Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State

Satya N. Majumdar 1, Oriol Bohigas 1, Arul Lakshminarayan 2, 3

Journal of Statistical Physics 131 (2008) 33-49

A recent conjecture regarding the average of the minimum eigenvalue of the reduced density matrix of a random complex state is proved. In fact, the full distribution of the minimum eigenvalue is derived exactly for both the cases of a random real and a random complex state. Our results are relevant to the entanglement properties of eigenvectors of the orthogonal and unitary ensembles of random matrix theory and quantum chaotic systems. They also provide a rare exactly solvable case for the distribution of the minimum of a set of N {\em strongly correlated} random variables for all values of N (and not just for large N).

• 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
CNRS : UMR8626 – Université Paris XI - Paris Sud
• 2. Max-Planck-Institut für Physik komplexer Systeme,
Max-Planck-Institut
• 3. Department of Physics,