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]
  • 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

Algebraic groups

Symmetric spaces and multiplicities

Media

Anime

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

Musics

Touhou


  1. The Casimir eigenvalue is \(\av{\lambda,\lambda + 2\delta}\), generalizing the \(j(j+1)\) of the squared angular momentum, where \(\delta = \frac{1}{2}\sum_{\alpha \in \Sigma_+} \alpha\) is the ubiquitous half-sum of positive roots (Knapp, Proposition 5.28, p.295). For highest weight states, \(\sum_j \av{X_j}^2 = \av{\lambda,\lambda}\), making the variance \(\av{\lambda,2\delta}\).

  2. One we already discussed is that the ground state would be the lowest weight state but we can make it the highest weight state by a trivial relabeling. The other is that the set of Hermitian matrices are not closed under the commutator bracket so Hamiltonians don’t form a Lie algebra. Rather it is \(i\) times the Hamiltonians that are in \(\h\). These exponentiate directly into group elements.

  3. For each \(\alpha\), the Lie algebra generated by \(\{ E_{\pm \alpha}, H \}\), where \(E_{\pm \alpha} \in \g_{\pm \alpha}\) and \(H \propto [E_{\alpha},E_{-\alpha}] \in h\) is isomorphic to the \(\sl(2,\C)\) Lie algebra. Note that \(\sl(2,\C)\) is what physicists call “\(\mathfrak{su}(2)\)” but it is actually the complexification of the real Lie algebra \(\mathfrak{su}(2)\).



Archieves by date, by category, by tag.