We prove the Rényi quantum null energy condition for every integer Rényi order
n≥2. In local Poincaré-invariant quantum field theory, this states that the sandwiched Rényi divergence (SRD) of an excited state relative to the vacuum is convex along a future-directed null deformation. We assume that the deformation is realized by the canonical translations of a half-sided modular inclusion and that the divergence is finite for the initial, undeformed algebra. The proof establishes convexity of the SRD along the induced positive contraction semigroup on Haagerup
Ln(M), for every non-zero positive initial element. We also establish a Rényi ant formula at every even integer order and, under an additional regularity assumption, at odd integer orders. The Rényi ant formula identifies the negative first null derivative of SRD with a multiple of the infimum of the canonical null-translation energy over vector purifications of an escort state constructed from the excited state and the vacuum.
Abstract The sandwiched Rényi divergences of two finite-dimensional density operators quantify their asymptotic distinguishability in the strong converse domain. This establishes the sandwiched Rényi divergences as the operationally relevant ones among the infinitely many quantum extensions of the classical Rényi divergences for Rényi parameter
α>1 α > 1 . The known proof of this goes by showing that the sandwiched Rényi divergence coincides with the regularized measured Rényi divergence, which in turn is proved by asymptotic pinching, a fundamentally finite-dimensional technique. Thus, while the notion of the sandwiched Rényi divergences was extended recently to density operators on an infinite-dimensional Hilbert space (in fact, even for states of an arbitrary von Neumann algebra), these quantities were so far lacking an operational interpretation similar to the finite-dimensional case, and it has also been open whether they coincide with the regularized measured Rényi divergences. In this paper we fill this gap by answering both questions in the positive for density operators on an infinite-dimensional Hilbert space, using a simple finite-dimensional approximation technique. We also initiate the study of the sandwiched Rényi divergences, and the related problem of the strong converse exponent, for pairs of positive semi-definite operators that are not necessarily trace-class (this corresponds to considering weights in a general von Neumann algebra setting). This is motivated by the need to define conditional Rényi entropies in the infinite-dimensional setting, while it might also be interesting from the purely mathematical point of view of extending the concept of Rényi (and other) divergences to settings beyond the standard one of positive trace-class operators (positive normal functionals in the von Neumann algebra setting). In this spirit, we also discuss the definition and some properties of the more general family of Rényi
(α,z) ( α , z ) -divergences of positive semi-definite operators on an infinite-dimensional separable Hilbert space.