We show a relation between a quantum channel $Φ$ and its conjugate $Φ^C$ , which implies that the $p\to p$ Schatten norm of the channel is the same as the $1\to p$ completely bounded norm of the conjugate. This relation is used to give an alternative proof of the multiplicativity of both norms.We show a relation between a quantum channel $Φ$ and its conjugate $Φ^C$, which implies that the $p\to p$ Schatten norm of the channel is the same as the $1\to p$ completely bounded norm of the conjugate. This relation is used to give an alternative proof of the multiplicativity of both norms.

arXiv:quant-ph/0601071Jan 11, 20063 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.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:2603.16722Mar 18, 20260 citations

Channel capacities via p-summing norms

Marius Junge, Carlos Palazuelos

arXiv:1305.1020Advances in MathematicsDec 31, 201410 citations

Imperfect measurements and conjugate observables

Adarsh S, P. N. Bala Subramanian, T. P. Sreeraj

arXiv:2401.05013PramanaApr 16, 20260 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 $α\geq \frac{1}{2}$ .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:2604.14055Apr 16, 20260 citations

Quantumness of channels

Javid Naikoo, Subhashish Banerjee, R. Srikanth

arXiv:1911.07677Quantum Information ProcessingJan 1, 20216 citations

We show a relation between a quantum channel

Φ

and its conjugate

Φ^C

, which implies that the

p\to p

Schatten norm of the channel is the same as the

1\to p

completely bounded norm of the conjugate. This relation is used to give an alternative proof of the multiplicativity of both norms.

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.

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}