Distribution of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>G</mml:mi></mml:math>concurrence of random pure states

Abstract

The average entanglement of random pure states of an $N\ifmmode\times\else\texttimes\fi{}N$ composite system is analyzed. We compute the average value of the determinant $D$ of the reduced state, which forms an entanglement monotone. Calculating higher moments of the determinant, we characterize the probability distribution $P(D)$. Similar results are obtained for the rescaled $N\text{th}$ root of the determinant, called the $G$ concurrence. We show that in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ this quantity becomes concentrated at a single point ${G}_{\ensuremath{\star}}=1∕e$. The position of the concentration point changes if one consider an arbitrary $N\ifmmode\times\else\texttimes\fi{}K$ bipartite system, in the joint limit $N,K\ensuremath{\rightarrow}\ensuremath{\infty}$, with $K∕N$ fixed.