January 2019 Archive

This is an archive of what I found useful, interesting, or simply enjoyable from January 2019 (and a bit from December 2018).

Research

Generalized coherent states

• Coherent states: Theory and some applications [paywalled], Zhang, Feng and Gilmore 1990

  The paper talks about group generalizations of the notion of coherent states, mainly of Perelomov and Gilmore. Naturally, it briefly mentions other approaches but is quick to point out their drawbacks.

  1. Coherent states as eigenstates of lowering operators

  > [...] the adoption of this definition [...] has two major drawbacks, the first mathematical and the second physical: (a) Coherent states cannot be defined in Hilbert spaces of finite dimensionality in this way. In particular, this would preclude the construction of coherent states for compact Lie groups (Perelomov, 1972). Furthermore, states defined in this way have few useful properties and in particular they are not computationally useful. (b) The states so defined do not correspond to physically realizable states, except under the special circumstance that the commutator of the annihilation operator (or lowering step operator) and its Hermitian adjoint is a multiple of the identity operator. Therefore, under these conditions one restricts oneself to the electromagnetic field.
  

  1. Coherent states as minimum-uncertainty states

  > [...] this generalization has several limitations.  First of all, these coherent states can only be constructed for the classically integrable systems in which there exists a set of canonical coordinates and momenta such that the respective Hamiltonians can be reduced to quadrature. This condition requires a flatness condition on the Lie algebra, which reduces the commutation relations to those of the standard photon creation and annihilation operators. Secondly, the wave packets with minimum uncertainty are not unique (see Fig. 1). Different ones may have different properties. Such states may also be incomplete, or even if they are complete it is not certain that a resolution of unity of the standard form exists (Klauder and Skagerstam, 198S). Thus minimum-uncertainty states appear to have few, if any, useful properties. For example, minimum-uncertainty states do not evolve into minimum-uncertainty states when they are driven by a Hamiltonian linear in the generators in the Lie group.
  

  Revisiting this paper after a couple of years, now I would say that the group approach in which generalize coherent states (GCSs) are defined to be the orbit of a highest weight state does loosely generalize 1. and 2. as well.

  Given a semisimple Lie algebra $\g$ and its root space decomposition \(\g = \overbrace{\bigoplus_{\alpha \in \Sigma^+} \g_{-\alpha}}^{\displaystyle\n^-} \oplus \h \oplus \overbrace{\bigoplus_{\alpha \in \Sigma^+} \g_{\alpha}}^{\displaystyle\n^+},\) where $\h$ is a Cartan subalgebra, a highest weight state $\ket{\psi}$ is by definition the 0-eigenvector of the nilpotent subalgebra $\n^+$ of all positive roots which are analogues of the raising operators. But one might as well shuffle the labels between positive and negative roots and call elements in $\n^+$ “lowering operators”. What’s important is that in a given irrep, although there is only one highest weight vector with respect to a fixed Cartan subalgebra, every state in the orbit of that vector is a highest weight vector with respect to some choice of a Cartan subalgebra. (In fact, a GCS in finite dimensions can be defined as an eigenvector of a Borel subalgebra $\h \oplus \n^+$. This is a part of the geometric highest weight theory.)

  Even though minimum-uncertainty states don’t generalize well, minimum variance states do, and the same trick of shifting Cartan subalgebras around also does the work here. Let ${X_j}$ be an orthogonal basis of $\g$. A vector $\psi$ in an irrep of highest weight $\lambda$ is a GCS iff the total variance \(\mathrm{Var}(\psi) := \sum_j \av{X_j^2}_{\psi} - \sum_j \av{X_j}_{\psi}^2\) is minimal. The first term is the Casimir eigenvalue which is fixed for a given irrep ¹. The only states that maximize the second term are, again, highest weight states with respect to some choices of Cartan subalgebras, and these are precisely the orbit of the unique highest weight state with respect to a fixed Cartan subalgebra.

  The comparison here with the traditional coherent states is a bit wonky as the Heisenberg group is nilpotent and not semisimple. Zhang et al. takes the symmetry algebra of single-mode coherent states to be generated by ${a,a\dgg,a\dgg a,1}$, where 1 is the “scalar” in the commutator $[a\dgg,a] = 1$ and $a\dgg a$ is the number operator. 1 and $a\dgg a$ each has some property of an element of a Cartan subalgebra but not all. For instance, $a$ and $a\dgg$ are eigenvectors of the adjoint action of $a\dgg a$ but $[a\dgg,a]$ is not proportional to $a\dgg a$.

  I learned these interesting characterizations of GCSs from:

• Coherent states, entanglement, and geometric invariant theory, Klyachko 2002 [Extended (and corrected) version]
  • A system of quadrics describing the orbit of the highest weight vector, Lichtenstein 1982 [MO]
• Generalizations of entanglement based on coherent states and convex sets, Barnum, Knill, Ortiz and Viola 2003

More GCS:

• Quantum circuit synthesis for generalized coherent states, Somma arXiv:1811.08479

  A highest weight state with respect to a choice of a Cartan subalgebra $\h$ has a physical interpretation as the ground state of a Hamiltonian in $\h$. (That statement is manifestly false and requires two adjustments ² but it has a nicer ring to it if you’re a physicist.) Suppose that your quantum computation starts in the ground state of a fixed $\h_0 \subset \g$ and after some time you don’t know what the state is anymore but you know that it is a GCS $\ket{\psi}$ under the group $G$. This could happen, for example, if the quantum gates are limited to those in $G$. This paper shows how to reconstruct a quantum circuit that prepares the GCS from the initial ground state by measuring observables ${X_j}$ in $\g$. The number of measurements and the reconstruction process are efficient in the dimension of the Lie algebra.

  Two neat tricks: The Hamiltonian \(H = - \sum_j X_j \av{X_j}_{\psi}\) has $\ket{\psi}$ as the ground state because the expectation value \(- \av{H} = \sum_j \av{X_j}^2\) is exactly the quantity maximized only by $\ket{\psi}$. One then diagonalizes the Hamiltonian to be in $\h_0$. Any complex semisimple $\g$ is a bunch of overlapping $\sl(2,\C)$ subalgebras ³, and the task turns out to break up into sequentially diagonalizing each $\sl(2,\C)$ subalgebra: find a rotation that brings a linear combination of Paulis to a pure, fixed Pauli, $Z$ for example.

• A note on coherent state representations of Lie groups, Onofri 1975

  Let $G$ be a real Lie group, $T$ a maximal torus, $K \subset G$ the (projective) stabilizer subgroup of a highest weight state, $G^{\C}$ the complexification of $G$, and $B \subset P \subset G^{\C}$ the Borel and a parabolic subgroup respectively.

  An argument for the (canonical) diffeomorphism $G /T \cong G^{\C} /B$ can be found in Hogreve et al. 1990, Section 2.3, p.472: the real span of $\g$ and $\mathfrak{b}$ is all of $\g^{\C}$, so the $G$-orbit is open in the $G^{\C}$ orbit ($G^{\C} /B$). On the other hand, the orbit is closed because of the compactness, thus they are the same. The argument extends to $G /K \cong G^{\C} /P$ by defining $K = P \cap G$, then $K$ contains a maximal torus $T$.

Representation theory

• Structure theory of semisimple Lie groups, Knapp 1997

Algebraic groups

• Lie Groups and Linear Algebraic Groups I. Complex and Real Groups, Borel
• Algebraic Groups and Arithmetic Groups, Milne 2005
• Introduction to actions of algebraic groups, Brion 2009
• Structure theory of reductive groups through examples, Makisumi 2011
• Peter-Weyl for compact Lie groups vs algebraic groups, MSE (Procesi, for example, has both verions: compare Chapter 7, p. 180 and the Peter-Weyl theorem in Chapter 8, p. 213)
  • Distributions on a linear algebraic group, Wikipedia

Symmetric spaces and multiplicities

• Foundations of Differential Geometry, Vol. 2, Kobayashi and Nomizu 1963
• Perspectives on Invariant Theory: Schur Duality, Multiplicity-Free Actions and Beyond in The Schur Lectures, Howe 1992
• Notes on multiplicity-free actions, Benson and Ratcliff
• Multiplicity-free products and restriction of Weyl characters, Stembridge 2003
• On spherical double cones, Littelmann 1994
• The Cartan-Helgason theorem [Wikipedia]
  • Harmonic Analysis on Semi-Simple Lie Groups I, Warner 1972
  • Groups and Geometric Analysis, Helgason 1984 (p. 535)
  • Symmetry, Representations, and Invariants, Goodman and Wallach 2009 (Chapter 12)

Media

Links

• Added to the list of great scientists we’ve lost in 2018 are Judith Rich Harris of the Nurture Assumption fame and Roy Glauber. Glauber and E.C.G. Sudarshan (who also passed away earlier in 2018), are both monumental figures that helped define the field of quantum optics that we know today.
• Apparently living in an area with 22 $\mu g/m^3$ of PM2.5 is equivalent to smoking a cigarette per day.
• Average individual consumption is a better proxy for wealth than GDP.
• Bullshit Jobs, Graeber 2013 [SSC subreddit]
  • Post-Work, Guardian
• Fried rice syndrome [Wikipedia]
• Parachute use to prevent death and major trauma when jumping from aircraft: randomized controlled trial, Yeh et al. 2018 (The now classic 2003 paper by Smith and Pell was a literature review. Here they actually did the experiment.)

• The perils of short-termism: Civilization’s greatest threat, BBC

  To paraphrase the investor Esther Dyson: in politics the dominant time frame is a term of office, in fashion and culture it’s a season, for corporations it’s a quarter, on the internet it’s minutes, and on the financial markets mere milliseconds.

• When is it acceptable to offset a bad deed by a good deed? A simple rule: you can offset axiology, but not morality.
• Science is getting less bang for its buck, Collison and Nielsen, Atlantic
• What makes people intellectually active?, LW
  • William Shockley observed that the distribution of productivity at Bell labs was log-normal, which could arise from a product of i.i.d. distributions of several different abilities e.g. the ability to recognize a good problem, solve it, write about it, persevere in the battle with reviewers, write a grant proposal, bring in money etc. (As a side note, the median of the number of citations is apparently zero!)
• Physics Forums has a nice Insight article on Klein geometries.
  • Differential Geometry: Cartan’s Generalization of Klein’s Erlangen program, Sharpe 1997 (More examples of planar Klein’s geometries. There are 23 families of two-dimensional Klein’s geometries)
  • Desargues’ theorem in projective geometry is quoted in several treatises of perspective drawing according to this paper.
• God’s Number is 20: the longest shortest sequence to solve a Rubik’s cube is 20 moves in the half-turn metric (any non-trivial turn of a face counts as one move).

Anime

• Guilty Gear Xrd’s Art Style : The X Factor Between 2D and 3D, Junya C. Motomura, Lead character modeler

  Team Red of Arc System Works was the only one that got anime-like cel-shading right. (They were thus given the job to handle Dragon Ball FighterZ and now Granblue Fantasy Versus). Each character has its own fixed light source and the team manually adjusts the normal vector and the cel-shading threshold of each major polygon face (16:10) to make sure that every close-up frame looks good.

• Good Touhou anime parody still exists in 2019. (Onegai! Cinderella, The Idolm@ster Cinderella Girls)
• China’s responses to Japanese anime invasion

Physics/quantum

• Trump signs the $1.2 billion National Quantum Initiative Act.
• In what sense is quantum computing a science?, Nielsen
• A sensible rebuttal to what I think is a fundamentally misguided piece, The Case Against Quantum Computing that appeared in IEEE Spectrum last November.
• Boaz Barak’s blog, Windows on Theory, is having a series of guest blog posts on connections between computer science and physics. The topics range from statistical-physics-inspired algorithms to quantum information and black hole information paradox.
• George Smoot thinks that the region of the universe around us lacks “low-metallica” stars, giant stars that can collapse and give rise to >20 solar-mass black holes that LIGO and Virgo have been detecting, and so surmises that the events we have seen took place “much farther away—on the order of 10 billion light-years distant—magnified and made visible through gravitational lensing”.

Meditation

• Mind the Hype: A Critical Evaluation and Prescriptive Agenda for Research on Mindfulness and Meditation, Van Dam et al. 2017
• Where’s the Proof That Mindfulness Meditation Works?, SA [reddit comment]
• Altered traits: science reveals how meditation changes your mind, brain, and body, Goleman and Davidson 2017 [Dan Harris’ podcast]
• Sequence introduction: non-agent and multiagent models of mind, Sotala, LW
  • Book Summary: Consciousness and the Brain
  • Building up to an Internal Family Systems model
• Thomas Metzinger formalizes necessary conditions for suffering: 1. conscious experience 2. self-model 3. negative values in the self-model 4. identification with the self-model. Buddhism’s “no self” strives to eliminate 2 (and consequently 3 and 4).

Musics

• Uchiage hanabi, DAOKO X Yonezu Kenshi
• Shadowgraph, Myth & Roid {Boogiepop wa warawanai 2019} [OP]
• 99.9, MOB CHOIR feat. sajou no hana {Mob Psycho 100 II} [OP]
• Sayonara Gokko, amazarashi {Dororo 2019} [ED]
• Love Dramatic feat. Rikka Ihara, Masayuki Suzuki {Kaguya-sama wa kokurasetai} [OP]

Touhou

• 弦魔術狂想曲 (Mary the Magician), FAR EAST OF EAST -XVI- (C94), TatshMusicCircle
• Never Ending Night (Spring Lane ~ Colorful Path), 東方零響奇譚 ~ Sophisticated Insanity (C79), Zero Symphonic Battalion (Hiroki Kikuta, Jillian Aversa)
• 最終制御室～Final control room～ (Last Remote), 東方幻霊殿 (C82), Ryuuha Mikakutei
• 空へと沈む鬼と針の城 (The Shining Needle Castle Sinking in the Air), 逆弾奏結界 反逆諧謔曲 Reversal Scherzo (C85), dBu
• 魔界地方遊戯ソリテェア (Makai Provincial City Esoteria), Lotus (C77), Sound.AVE
• Sealed Sword & Alice in Wonderland (The Grimoire of Alice & Alice in Wonderland), Ancient Treasures (C79), Oriens

• 2022 1
• 2021 1
• 2019 7
• 2017 1