The additivity of minimum output entropies is a central problem in quantum information theory. Nonadditivity is known for every Rényi order
p>1, at the von Neumann point
p=1, and for sufficiently small positive
p, while much of the interval
0<p<1 has remained open. In this work, we prove that for every
p>0 there exist finite-dimensional quantum channels whose minimum output
p-Rényi entropies violate additivity. Our proof combines two constructions: random projection-induced channels yield nonadditivity for
p>3/4, and antisymmetric postprocessing extends the violation to all positive Rényi orders. Our estimates also improve the output-dimension bound obtained by Belinschi, Collins, and Nechita for additivity violation of minimum output von Neumann entropy.
One of the main recent achievements of quantum information theory [1] is the proof of the equivalence of several (sub-, super-) additivity conjectures for finite-dimensional quantum channels and systems [2] (see the survey in [3]). The main purpose of this paper is to give a generalization of this result to the infinite-dimensional case. Let H and H ′ be separable Hilbert spaces. A quantum channel is a linear completely positive trace-preserving map Φ: T(H ) 7→ T(H ′), where T(H ) is the ideal of all trace class operators on H . In particular, Φ generates an affine map of the convex set S(H ) of states (that is, density operators) on the space H to the set S(H ′) of states on the space H ′ [1]. Important characteristics of a quantum channel are the output entropy HΦ(ρ) = H(Φ(ρ)), a lower-semicontinuous concave function on the input state space with values in [0,+∞], and its convex closure (see [4]), denoted by b HΦ(ρ) and called the b H-function of the channel Φ. It was shown in [5] that the convex closure of the output entropy of an arbitrary quantum channel Φ is given by the expression b HΦ(ρ) = inf μ Z