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Linear Algebra and its Applications

Reverse of the Ruskai-Stillinger bounds for the quantum relative entropy

Mina Sadeghinezhad, Hamed Najafi

Published Jun 1, 2026• 0 citations
Publisher
DOI
10.1016/j.laa.2026.05.028

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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).
The relative entropy description of Holevo-Schumacher-Westmoreland (HSW) classical channel capacity is applied to single qubit channels. A simple formula for the relative entropy of qubit density matrices in the Bloch sphere representation is derived. This formula is combined with the King-Ruskai-Szarek-Werner qubit channel ellipsoid picture to analyze several unital and non-unital qubit channels in detail. An alternate proof that the optimal HSW signalling states for single qubit unital channels are those states with the minimum channel output entropy is presented. The derivation is based on the symmetries of the qubit relative entropy formula and the King-Ruskai-Szarek-Werner qubit channel ellipsoid picture. A proof is given that the average output density matrix of any set of optimal HSW signalling states for a (qubit or non-qubit) quantum channel is unique.