Abstract

Any set of pure states living in a given Hilbert space possesses a natural and unique metric—the Haar measure—on the group U(N) of unitary matrices. However, there is no specific measure induced on the set of eigenvalues, Δ, of any density matrix ρ. Therefore, a general approach to the global properties of mixed states depends on the specific metric defined on Δ. In the present work we shall employ a simple measure on Δ that has the advantage of possessing a clear geometric visualization whenever discussing how arbitrary states are distributed according to some measure of mixedness. The degree of mixture will be that of the participation ratio R=1/Tr(ρ2) and the concomitant maximum eigenvalue, λm. The cases studied will be the qubit–qubit system and the qubit–qutrit system, whereas some discussion will be made on higher-dimensional bipartite cases in both the R-domain and the λm-domain.

© 2014 Optical Society of America

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