A note on the infinite-dimensional quantum Strassen’s theorem

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In Ref. 3, the quantum Strassen’s theorem has been extended to the infinite-dimensional case. This theorem consists in the solution of the coupling problem for two states on the algebra of bounded operators on two Hilbert spaces [Formula: see text], [Formula: see text] with the additional constraint that the coupling state has support in a pre-assigned sub-space of [Formula: see text]. In this paper, we give an alternative proof of the main theorem in Ref. 3 that allows such extension.