Symmetry, invariants, topology. Basic tools

Michel, Louis

doi:10.1016/s0370-1573(00)00088-0

Cited by 83 publications

(100 citation statements)

References 58 publications

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“…Analyzing their standard action on S 2 (see, e.g., [81,88,93,96,132]), one can find in all cases the orbits with maximal stabilizers. Gathering this information together, we get all highly symmetric finite subsets of S 2 , and so all HS-POVMs in two dimensions.…”

Section: Or the Povm Is Informationally Complete And According To Thmentioning

confidence: 99%

“…As the POVM is highly symmetric, this set contains S itself. According to the Michel theory of critical orbits of group actions [86,88], the elements of the maximal stratum, being critical points for the entropy of measurement H , which is a Sym(S)-invariant function, are natural candidates for the minimizers. Studying their character, we see that H has local minima at the inert states σ ⊥ j ( j = 1, .…”

Section: Introductionmentioning

confidence: 99%

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Highly symmetric POVMs and their informational power

Słomczyński

Szymusiak

2015

Quantum Inf Process

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We discuss the dependence of the Shannon entropy of normalized finite rank-1 POVMs on the choice of the input state, looking for the states that minimize this quantity. To distinguish the class of measurements where the problem can be solved analytically, we introduce the notion of highly symmetric POVMs and classify them in dimension 2 (for qubits). In this case, we prove that the entropy is minimal, and hence, the relative entropy (informational power) is maximal, if and only if the input state is orthogonal to one of the states constituting a POVM. The method used in the proof, employing the Michel theory of critical points for group action, the Hermite interpolation, and the structure of invariant polynomials for unitary-antiunitary groups, can also be applied in higher dimensions and for other entropy-like functions. The links between entropy minimization and entropic uncertainty relations, the Wehrl entropy, and the quantum dynamical entropy are described.

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Section: Or the Povm Is Informationally Complete And According To Thmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Highly symmetric POVMs and their informational power

Słomczyński

Szymusiak

2015

Quantum Inf Process

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“…As a consequence, these points must be stationary or critical points of any G-invariant function f (x) on P. When P is a reduced phase space and f (x) is a reduced Hamiltonian H eff , these points correspond to RE of the initial system. This makes finding critical orbits the primary purpose of our group action study [48,49,50]. In general, we also look for invariant subspaces of the reduced phase space P-and especially for the invariant subspaces whose stabilizer is a purely spatial symmetry subgroup of G.…”

Section: Group Actionmentioning

confidence: 99%

“…Once the action of the dynamical symmetry on the initial phase space C k of the k-oscillator is defined explicitly in (6.2) we can compute the Molien generating function g(λ), a heuristic tool [48,49,50,84] suggesting certain structural characteristics of the ring of invariant polynomials in (z,z). The function g(λ) can be obtained directly from the Molien theorem [83] …”

Section: Generating Function For Oscillator Symmetrymentioning

confidence: 99%

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Analysis of Rotation--Vibration Relative Equilibria on the Example of a Tetrahedral Four Atom Molecule

Efstathiou¹,

Sadovskií²,

Zhilinskii³

2004

SIAM J. Appl. Dyn. Syst.

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Abstract. We study relative equilibria (RE) of a nonrigid molecule, which vibrates about a well-defined equilibrium configuration and rotates as a whole. Our analysis unifies the theory of rotational and vibrational RE. We rely on the detailed study of the symmetry group action on the initial and reduced phase space of our system and consider the consequences of this action for the dynamics of the system. We develop our approach on the concrete example of a four-atomic molecule A4 with tetrahedral equilibrium configuration, a dynamical system with six vibrational degrees of freedom. 1. Introduction. This paper unifies modern methods of classical theory of symmetric Hamiltonian dynamical systems and quantum theory of molecules (and other isolated finiteparticle systems). Considerable progress was achieved in both directions in the last decades and deep relations between these seemingly distant theories became evident. Significant effort by mathematicians and molecular physicists to converge the two fields resulted in the qualitative theory of highly excited quantum molecular systems based on recent mathematical developments. We join the two approaches and demonstrate what kind of concrete results can be immediately obtained in molecular systems [1,2,3,4] by applying powerful methods of symmetric Hamiltonian systems [5,6,7,8,9,10,11]. We choose a concrete problem of rotation-vibration of a four-atomic molecule with tetrahedral equilibrium configuration [12,13] in order to explain the details of our approach.

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Lowering the symmetry group of carbon tetrachloride by high pressure

Chen

Sun

et al. 2012

Physica Status Solidi (b)

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Using the pressure‐induced polarized Raman spectra of carbon tetrachloride up to 22 GPa, a significant anomalous polarization (the depolarization ratio >0.75) was observed. We analyzed it by evaluating the antisymmetric anisotropy distribution of modes in Td and explored the corresponded molecular structure having been translated from Td into a lower symmetry. The P21/c and Pa3 structures of carbon tetrachloride were optimized as a function of pressure to further confirm that the molecules deviated from regular tetrahedra. The C2v and D2 were analyzed to be candidates of the lower symmetry based on group and Landau theory.

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Symmetry, invariants, topology. Basic tools

Cited by 83 publications

References 58 publications

Highly symmetric POVMs and their informational power

Highly symmetric POVMs and their informational power

Analysis of Rotation--Vibration Relative Equilibria on the Example of a Tetrahedral Four Atom Molecule

Lowering the symmetry group of carbon tetrachloride by high pressure

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