just about information theory):
Many thanks to falcxne for allowing use of his songs “Said You Wouldn’t Change” (from the album Timbits III), “I Hate Homework (Interlude)” (from Timbits II) and “NYC (Interlude)” (from Timbits III); you can find these and more at https://falcxne.Contact mehttps://ericphanson.com/contact/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/contact/This is an attempt to have a LaTeX-friendly way to contact me.
Write your message in the box below. The “Editor (popout)” button should pop-out a more full-featured editor (via StackEdit), but it might not work on mobile. When you’re done, click the “x” in the top-left to leave the editor, and press “Submit”. That will email me the message via a service called formspree. I am not affiliated with them or with StackEdit, so please don’t type in anything secret.RIS full dipole codehttps://ericphanson.com/code/ris_full_dipole/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/code/ris_full_dipole/Files Mathematica file: RIS_full_dipole.nb
PDF: RIS_full_dipole.pdf
Last updated 8 April 2016.
Back to code.Searchhttps://ericphanson.com/search/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/search/ Search the websiteThankshttps://ericphanson.com/thanks/Mon, 01 Jan 0001 00:00:00 +0000https://ericphanson.com/thanks/Thanks for sending me a message! I’ll try to get back to you soon.