The difference between two random mixed quantum states: exact and asymptotic spectral analysis

Mejía, José Fernando; Zapata, Camilo; Botero, Alonso

doi:10.1088/1751-8121/50/2/025301

Cited by 21 publications

(28 citation statements)

References 72 publications

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“…This is a new result in this note, and it will be used to derive the pdf of eigenvalues of the sum of two random Hermitian matrices with given spectra. To keep accordance with the notation in the present literature, the notation adopted here is only a little different from that in [14]. Denote x = (x 1 , .…”

Section: The Pdf Of Diagonals Of the Sum Of Two Random Hermitian Matrmentioning

confidence: 99%

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A variant of Horn's problem and the derivative principle

Zhang

Xiang

2020

Linear Algebra and its Applications

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Identifying the spectrum of the sum of two given Hermitian matrices with fixed eigenvalues is the famous Horn's problem. In this note, we investigate a variant of Horn's problem, i.e., we identify the probability density function (abbr. pdf) of the diagonals of the sum of two random Hermitian matrices with given spectra. We then use it to re-derive the pdf of the eigenvalues of the sum of two random Hermitian matrices with given eigenvalues via derivative principle, a powerful tool used to get the exact probability distribution by reducing to the corresponding distribution of diagonal entries. We can recover Jean-Bernard Zuber's recent results on the pdf of the eigenvalues of two random Hermitian matrices with given eigenvalues. Moreover, as an illustration, we derive the analytical expressions of eigenvalues of the sum of two random Hermitian matrices from GUE(n) or Wishart ensemble by derivative principle, respectively. We also investigate the statistics of exponential of random matrices and connect them with Golden-Thompson inequality, and partly answer a question proposed by Forrester. Some potential applications in quantum information theory, such as uniform average quantum Jensen-Shannon divergence and average coherence of uniform mixture of two orbits, are discussed.Mathematics Subject Classification. 22E70, 81Q10, 46L30, 15A90, 81R05 n ) satisfying the following constraint:This problem has an affirmative answer [4], in terms of linear inequalities which are now calledHorn's inequalities. The solutions form a convex polytope whose describing inequalities have been conjectured by Horn in 1962 [8]. Note that the convex polytope for the solution of Horn's problem is, in general, nontrivial. Hence each point in this convex polytope corresponds to a possible eigenvalue vector for the sum, except the trivial cases (for example, when one of the matrices is scalar).Although Horn's problem is apparently an elementary problem (a complete answer to Horn's problem takes almost a century), it turns out to be connected with many areas of mathematics:linear algebra of course [11], but also combinatorics, algebraic geometry [11], symplectic geometry, and even probability theory, etc. For instance, Alekseev et. al give a symplectic proof of the Horn inequalities on eigenvalues of a sum of two Hermitian matrices with given spectra [1], and Zuber investigate the probability distribution over the Horn's polytope [18]. Besides, some researchers give a description of the Duistermaat-Heckman measure on the Horn polytope. Mathematical speaking, the eigenvalue distributions involved are so-called Duistermaat-Heckman measures [5], which are defined using the push-forward of the Liouville measure on a symplectic manifold along the moment map. We follow the notations along the paper [5], consider the problem of describing the sum of two coadjoint orbits O a + O b , where K acts on O a × O b diagonally with moment map (A, B) → A + B and we have [5]: Let a ∈ t * >0 and bis the length of the Weyl group element w; and DH is the Duisterma...

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Section: The Pdf Of Diagonals Of the Sum Of Two Random Hermitian Matrmentioning

confidence: 99%

“…Later, Mejía, Zapata, and Botero rederived this result in Random Matrix Theory (RMT) [14], and they used this result to study the difference between two random mixed quantum states. The following version of the derivative principle is from [14].…”

Section: The Pdf Via Derivative Principlementioning

confidence: 99%

A variant of Horn's problem and the derivative principle

Zhang

Xiang

2020

Linear Algebra and its Applications

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“…Derivative principle,[23]). Let Z be a random matrix drawn from a unitarily invariant random matrix ensemble, ̺ Z the joint eigenvalue distribution for Z and p Z the joint distribution of the diagonal elements of Z. Then…”

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confidence: 99%

Average coherence and its typicality for random mixed quantum states

Zhang¹

2017

J. Phys. A: Math. Theor.

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Wishart ensemble is a useful and important random matrix model used in diverse fields. By realizing induced random mixed quantum states as Wishart ensemble with the fixed-trace one, using matrix integral technique we give a fast track to the average coherence for random mixed quantum states induced via partial-tracing of the Haar-distributed bipartite pure states. As a direct consequence of this result, we get a compact formula of the average subentropy of random mixed states. These obtained compact formulae extend our previous work.

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“…Due to the constraint, the eigenvalues scales as 1 n , so we shall instead use 21,36 (x, ρ) = 1 n i δ(x − nλ i )…”

Section: Average Of Expectation Valuesmentioning

confidence: 99%

Measuring the distance between quantum many-body wave functions

Chen

Zhou

2018

J. Stat. Mech.

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We study the distance of two wave functions under chaotic time evolution. The two initial states are differed only by a local perturbation. To be entitled "chaos" the distance should have a rapid growth afterwards. Instead of focusing on the entire wave function, we measure the distance d 2 (t) by investigating the difference of two reduced density matrices of the subsystem A that is spatially separated from the local perturbation. This distance d 2 (t) grows with time and eventually saturates to a small constant. We interpret the distance growth in terms of operator scrambling picture, which relates d 2 (t) to the square of commutator C(t) (out-of-time-order correlator) and shows that both these quantities measure the area of the operator wave front in subsystem A. Among various one-dimensional spin-1 2 models, we numerically show that the models with non-local power-law interaction can have an exponentially growing regime in d 2 (t) when the local perturbation and subsystem A are well separated. This regime is absent in the spin-1 2 chain with local interaction only. After sufficiently long time evolution, d 2 (t) relaxes to a small constant, which decays exponentially as we increase the system size and is consistent with eigenstate thermalization hypothesis. Based on these results, we demonstrate that d 2 (t) is a useful quantity to characterize both quantum chaos and quantum thermalization in many-body wave functions. arXiv:1712.06054v2 [cond-mat.stat-mech]

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The difference between two random mixed quantum states: exact and asymptotic spectral analysis

Cited by 21 publications

References 72 publications

A variant of Horn's problem and the derivative principle

A variant of Horn's problem and the derivative principle

Average coherence and its typicality for random mixed quantum states

Measuring the distance between quantum many-body wave functions

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