Bounds for The Reduced Relative Entropies

Markus B. Fröb, Leonardo Sangaletti

Abstract We consider the generalization of the Araki–Uhlmann formula for relative entropy to Petz–Rényi relative entropy. We compute this entropy for a free scalar field in the Minkowski wedge between the vacuum and a coherent state, as well as for the free chiral current in a thermal state. In contrast to the relative entropy which in these cases only depends on the symplectic form and thus reduces to the classical entropy of a wave packet, the Petz–Rényi relative entropy also depends on the symmetric part of the two-point function and is thus genuinely quantum. We also consider the relation with standard subspaces, where we define the Rényi entropy of a vector and show that it admits an upper bound given by the entropy of the vector.

Ismail Nikoufar, Maryam Fazlolahi, Shigeru Furuichi

Ingemar Bengtsson, Karol Zyczkowski

Markus B. Fröb, Albert Much, Kyriakos Papadopoulos

We consider the relative entropy between the vacuum and a coherent state in the Rindler wedge for an interacting λ ϕ 4 theory to first order in λ . We construct the perturbatively interacting Weyl algebra of the wedge, and employ Tomita-Takesaki modular theory and the Araki-Uhlmann formula to compute the relative entropy. We verify that the relative entropy reduces to the classical (interacting) boost Noether charge, analogously to the free theory, and that the Bekenstein bound holds.

Ingemar Bengtsson, Karol Życzkowski

Spencer Hill, Fady Alajaji, Tamás Linder +1 more

In relative entropy coding, a sender aims to design a stochastic code such that, on input $X \sim P_X$, the receiver can generate a sample $Y \sim P_{Y \mid X}$. It is a standard result that (1) this requires at least $I(X; Y)$ bits, (2) the lower bound is achievable within a logarithmic gap, and (3) this gap cannot be reduced in general. The necessity of the gap suggests that the mutual information is not the correct information measure to quantify the rate of relative entropy coding. A potential alternative emerged in the work of Flamich et al. (2025), who proved a tighter lower bound of $I_F(X \to Y)$, a quantity we call the functional information. In this paper, we construct the ring toss code, an encoding method for rejection sampling which uses at most $I_F(X \to Y) + \log e$ bits, hence obtaining a tight one-shot characterization of relative entropy coding. For the trivial channel $Y = X$, our result recovers the noiseless source coding theorem within a small constant. For a general channel, it implies that the classical mutual information lower bound is achievable within $\log(I(X; Y) + 1) + 2.45$ bits in general and within $1.45$ bits for singular channels, which are both the tightest bounds of their kind to date. Moreover, our one-shot result also recovers Sriramu and Wagner's asymptotic results on the second-order redundancy of relative entropy codes.