Some features of the conditional <InlineEquation ID="Equ1"> <EquationSource Format="TEX">${\mathsf q}$</EquationSource> </InlineEquation>-entropies of composite quantum systems

The study of conditional q-entropies in composite quantum systems has recently been the focus of considerable interest, particularly in connection with the problem of separability. The q-entropies depend on the density matrix <InlineEquation ID="Equ2"> <EquationSource Format="TEX">$\rho$</EquationSource> </InlineEquation> through the quantity <InlineEquation ID="Equ3"> <EquationSource Format="TEX">$\omega_q={\rm Tr}\rho^q$</EquationSource> </InlineEquation>, and admit as a particular instance the standard von Neumann entropy in the limit case <InlineEquation ID="Equ4"> <EquationSource Format="TEX">$q\rightarrow 1$</EquationSource> </InlineEquation>. A comprehensive numerical survey of the space of pure and mixed states of bipartite systems is here performed, in order to determine the volumes in state space occupied by those states exhibiting various special properties related to the signs of their conditional q-entropies and to their connections with other separability-related features, including the majorization condition. Different values of the entropic parameter q are considered, as well as different values of the dimensions N <Subscript>1</Subscript> and N <Subscript>2</Subscript> of the Hilbert spaces associated with the constituting subsystems. Special emphasis is paid to the analysis of the monotonicity properties, both as a function of q and as a function of N <Subscript>1</Subscript> and N <Subscript>2</Subscript>, of the various entropic functionals considered. Copyright Springer-Verlag Berlin/Heidelberg 2003