Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State

Majumdar, Satya N.; Bohigas, O.; Lakshminarayan, Arul

doi:10.1007/s10955-008-9491-5

Cited by 54 publications

(95 citation statements)

References 41 publications

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“…for k = 1, ..., r. The g jk and H jk within the Pfaffian in (26) are as in (19), (22) and (23). In reference [46], the r-point correlation function for the unrestricted Bures-Hall ensemble has been derived by exploiting its relationship with the Cauchy twomatrix ensemble.…”

Section: K=mentioning

confidence: 99%

Bures–Hall ensemble: spectral densities and average entropies

Sarkar¹,

Kumar²

2019

J. Phys. A: Math. Theor.

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We consider an ensemble of random density matrices distributed according to the Bures measure. The corresponding joint probability density of eigenvalues is described by the fixed trace Bures-Hall ensemble of random matrices which, in turn, is related to its unrestricted trace counterpart via a Laplace transform. We investigate the spectral statistics of both these ensembles and, in particular, focus on the level density, for which we obtain exact closed-form results involving Pfaffians. In the fixed trace case, the level density expression is used to obtain an exact result for the average Havrda-Charvát-Tsallis (HCT) entropy as a finite sum. Averages of von Neumann entropy, linear entropy and purity follow by considering appropriate limits in the average HCT expression. Based on exact evaluations of the average von Neumann entropy and the average purity, we also conjecture very simple formulae for these, which are similar to those in the Hilbert-Schmidt ensemble.

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Section: K=mentioning

confidence: 99%

Bures–Hall ensemble: spectral densities and average entropies

Sarkar¹,

Kumar²

2019

J. Phys. A: Math. Theor.

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“…It is seen from the figure that whereas λ 1 monotonically decreases towards its random matrix average of ≈ 4/N , other principal eigenvalues, such as the second or third largest, display a nonmonotonic approach to their respective averages. The smallest eigenvalue, λ N grows to about 1/N 3 , which is the random matrix average [102,103]. For Λ 1, the probability density of the set of λ i follows the Marčenko-Pastur distribution given in Eq.…”

Section: Eigenvalue Moments Of the Reduced Density Matrixmentioning

confidence: 99%

“…(9). The largest alone is distributed according to the Tracy-Widom density (after appropriate scaling and shift) [104], whereas the smallest is known to be exponential [103]. Numerical results for the transition towards these results are presented in Sect.…”

Section: Eigenvalue Moments Of the Reduced Density Matrixmentioning

confidence: 99%

Eigenstate entanglement between quantum chaotic subsystems: Universal transitions and power laws in the entanglement spectrum

et al. 2018

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We derive universal entanglement entropy and Schmidt eigenvalue behaviors for the eigenstates of two quantum chaotic systems coupled with a weak interaction. The progression from a lack of entanglement in the noninteracting limit to the entanglement expected of fully randomized states in the opposite limit is governed by the single scaling transition parameter, Λ. The behaviors apply equally well to few-and many-body systems, e.g. interacting particles in quantum dots, spin chains, coupled quantum maps, and Floquet systems as long as their subsystems are quantum chaotic, and not localized in some manner. To calculate the generalized moments of the Schmidt eigenvalues in the perturbative regime, a regularized theory is applied, whose leading order behaviors depend on √ Λ. The marginal case of the 1/2 moment, which is related to the distance to closest maximally entangled state, is an exception having a √ Λ ln Λ leading order and a logarithmic dependence on subsystem size. A recursive embedding of the regularized perturbation theory gives a simple exponential behavior for the von Neumann entropy and the Havrda-Charvát-Tsallis entropies for increasing interaction strength, demonstrating a universal transition to nearly maximal entanglement. Moreover, the full probability densities of the Schmidt eigenvalues, i.e. the entanglement spectrum, show a transition from power laws and Lévy distribution in the weakly interacting regime to random matrix results for the strongly interacting regime. The predicted behaviors are tested on a pair of weakly interacting kicked rotors, which follow the universal behaviors extremely well.

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“…. , λ n−1 , s) factorises Majumdar, Bohigas and Lakshminarayan [34] computed the distribution and moments of the smallest eigenvalue λ ↓ n of complex fixed-trace Wishart matrices in terms of elementary functions in the case m = n. Let (λ 1 , . .…”

Section: Discussionmentioning

confidence: 99%

Volume of the set of LOCC-convertible quantum states

Cunden

Facchi

Florio

et al. 2020

J. Phys. A: Math. Theor.

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The class of quantum operations known as Local Operations and Classical Communication (LOCC) induces a partial ordering on quantum states. We present the results of systematic numerical computations relating to the volume (with respect to the unitarily invariant measure) of the set of LOCC-convertible bipartite pure states, where the ordering is characterised by an algebraic relation known as majorization. The numerical results, which exploit a tridiagonal model of random matrices, provide a quantitative evidence that the proportion of LOCC-convertible pairs vanishes in the limit of large dimension, and therefore support a previous conjecture by Nielsen. In particular, we show that the problem is equivalent to the persistence of a non-Markovian stochastic process and the proportion of LOCC-convertible pairs decays algebraically with a nontrivial persistence exponent. We extend this analysis by investigating the distribution of the maximal probability of successful LOCCconversions. Here the asymptotics is somehow surprising: in the limit of large dimensions, for the overwhelming majority of pairs of states a perfect LOCC-conversion is not possible; nevertheless, for most of the states there exist local strategies that succeed in achieving the conversion with probability arbitrarily close to one. We present strong evidences of a universal scaling limit for the maximal probability of successful LOCC-conversions and we suggest a connection with the typical fluctuations of the smallest eigenvalue of Wishart random matrices. arXiv:1910.04646v2 [quant-ph] 8 Nov 2019Volume of the set of LOCC-convertible quantum states

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Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State

Cited by 54 publications

References 41 publications

Bures–Hall ensemble: spectral densities and average entropies

Bures–Hall ensemble: spectral densities and average entropies

Eigenstate entanglement between quantum chaotic subsystems: Universal transitions and power laws in the entanglement spectrum

Volume of the set of LOCC-convertible quantum states

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