ON AN INEQUALITY CONJECTURED BY BESENYEI AND PETZ

In this paper we investigate the inequality Tr(T circle times I-2)rho(12)( log(q)rho(12) - log(q)rho(1)circle times I-2 - I-1 circle times log(q)rho(2)) >= 0, where rho(12) is a density matrix and 0 <= T is an element of M-m(C). This inequality was conjectured by Besenyei and Petz in 2013, where it was proved to hold for the density matrices in M-2(C) circle times M-2(C). Here we prove this inequality for the density matrices in M-m(C) circle times M-n(C) using some elementary matrix methods. We also obtain some new inequalities related to the operators (matrices) in this inequality.