Paradan’s wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator

Boysal, Arzu; Vergne, Michèle

doi:10.5802/aif.2475

Cited by 9 publications

(22 citation statements)

References 9 publications

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“…In a recent paper [47], Boysal and Vergne have analyzed general push-forward measures of this form under the assumption that the vectorsω k span a proper convex cone (i.e., a convex cone of maximal dimension that does not contain any straight line). This ensures that the measure is locally finite and absolutely continuous with respect to Lebesgue measure on t * .…”

Section: Algorithms For Duistermaat-heckman Measuresmentioning

confidence: 99%

Eigenvalue Distributions of Reduced Density Matrices

Christandl

Doran

Kousidis

et al. 2014

Commun. Math. Phys.

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Given a random quantum state of multiple distinguishable or indistinguishable particles, we provide an effective method, rooted in symplectic geometry, to compute the joint probability distribution of the eigenvalues of its one-body reduced density matrices. As a corollary, by taking the distribution's support, which is a convex moment polytope, we recover a complete solution to the one-body quantum marginal problem. We obtain the probability distribution by reducing to the corresponding distribution of diagonal entries (i.e., to the quantitative version of a classical marginal problem), which is then determined algorithmically. This reduction applies more generally to symplectic geometry, relating invariant measures for the coadjoint action of a compact Lie group to their projections onto a Cartan subalgebra, and can also be quantized to provide an efficient algorithm for computing bounded height Kronecker and plethysm coefficients.Comment: 51 pages, 7 figure

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Section: Algorithms For Duistermaat-heckman Measuresmentioning

confidence: 99%

Eigenvalue Distributions of Reduced Density Matrices

Christandl

Doran

Kousidis

et al. 2014

Commun. Math. Phys.

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“…We will first define a pairing that induces this duality. If X is unimodular, this pairing agrees with the one defined in (7). Then we will show that P C (X) is canonically isomorphic to C[Λ]/ J ∇ C (X).…”

Section: Results On Periodic P-spaces and Arithmetic Matroidsmentioning

confidence: 57%

“…• In Section 8 we will discuss a wall-crossing formula for splines due to Boysal-Vergne [7] and use it to prove that the space P − (X) can be characterised as the space of differential operators with periodic coefficients that leave the spline continuous (R4).…”

Section: Introductionmentioning

confidence: 99%

Splines, lattice points, and arithmetic matroids

Lenz

2015

J Algebr Comb

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Abstract. Let X be a (d × N )-matrix. We consider the variable polytope Π X (u) = {w ≥ 0 : Xw = u}. It is known that the function T X that assigns to a parameter u ∈ R d the volume of the polytope Π X (u) is piecewise polynomial. The Brion-Vergne formula implies that the number of lattice points in Π X (u) can be obtained by applying a certain differential operator to the function T X . In this article we slightly improve the Brion-Vergne formula and we study two spaces of differential operators that arise in this context: the space of relevant differential operators (i. e. operators that do not annihilate T X ) and the space of nice differential operators (i. e. operators that leave T X continuous). These two spaces are finite-dimensional homogeneous vector spaces and their Hilbert series are evaluations of the Tutte polynomial of the arithmetic matroid defined by the matrix X. They are closely related to the P-spaces studied by ArdilaPostnikov and Holtz-Ron in the context of zonotopal algebra and power ideals.

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“…where σ 1 = 1 4 ∑ 4 j=1 ρ(r j ) and σ 2 = ρ(r 5 ). From this, we see that p (5) Thus we get the result.…”

Section: Derivation Of P (5) (R)mentioning

confidence: 74%

“…The first proof. In the present proof, we follow up the method used in [8], where Paradan's wallcrossing formula [5] are heavily used. The measure H −h 1 * H −h 1 −h 2 * H −h 2 is, in fact, the non-Abelian Duistermaat-Heckman measure that is on the closures of the regular chambers containing the vertex (0, 0) given by the convolution…”

Section: The Mixture Of Two Qutrit Statesmentioning

confidence: 99%

Duistermaat–Heckman measure and the mixture of quantum states

Zhang

Jiang

2019

J. Phys. A: Math. Theor.

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In this paper, we present a general framework to solve a fundamental problem in Random Matrix Theory (RMT), i.e., the problem of describing the joint distribution of eigenvalues of the sum A + B of two independent random Hermitian matrices A and B. Some considerations about the mixture of quantum states are basically subsumed into the above mathematical problem. Instead, we focus on deriving the spectral density of the mixture of adjoint orbits of quantum states in terms of Duistermaat-Heckman measure, originated from the theory of symplectic geometry. Based on this method, we can obtain the spectral density of the mixture of independent random states. In particular, we obtain explicit formulas for the mixture of random qubits. We also find that, in the two-level quantum system, the average entropy of the equiprobable mixture of n random density matrices chosen from a random state ensemble (specified in the text) increases with the number n. Hence, as a physical application, our results quantitatively explain that the quantum coherence of the mixture monotonously decreases statistically as the number of components n in the mixture. Besides, our method may be used to investigate some statistical properties of a special subclass of unital qubit channels.Mathematics Subject Classification. 22E70, 81Q10, 46L30, 15A90, 81R05

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Paradan’s wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator

Cited by 9 publications

References 9 publications

Eigenvalue Distributions of Reduced Density Matrices

Eigenvalue Distributions of Reduced Density Matrices

Splines, lattice points, and arithmetic matroids

Duistermaat–Heckman measure and the mixture of quantum states

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