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arXiv:2604.14055

Two-Indexed Schatten Quasi-Norms with Applications to Quantum Information Theory

Jan Kochanowski, Omar Fawzi, Cambyse Rouzé

quant-phcs.ITmath.FAmath.OASubmitted Apr 15, 2026• Announced Apr 16, 2026• 0 citations• Revised Sep 22, 2026 (v2)
v1 · Apr 15, 2026v2 · Sep 22, 2026
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Abstract

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 $|\frac{1}{q} - \frac{1}{p}| \leq 1$. 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 $q \to p$ completely bounded co-quasi-norm is super-multiplicative for tensor products of quantum channels for $q \geq p>0$, 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 $α\geq \frac{1}{2}$ 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 $α\geq\frac{1}{2}$ extending another important result of [Devetak, Junge, King, Ruskai, 2006], and additivity of the maximum output Rényi-$α$ entropy for $α\geq \frac{1}{2}$.

Comments
63 pages Added Theorem 4.3 and Corollary 4.3 answering one of the initial open questions Expanded the statements of Theorem 3.3 and Lemma 3.3 and corrected some minor mathematical inconsistencies Simplified Lemma C.1

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We define 2-indexed (q,p)(q,p)(q,p)-Schatten quasi-norms for any q,p>0q,p > 0q,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 ∣1q−1p∣≤1|\frac{1}{q} - \frac{1}{p}| \leq 1∣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→pq \to pq→p completely bounded co-quasi-norm is super-multiplicative for tensor products of quantum channels for q≥p>0q \geq p>0q≥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≥12α\geq \frac{1}{2}α≥21​ or the Sandwiched Rényi Umlaut information for α<1α< 1α<1. Our multiplicativity results imply a tensorizing notion of reverse hypercontractivity, additivity of the completely bounded minimum output Rényi-ααα-entropy for α≥12α\geq\frac{1}{2}α≥21​ extending another important result of [Devetak, Junge, King, Ruskai, 2006], and additivity of the maximum output Rényi-ααα entropy for α≥12α\geq \frac{1}{2}α≥21​.
We introduce and study multi-indexed Schatten spaces which lift the commutative ℓq1 ⁣[ℓq2 ⁣[ℓq3 ⁣[… ]]]\ell_{q_1}\![\ell_{q_2}\![\ell_{q_3}\![\dots]]]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)\mathcal{Q}=(q_1,\dots,q_n)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 nnn-fold tensor product of Hilbert spaces. We do this for indices of the same sign satisfying the mutual compatibility condition ∣1qi−1qj∣≤1|\frac{1}{q_i}-\frac{1}{q_j}|\leq1∣qi​1​−qj​1​∣≤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≥12α\geq\frac12α≥21​. This extends the case α>1α>1α>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↑\tilde H^\uparrow_αH~α↑​, α∈[12,1)α\in[\frac12,1)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)α\in[1/2,1)α∈[1/2,1).
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∈[12,1)α\in[\frac{1}{2},1)α∈[21​,1). To prove this, we introduce the completely bounded 1→α1\toα1→α quasi-norms for completely positive maps, with α∈[12,1)α\in[\frac{1}{2},1)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>1α>1α>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 reformulate the Rényi entanglement of purification as a constrained minimum output Rényi entropy problem. Equivalently, for p>1p>1p>1, this formulation can be expressed in terms of a constrained maximal output Schatten ppp-norm. More precisely, for a completely positive map Ω:L(B′)→L(A)Ω:L(B')\to L(A)Ω:L(B′)→L(A), we consider the quantity υp(Ω)\upsilon_p(Ω)υp​(Ω) defined by optimizing ∥(Ω⊗idE)(σB′E)∥p\|(Ω\otimes \mathrm{id}_E)(σ^{B'E})\|_p∥(Ω⊗idE​)(σB′E)∥p​ over all bipartite states σB′Eσ^{B'E}σB′E whose B′B'B′-marginal is maximally mixed. We focus on the case p=2p=2p=2. First, we compute υ2\upsilon_2υ2​ 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)N:L(B')\to L(A)N:L(B′)→L(A) satisfies N†∘N=a idA+b Tr[⋅] IdN^{\dagger} \mathbin{\circ} N=a\,\mathrm{id}_A+b\,\mathrm{Tr}[\cdot]\,I_dN†∘N=aidA​+bTr[⋅]Id​ for some constants a,b≥0a,b\ge 0a,b≥0, where N†N^{\dagger}N† denotes the Hilbert-Schmidt adjoint of NNN, the quantity υ2(N)\upsilon_2(N)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 ΩΩΩ, multiplicativity of υp(Ω)\upsilon_p(Ω)υp​(Ω) implies multiplicativity for its complementary map. This yields the corresponding additivity statements for the associated Rényi-2 entanglement of purification.
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).
Quantum conditional mutual information (QCMI) is a fundamental measure of conditional independence, with central applications throughout quantum information theory and many-body physics. Despite its importance, finding a fully quantum Rényi generalization that satisfies all desirable structural properties has remained an open question. In this work, we introduce a quantum Rényi conditional mutual information possessing these desirable properties: it is monotone under local quantum channels, additive under tensor products, monotone in the Rényi parameter, and converges to the QCMI in the limit α→1α\to 1α→1. We establish bounds on this quantity in terms of relative entropies to the Petz-recovered state, showing that it provides a distinct measure of approximate conditional independence. As a consequence, we derive recovery-based upper and lower bounds on the QCMI in terms of quantum generalizations of the Kullback-Leibler divergence, improving upon the best known results in certain regimes. Finally, we establish bounds in terms of relative entropies to the sets of Markov and separable states and derive Rényi chain rules, which appear to be novel even in the classical setting.
In the framework of quantum mechanics over a quadratic extension of the ultrametric field of p-adic numbers, we introduce a notion of tensor product of p-adic Hilbert spaces. To this end, following a standard approach, we first consider the algebraic tensor product of p-adic Hilbert spaces. We next define a suitable norm on this linear space. It turns out that, in the p-adic framework, this norm is the analogue of the projective norm associated with the tensor product of real or complex normed spaces. Eventually, by metrically completing the resulting p-adic normed space, and equipping it with a suitable inner product, we obtain the tensor product of p-adic Hilbert spaces. That this is indeed the correct p-adic counterpart of the tensor product of complex Hilbert spaces is also certified by establishing a natural isomorphism between this p-adic Hilbert space and the corresponding Hilbert–Schmidt class. Since the notion of subspace of a p-adic Hilbert space is highly nontrivial, we finally study the tensor product of subspaces, stressing both the analogies and the significant differences with respect to the standard complex case. These findings should provide us with the mathematical foundations necessary to explore quantum entanglement in the p-adic setting, with potential applications in the emerging field of p-adic quantum information theory.