The large-<i>N</i>limit of matrix integrals over the orthogonal group

Zuber, Jean-Bernard

doi:10.1088/1751-8113/41/38/382001

Cited by 21 publications

(39 citation statements)

References 25 publications

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“…Finally, before leaving this section, we just mention that those operators K i,j are also related to the Laplacian over the set of matrices E β,n , as was noted recently by Zuber [34].…”

Section: Mp (Y ) = P (K)m (5-20)mentioning

confidence: 89%

Some properties of angular integrals

Bergère¹,

Eynard²

2009

J. Phys. A: Math. Theor.

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Abstract:We find new representations for Itzykson-Zuber like angular integrals for arbitrary β, in particular for the orthogonal group O(n), the unitary group U(n) and the symplectic group Sp(2n). We rewrite the Haar measure integral, as a flat Lebesge measure integral, and we deduce some recursion formula on n. The same methods gives also the Shatashvili's type moments. Finally we prove that, in agreement with Brezin and Hikami's observation, the angular integrals are linear combinations of exponentials whose coefficients are polynomials in the reduced variables (x i − x j )(y i − y j ).

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“…Finally, before leaving this section, we just mention that those operators K i,j are also related to the Laplacian over the set of matrices E β,n , as was noted recently by Zuber [34].…”

Section: Mp (Y ) = P (K)m (5-20)mentioning

confidence: 89%

Some properties of angular integrals

Bergère¹,

Eynard²

2009

J. Phys. A: Math. Theor.

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“…they are permutations of n elements, and s ( ) c n, are the so called Weingarten functions which depend on the number of elements appearing in the integral and on the particular permutation of those n elements. The analytic expression of the Weingarten functions is explicitely known for small values of n [51], and it can be computed for higher n with some effort.…”

Section: Appendix B the Twirling Channelmentioning

confidence: 99%

“…Weingarten functions for the case = = n m 4 The case = = n m 4 is particularly interesting to us since it appears in the computation of the variance of the skew information. We report here the Weingarten functions for n=4, taking them from [51]. First let us set some notation to deal with permutations.…”

Section: Appendix B the Twirling Channelmentioning

confidence: 99%

Building versatile bipartite probes for quantum metrology

et al. 2016

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We consider bipartite systems as versatile probes for the estimation of transformations acting locally on one of the subsystems. We investigate what resources are required for the probes to offer a guaranteed level of metrological performance, when the latter is averaged over specific sets of local transformations. We quantify such a performance via the average skew information (AvSk), a convex quantity which we compute in closed form for bipartite states of arbitrary dimensions, and which is shown to be strongly dependent on the degree of local purity of the probes. Our analysis contrasts and complements the recent series of studies focused on the minimum, rather than the average, performance of bipartite probes in local estimation tasks, which was instead determined by quantum correlations other than entanglement. We provide explicit prescriptions to characterize the most reliable states maximizing the AvSk, and elucidate the role of state purity, separability and correlations in the classification of optimal probes. Our results can help in the identification of useful resources for sensing, estimation and discrimination applications when complete knowledge of the interaction mechanism realizing the local transformation is unavailable, and access to pure entangled probes is technologically limited.

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“…An important difference between (9) and (8) is that the factorizations in (9) take place in S n+m for some m ≥ 0, while all those in (8) take place in S n . Notice that our factorizations must satisfy condition iii), which is related to the distribution of the elements from the set {1, ..., n} among the cycles of the factors τ 1 , τ 2 .…”

Section: Factorizationsmentioning

confidence: 99%

Expansion of polynomial Lie group integrals in terms of certain maps on surfaces, and factorizations of permutations

Novaes¹

2017

J. Phys. A: Math. Theor.

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Using the diagrammatic approach to integrals over Gaussian random matrices, we find a representation for polynomial Lie group integrals as infinite sums over certain maps on surfaces. The maps involved satisfy a specific condition: they have some marked vertices, and no closed walks that avoid these vertices. We also formulate our results in terms of permutations, arriving at new kinds of factorization problems.

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The large-Nlimit of matrix integrals over the orthogonal group

Cited by 21 publications

References 25 publications

Some properties of angular integrals

Some properties of angular integrals

Building versatile bipartite probes for quantum metrology

Expansion of polynomial Lie group integrals in terms of certain maps on surfaces, and factorizations of permutations

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