Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Multiplicativity of completely bounded p-norms implies a new additivity result

Devetak, I, Junge, M, King, C, Ruskai, M B

Published Jun 23, 2005• 78 citations
OpenAlex

Abstract

We prove additivity of the minimal conditional entropy associated with a quantum channel Phi, represented by a completely positive (CP), trace-preserving map, when the infimum of S(gamma_{12}) - S(gamma_1) is restricted to states of the form gamma_{12} = (I \\ot Phi)(| psi >< psi |). We show that this follows from multiplicativity of the completely bounded norm of Phi considered as a map from L_1 -> L_p for L_p spaces defined by the Schatten p-norm on matrices; we also give an independent proof based on entropy inequalities. Several related multiplicativity results are discussed and proved. In particular, we show that both the usual L_1 -> L_p norm of a CP map and the corresponding completely bounded norm are achieved for positive semi-definite matrices. Physical interpretations are considered, and a new proof of strong subadditivity is presented.

Related papers

New multiplicativity results for qubit maps

Christopher King, Nilufer Koldan

Let Phi be a trace-preserving, positivity-preserving linear map on the algebra of complex 2x2 matrices, and let Omega be any finite-dimensional completely positive map. For p=2 and for p >= 4, we prove that the maximal p-norm of the product map (Phi \\otimes Omega) is the product of the maximal p-norms of Phi and Omega. Restricting Phi to the class of completely positive maps, this settles the multiplicativity question for all qubit channels in the range of values p >= 4.

•Dec 21, 2005•9 citations

New multiplicativity results for qubit maps

Christopher King, Nilufer Koldan

Let Φ be a trace-preserving, positivity-preserving (but not necessarily completely positive) linear map on the algebra of complex 2×2 matrices, and let Ω be any finite-dimensional completely positive map. For p=2 and p⩾4, we prove that the maximal p-norm of the product map Φ⊗Ω is the product of the maximal p-norms of Φ and Ω. Restricting Φ to the class of completely positive maps, this settles the multiplicativity question for all qubit channels in the range of values p⩾4.

arXiv:quant-ph/0512185•Journal of Mathematical Physics•Apr 1, 2006•6 citations

Completely Bounded Qusi-Norms, Their Mutiplicativity, and New Additivity Results of Quantum Channels

Ke Li, Quanhua Xu

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 $α\in[\frac{1}{2},1)$. To prove this, we introduce the completely bounded $1\toα$ quasi-norms for completely positive maps, with $α\in[\frac{1}{2},1)$, 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.

quant-pharXiv:2603.16722•Mar 18, 2026•0 citations

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

Jan Kochanowski, Omar Fawzi, Cambyse Rouzé

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 $α\geq \frac{1}{2}$.

quant-pharXiv:2604.14055•Apr 16, 2026•0 citations

Minimal entropy of states emerging from noisy quantum channels

Christopher King, Mary Beth Ruskai

We consider the minimal entropy of qubit states transmitted through two uses of a noisy quantum channel, which is modeled by the action of a completely positive trace-preserving (or stochastic) map. We provide strong support for the conjecture that this minimal entropy is additive, namely, that the minimum entropy can be achieved when product states are transmitted. Explicitly, we prove that for a tensor product of two unital stochastic maps on qubit states, using an entanglement that involves only states which emerge with minimal entropy cannot decrease the entropy below the minimum achievable using product states. We give a separate argument, based on the geometry of the image of the set of density matrices under stochastic maps, which suggests that the minimal entropy conjecture holds for nonunital as well as for unital maps. We also show that the maximal norm of the output states is multiplicative for most product maps on n-qubit states, including all those for which at least one map is unital. For the class of unital channels on C/sup 2/, we show that additivity of minimal entropy implies that the Holevo (see IEEE Trans. Inform. Theory, vol.44, p.269-73, 1998 and Russian Math. Surv., p.1295-1331, 1999) capacity of the channel is additive over two inputs, achievable with orthogonal states, and equal to the Shannon capacity. This implies that superadditivity of the capacity is possible only for nonunital channels.

arXiv:quant-ph/9911079•IEEE Transactions on Information Theory•Jan 1, 2001•345 citations

The additivity problem for constrained quantum channels

Александр Семенович Холево, Максим Евгеньевич Широков

One of the central problems of quantum information theory [1] is the additivity conjecture for the channel capacity of quantum communication channels [2]–[4]. Recently Shor showed [5] that the validity of this conjecture for all quantum channels was equivalent to a whole series of (super)additivity properties for other important characteristics such as minimal output entropy and entanglementof formation. The present paper introduces the new property of strong additivity (additivity for channels with input constraints) with several equivalent formulations, establishes that it is satisfied for some nontrivial classes of channels and notes that the validity of the additivity conjecture for all channels implies strong additivity. Let H, H′ be finite-dimensional unitary spaces. By a quantum channel we mean a completely positive trace-preserving linear map Φ: B(H) → B(H′), where B(H) is the algebra of all operators on H. In particular, Φ generates an affine map of the convex set S(H) of states (density operators) on the space H into the set S(H′) of states on H′ [1]. The quantum analogue of Shannon’s theorem, established in [4], gives the following expression for the classical channel capacity:

Russian Mathematical Surveys•Apr 30, 2004•0 citations
We prove additivity of the minimal conditional entropy associated with a quantum channel Phi, represented by a completely positive (CP), trace-preserving map, when the infimum of S(gamma_{12}) - S(gamma_1) is restricted to states of the form gamma_{12} = (I \ot Phi)(| psi >< psi |). We show that this follows from multiplicativity of the completely bounded norm of Phi considered as a map from L_1 -> L_p for L_p spaces defined by the Schatten p-norm on matrices; we also give an independent proof based on entropy inequalities. Several related multiplicativity results are discussed and proved. In particular, we show that both the usual L_1 -> L_p norm of a CP map and the corresponding completely bounded norm are achieved for positive semi-definite matrices. Physical interpretations are considered, and a new proof of strong subadditivity is presented.
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 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 consider the minimal entropy of qubit states transmitted through two uses of a noisy quantum channel, which is modeled by the action of a completely positive trace-preserving (or stochastic) map. We provide strong support for the conjecture that this minimal entropy is additive, namely, that the minimum entropy can be achieved when product states are transmitted. Explicitly, we prove that for a tensor product of two unital stochastic maps on qubit states, using an entanglement that involves only states which emerge with minimal entropy cannot decrease the entropy below the minimum achievable using product states. We give a separate argument, based on the geometry of the image of the set of density matrices under stochastic maps, which suggests that the minimal entropy conjecture holds for nonunital as well as for unital maps. We also show that the maximal norm of the output states is multiplicative for most product maps on n-qubit states, including all those for which at least one map is unital. For the class of unital channels on C/sup 2/, we show that additivity of minimal entropy implies that the Holevo (see IEEE Trans. Inform. Theory, vol.44, p.269-73, 1998 and Russian Math. Surv., p.1295-1331, 1999) capacity of the channel is additive over two inputs, achievable with orthogonal states, and equal to the Shannon capacity. This implies that superadditivity of the capacity is possible only for nonunital channels.
One of the central problems of quantum information theory [1] is the additivity conjecture for the channel capacity of quantum communication channels [2]–[4]. Recently Shor showed [5] that the validity of this conjecture for all quantum channels was equivalent to a whole series of (super)additivity properties for other important characteristics such as minimal output entropy and entanglementof formation. The present paper introduces the new property of strong additivity (additivity for channels with input constraints) with several equivalent formulations, establishes that it is satisfied for some nontrivial classes of channels and notes that the validity of the additivity conjecture for all channels implies strong additivity. Let H, H′ be finite-dimensional unitary spaces. By a quantum channel we mean a completely positive trace-preserving linear map Φ: B(H) → B(H′), where B(H) is the algebra of all operators on H. In particular, Φ generates an affine map of the convex set S(H) of states (density operators) on the space H into the set S(H′) of states on H′ [1]. The quantum analogue of Shannon’s theorem, established in [4], gives the following expression for the classical channel capacity: