Maximal energy extraction via quantum measurement

We study the maximal amount of energy that can be extracted from a finite quantum system by means of projective measurements. For this quantity we coin the expression 'metrotropy' M, in analogy with 'ergotropy' W, which is the maximal amount of energy that can be extracted by means of unitary operations. The study is restricted to the case when the system is initially in a stationary state, and therefore the ergotropy is achieved by means of a permutation of the energy eigenstates. We show that (i) the metrotropy is achieved by means of an even combination of the identity and an involution permutation; (ii) it is M <= W/2, with the bound being saturated when the permutation that achieves the ergotropy is an involution.