Smallest eigenvalue distribution of the fixed trace Laguerre beta-ensemble

Abstract

In this paper we study entanglement of the reduced density matrix of a bipartite quantum system in a random pure state. It transpires that this involves the computation of the smallest eigenvalue distribution of the fixed trace Laguerre ensemble of $N\times N$ random matrices. We showed that for finite $N$ the smallest eigenvalue distribution may be expressed in terms of Jack polynomials. Furthermore, based on the exact results, we found, a limiting distribution, when the smallest eigenvalue is suitably scaled with $N$ followed by a large $N$ limit. Our results turn out to be the same as the smallest eigenvalue distribution of the classical Laguerre ensembles without the fixed trace constraint. This suggests in a broad sense, the global constraint does not influence local correlations, at least, in the large $N$ limit. Consequently, we have solved an open problem: The determination of the smallest eigenvalue distribution of the reduced density matrix---obtained by tracing out the environmental degrees of freedom---for a bipartite quantum system of unequal dimensions.

14 pages;we have revised our paper, so as that state in clearer way what we intend to do and the results we found

References in corpus (4)

• Aspects of generic entanglement
• Asymptotics of random density matrices
• Entanglement of random vectors
• Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State

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• Entangled random pure states with orthogonal symmetry: exact results
• Kurtosis of von Neumann entanglement entropy
• Largest Schmidt eigenvalue of entangled random pure states and conductance distribution in chaotic cavities
• Violations of Bell inequalities from random pure states
• Quantum interpolating ensemble: Biorthogonal polynomials and average entropies
• Volume of the set of LOCC-convertible quantum states
• Three Problems Related to the Eigenvalues of Complex Non-central Wishart Matrices with a Rank-1 Mean
• Quantum Generative Training Using Rényi Divergences
• Volume-law entanglement entropy of typical pure quantum states
• Entanglement transitions induced by large deviations