The primary entropic measures for quantum states are additive under the tensor product. In the analysis of quantum information processing tasks, the minimum entropy of a set of states, e.g., the minimum output entropy of a channel, often plays a crucial role. A fundamental question in quantum information and cryptography is whether the minimum output entropy remains additive under the tensor product of channels. Here, we establish a general additivity statement for the optimized sandwiched Rényi entropy of quantum channels. For that, we generalize the results of Devetak et al. (Commun Math Phys 266(1):37-63, 2006) to multi-index Schatten norms. As an application, we strengthen the additivity statement of Van Himbeeck and Brown (A tight and general finite-size security proof for quantum key distribution, 2025) thus allowing the analysis of time-adaptive quantum cryptographic protocols. In addition, we establish chain rules for Rényi conditional entropies that are similar to the ones used for the generalized entropy accumulation theorem of Metger et al. (Commun Math Phys 405(11):261, 2024).
The primary entropic measures for quantum states are additive under the tensor product. In the analysis of quantum information processing tasks, the minimum entropy of a set of states, e.g., the minimum output entropy of a channel, often plays a crucial role. A fundamental question in quantum information and cryptography is whether the minimum output entropy remains additive under the tensor product of channels. Here, we establish a general additivity statement for the optimized sandwiched Rényi entropy of quantum channels. For that, we generalize the results of Devetak et al. (Commun Math Phys 266(1):37-63, 2006) to multi-index Schatten norms. As an application, we strengthen the additivity statement of Van Himbeeck and Brown (A tight and general finite-size security proof for quantum key distribution, 2025) thus allowing the analysis of time-adaptive quantum cryptographic protocols. In addition, we establish chain rules for Rényi conditional entropies that are similar to the ones used for the generalized entropy accumulation theorem of Metger et al. (Commun Math Phys 405(11):261, 2024).
We introduce and study multi-indexed Schatten spaces which lift the commutative ℓq1[ℓq2[ℓq3[…]]] -spaces to the quantum toolbox. They extend Pisier's operator-valued Schatten norms and the recently introduced two-indexed Schatten quasi-norms. For every ordered tuple Q=(q1,…,qn) of non-zero indices we define a quasi-norm (for positive indices) or an anti-norm (for negative indices) on operators on an n-fold tensor product of Hilbert spaces. We do this for indices of the same sign satisfying the mutual compatibility condition ∣qi1−qj1∣≤1. Beyond establishing desirable properties of these quasi- and anti-norms via factorization formulas that combine operator space and matrix-analytic techniques, we lift several central tools and results of the seminal [Devetak, Junge, King, Ruskai, CMP 2006] to the multi-indexed and quasi-normed setting. These include a general non-commutative Minkowski inequality relating swapped systems and two identity-removal equalities. As a consequence, complete hypercontractivity and complete reverse hypercontractivity tensorize for all compatible indices, including negative ones.
Concerning quantum information theory we prove that the completely bounded minimal and the maximal output conditional entropies are additive under tensor products of quantum channels for all α≥21. This extends the case α>1 of [Fawzi, Kochanowski, Rouzé, Van Himbeeck, CMP 2026] and the unconditional results of [Devetak, Junge, King, Ruskai, CMP 2006]. Beyond tensor products, we establish chain rules for H~α↑, α∈[21,1), for compositions of possibly signaling channels, extending upon the smooth max-entropy chain rule used in device-independent entanglement certification to the whole interval α∈[1/2,1).
We define 2-indexed (q,p)-Schatten quasi-norms for any q,p>0 on operators on a tensor product of Hilbert spaces, naturally extending the norms defined by Pisier's theory of operator-valued Schatten spaces. We establish several desirable properties of these quasi-norms, such as relational consistency and the behavior on block diagonal operators, assuming that ∣q1−p1∣≤1. In fact, we show that this condition is essentially necessary for natural properties to hold. Furthermore, for linear maps between spaces of such quasi-norms, we introduce completely bounded quasi-norms and co-quasi-norms. We prove that the q→p completely bounded co-quasi-norm is super-multiplicative for tensor products of quantum channels for q≥p>0, extending an influential result of [Devetak, Junge, King, Ruskai, 2006]. Our proofs rely on elementary matrix analysis and operator convexity tools and do not require operator space theory. On the applications side, we demonstrate that these quasi-norms can be used to express relevant quantum information measures such as Rényi conditional entropies for α≥21 or the Sandwiched Rényi Umlaut information for α<1. Our multiplicativity results imply a tensorizing notion of reverse hypercontractivity, additivity of the completely bounded minimum output Rényi-α-entropy for α≥21 extending another important result of [Devetak, Junge, King, Ruskai, 2006], and additivity of the maximum output Rényi-α entropy for α≥21.
Security proofs in quantum cryptography rely on conditional entropies. In a many-round protocol, their estimation is a challenging task; one must account for the most general attacks by an eavesdropper, including those that are not independently and identically distributed (i.i.d.) across all rounds. Chain rules address this problem by relating the conditional entropy of a structured, but non-i.i.d. process to a sum of entropy contributions from each round. They are a key ingredient in entropy accumulation theorems (EATs), which provide a versatile security proof framework for many protocols in quantum cryptography. Recently, chain rules in the setting of trusted devices have lead to tight i.i.d. reductions at a finite number of rounds, and whether analogous results can be recovered in the device-independent (DI) setting has not been addressed. Surprisingly, we show that a natural tightening of the chain rule of Dupuis et al. [Commun. Math. Phys. 379, 867-913, (2020)] that would answer this question affirmatively cannot hold, highlighting a limitation of the current DI security proof approach. Nonetheless, we show that an intermediate improvement is possible by proving a new chain rule in this setting. Following the framework of Arqand et al. [Phys. Rev. X 15, 041013 (2025)], we use our chain rule to provide a slightly tighter version of the Rényi EAT in certain contexts. In addition, we provide a self-contained framework that unifies existing chain rules and compares their applications, framing our results in a broader context.
We obtain two new additivity results of quantum channels. The first one is the additivity of the channel Rényi information associated with the sandwiched Rényi divergence of order α∈[21,1). To prove this, we introduce the completely bounded 1→α quasi-norms for completely positive maps, with α∈[21,1), and show that it is multiplicative. The additivity/multiplicativity derived here extends and complements the results of Devetak {\it et al} (Commun Math Phys 266:37-63, 2006) and Gupta and Wilde (Commun Math Phys 334:867-887, 2015), which deal with the case α>1. The second one is the additivity of the channel dispersion, which is a quantity related to the second-order behavior of quantum information tasks.
We derive a novel chain rule for a family of channel conditional entropies, covering von Neumann and sandwiched Rényi entropies. In the process, we show that these channel conditional entropies are equal to their regularized version, and more generally, additive across tensor products of channels. For the purposes of cryptography, applying our chain rule to sequences of channels yields a new variant of Rényi entropy accumulation, in which we can impose some specific forms of marginal-state constraint on the input states to each individual channel. This generalizes a recently introduced security proof technique that was developed to analyze prepare-and-measure QKD with no limitations on the repetition rate. In particular, our generalization yields ``fully adaptive'' protocols that can in principle update the entropy estimation procedure during the protocol itself, similar to the quantum probability estimation framework.
We reformulate the Rényi entanglement of purification as a constrained minimum output Rényi entropy problem. Equivalently, for p>1, this formulation can be expressed in terms of a constrained maximal output Schatten p-norm. More precisely, for a completely positive map Ω:L(B′)→L(A), we consider the quantity υp(Ω) defined by optimizing ∥(Ω⊗idE)(σB′E)∥p over all bipartite states σB′E whose B′-marginal is maximally mixed. We focus on the case p=2. First, we compute υ2 for the transpose-depolarizing channel and prove that it is multiplicative under tensor powers. We then establish a general multiplicativity criterion: whenever a completely positive map N:L(B′)→L(A) satisfies N†∘N=aidA+bTr[⋅]Id for some constants a,b≥0, where N† denotes the Hilbert-Schmidt adjoint of N, the quantity υ2(N) is multiplicative under tensor powers. Examples of channels satisfying this criterion include the transpose-depolarizing channel, the depolarizing channel, and their respective complementary channels. Furthermore, we show that, for every completely positive map Ω, multiplicativity of υp(Ω) implies multiplicativity for its complementary map. This yields the corresponding additivity statements for the associated Rényi-2 entanglement of purification.