Characterizing scalable measures of quantum resources

Abstract

The question of how quantum resources, like entanglement and coherence, depend on the number of copies is addressed. We define scalable quantities E (ρ ⊗N ) as physically consistent measures that can be described as functions of the variables {E (ρ ⊗i 1 ), E (ρ ⊗i 2 ), . . . , E (ρ ⊗iq ); N } for N > ij . If analyticity around vanishing resources is assumed, recursive relations can be derived for the Maclaurin series of E (ρ ⊗N ), which enable us to determine the possible functional forms. In particular, we find that if E (ρ ⊗2 n ) depends only on E (ρ), E (ρ ⊗2 ), and n, then it is completely determined by Fibonacci polynomials, to leading order. We show that the one-shot distillable (OSD) entanglement is well described as a scalable measure for several families of states. For a particular two-qutrit state ̺, we determine the OSD entanglement for ̺ ⊗96 from smaller tensorings, to leading order, with an accuracy of 97% and no extra computational effort.