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We obtain a new inequality for arbitrary Hermitian matrices. We describeparticular linear maps called the matrix portrait of arbitrary NxN matrices.The maps are obtained as analogs of partial tracing of density matrices ofmultipartite qudit systems. The structure of the maps is inspired by "portrait"map of the probability vectors corresponding to the action on the vectors bystochastic matrices containing either unity or zero matrix elements. We obtainnew entropic inequalities for arbitrary qudit states including a single quditand an discuss entangled single qudit state. We consider in detail the examplesof N=3 and 4. Also we point out a possible use of entangled states of systemswithout subsystems (e.g., a single qudit) as a resource for quantumcomputations.

The maps of matrices and portrait maps of density operators of composite and noncomposite systems