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On Norms of Completely Positive Maps

Stanisław J. Szarek

Published Jan 1, 2010• 4 citations
Publisher

Abstract

King and Ruskai asked whether the norm of a completely positive map acting between Schatten classes of operators is equal to that of its restriction to the real subspace of self-adjoint operators. Proofs have been promptly supplied by Watrous and Audenaert. Here we provide one more proof, in fact of a slightly more general fact, under the (slightly weaker) assumption of 2-positivity. The argument is elementary and self-contained.

DOI
10.1007/978-3-0346-0158-0_31

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Still more on norms of completely positive maps

Stanisław J. Szarek

King and Ruskai asked whether the norm of a completely positive map acting between Schatten classes of operators is equal to that of its restriction to the real subspace of self-adjoint operators. Proofs have been promptly supplied by Watrous and Audenaert. Here we provide one more proof, in fact of a slightly more general fact, under the (slightly weaker) assumption of 2-positivity. The argument is elementary and self-contained.

arXiv:quant-ph/0603110•Mar 13, 2006•2 citations

A Note on the p->q norms of Completely Positive Maps

Koenraad M. R. Audenaert

King and Ruskai asked whether the $p\to q$ norm of a completely positive map $Φ$, acting between Schatten $p$ and $q$ classes of self-adjoint operators, $||Φ||_{p\to q} = \max_{A=A^*} \frac{||Φ(A)||_q}{||A||_p}$, is equal to the $p\to q$ norm of that map when acting between Schatten classes of general, not necessarily self-adjoint, operators. The first proof has been given by Watrous. We give an alternative proof of this statement.

arXiv:math-ph/0505085•May 31, 2005•10 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

Notes on super-operator norms induced by Schatten norms

John Watrous

Let $\Phi$ be a super-operator, i.e., a linear mapping of the form $\Phi:\mathrm{L}(\mathcal{F})\rightarrow\mathrm{L}(\mathcal{G})$ for finite dimensional Hilbert spaces $\mathcal{F}$ and $\mathcal{G}$. This paper considers basic properties of the super-operator norms defined by $\|\Phi\|_{q\rightarrow p}= \sup\{\|\Phi(X)\|_p/\|X\|_q\,:\,X\not=0\}$, induced by Schatten norms for $1\leq p,q\leq\infty$. These super-operator norms arise in various contexts in the study of quantum information. In this paper it is proved that if $\Phi$ is completely positive, the value of the supremum in the definition of $\|\Phi\|_{q\rightarrow p}$ is achieved by a positive semidefinite operator $X$, answering a question recently posed by King and Ruskai~\cite{KingR04}. However, for any choice of $p\in [1,\infty]$, there exists a super-operator $\Phi$ that is the {\em difference} of two completely positive, trace-preserving super-operators such that all Hermitian $X$ fail to achieve the supremum in the definition of $\|\Phi\|_{1\rightarrow p}$. Also considered are the properties of the above norms for super-operators tensored with the identity super-operator. In particular, it is proved that for all $p\geq 2$, $q\leq 2$, and arbitrary $\Phi$, the norm $\|\Phi \|_{q\rightarrow p}$ is stable under tensoring $\Phi$ with the identity super-operator, meaning that $\|\Phi \|_{q\rightarrow p} = \|\Phi \otimes I\|_{q\rightarrow p}$. For $1\leq p < 2$, the norm $\|\Phi\|_{1\rightarrow p}$ may fail to be stable with respect to tensoring $\Phi$ with the identity super-operator as just described, but $\|\Phi\otimes I\|_{1\rightarrow p}$ is stable in this sense for $I$ the identity super-operator on $\mathrm{L}(\mathcal{H})$ for $\op{dim}(\mathcal{H}) = \op{dim}(\mathcal{F})$. This generalizes and simplifies a proof due to Kitaev \cite{Kitaev97} that established this fact for the case $p=1$.

arXiv:quant-ph/0411077•Quantum Information and Computation•Jan 1, 2005•96 citations

A simple proof of the Jamiolkowski criterion for complete positivity of linear maps of algebras of Hilbert-Schmidt operators

Salgado, D., J. L. Sánchez‐Gómez

We generalize a preceding simple proof of the Jamiolkowski criterion to check whether a given linear map between algebras of operators is completely positive or not. The generalization is performed to embrace all algebras of Hilbert-Schmidt class operators, thus possibly infinite-dimensional.

arXiv:math-ph/0411074•Nov 24, 2004•0 citations

Multi-Indexed Schatten Quasi- and Anti-Norms and Additivity of Conditional Rényi Entropies

Jan Kochanowski

We introduce and study multi-indexed Schatten spaces which lift the commutative $\ell_{q_1}\![\ell_{q_2}\![\ell_{q_3}\![\dots]]]$ -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 $\mathcal{Q}=(q_1,\dots,q_n)$ 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 $|\frac{1}{q_i}-\frac{1}{q_j}|\leq1$. 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 $α\geq\frac12$. 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 $\tilde H^\uparrow_α$, $α\in[\frac12,1)$, for compositions of possibly signaling channels, extending upon the smooth max-entropy chain rule used in device-independent entanglement certification to the whole interval $α\in[1/2,1)$.

quant-pharXiv:2610.04023•Oct 6, 2026•0 citations
King and Ruskai asked whether the norm of a completely positive map acting between Schatten classes of operators is equal to that of its restriction to the real subspace of self-adjoint operators. Proofs have been promptly supplied by Watrous and Audenaert. Here we provide one more proof, in fact of a slightly more general fact, under the (slightly weaker) assumption of 2-positivity. The argument is elementary and self-contained.
King and Ruskai asked whether the p→qp\to qp→q norm of a completely positive map ΦΦΦ, acting between Schatten ppp and qqq classes of self-adjoint operators, ∣∣Φ∣∣p→q=max⁡A=A∗∣∣Φ(A)∣∣q∣∣A∣∣p||Φ||_{p\to q} = \max_{A=A^*} \frac{||Φ(A)||_q}{||A||_p}∣∣Φ∣∣p→q​=maxA=A∗​∣∣A∣∣p​∣∣Φ(A)∣∣q​​, is equal to the p→qp\to qp→q norm of that map when acting between Schatten classes of general, not necessarily self-adjoint, operators. The first proof has been given by Watrous. We give an alternative proof of this statement.
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​.
Let Φ\PhiΦ be a super-operator, i.e., a linear mapping of the form Φ:L(F)→L(G)\Phi:\mathrm{L}(\mathcal{F})\rightarrow\mathrm{L}(\mathcal{G})Φ:L(F)→L(G) for finite dimensional Hilbert spaces F\mathcal{F}F and G\mathcal{G}G. This paper considers basic properties of the super-operator norms defined by ∥Φ∥q→p=sup⁡{∥Φ(X)∥p/∥X∥q : X≠0}\|\Phi\|_{q\rightarrow p}= \sup\{\|\Phi(X)\|_p/\|X\|_q\,:\,X\not=0\}∥Φ∥q→p​=sup{∥Φ(X)∥p​/∥X∥q​:X=0}, induced by Schatten norms for 1≤p,q≤∞1\leq p,q\leq\infty1≤p,q≤∞. These super-operator norms arise in various contexts in the study of quantum information. In this paper it is proved that if Φ\PhiΦ is completely positive, the value of the supremum in the definition of ∥Φ∥q→p\|\Phi\|_{q\rightarrow p}∥Φ∥q→p​ is achieved by a positive semidefinite operator XXX, answering a question recently posed by King and Ruskai~\cite{KingR04}. However, for any choice of p∈[1,∞]p\in [1,\infty]p∈[1,∞], there exists a super-operator Φ\PhiΦ that is the {\em difference} of two completely positive, trace-preserving super-operators such that all Hermitian XXX fail to achieve the supremum in the definition of ∥Φ∥1→p\|\Phi\|_{1\rightarrow p}∥Φ∥1→p​. Also considered are the properties of the above norms for super-operators tensored with the identity super-operator. In particular, it is proved that for all p≥2p\geq 2p≥2, q≤2q\leq 2q≤2, and arbitrary Φ\PhiΦ, the norm ∥Φ∥q→p\|\Phi \|_{q\rightarrow p}∥Φ∥q→p​ is stable under tensoring Φ\PhiΦ with the identity super-operator, meaning that ∥Φ∥q→p=∥Φ⊗I∥q→p\|\Phi \|_{q\rightarrow p} = \|\Phi \otimes I\|_{q\rightarrow p}∥Φ∥q→p​=∥Φ⊗I∥q→p​. For 1≤p<21\leq p < 21≤p<2, the norm ∥Φ∥1→p\|\Phi\|_{1\rightarrow p}∥Φ∥1→p​ may fail to be stable with respect to tensoring Φ\PhiΦ with the identity super-operator as just described, but ∥Φ⊗I∥1→p\|\Phi\otimes I\|_{1\rightarrow p}∥Φ⊗I∥1→p​ is stable in this sense for III the identity super-operator on L(H)\mathrm{L}(\mathcal{H})L(H) for $\op{dim}(\mathcal{H}) = \op{dim}(\mathcal{F})$. This generalizes and simplifies a proof due to Kitaev \cite{Kitaev97} that established this fact for the case p=1p=1p=1.
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).