Udaysinh T. Bhosale, Steven Tomsovic, Arul Lakshminarayan
The entanglement between two arbitrary subsystems of random pure states is
studied via properties of the density matrix's partial transpose,
$\rho_{12}^{T_2}$. The density of states of $\rho_{12}^{T_2}$ is close to the
semicircle law when both subsystems have dimensions which are not too small and
are of the same order. A simple random matrix model for the partial transpose
is found to capture the entanglement properties well, including a transition
across a critical dimension. Log-negativity is used to quantify entanglement
between subsystems and approximate analytic formulas for this are derived. The
skewness of the eigenvalue density of $\rho_{12}^{T_2}$ is derived
analytically, using the average of the third moment that is also shown to be
related to a generalization of the Kempe invariant. Extreme value statistics,
especially the Tracy-Widom distribution, is found to be useful in calculating
the fraction of entangled states at critical dimensions. These results are
tested in a quantum dynamical system of three coupled standard maps, where one
finds that if the parameters represent a strongly chaotic system, the results
are close to those of random states, although there are some systematic
deviations at critical dimensions.
View original:
http://arxiv.org/abs/1202.2982
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