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

An application of decomposable maps in proving multiplicativity of low dimensional maps

Motohisa Fukuda

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

In this paper we present a class of maps for which the multiplicativity of the\n maximal output p-norm holds when p is 2 and p is larger than or equal to 4. The class\n includes all positive trace-preserving maps from the matrix algebra on the\n three-dimensional space to that on the two-dimensional.

Related papers

An application of decomposable maps in proving multiplicativity of low dimensional maps

Motohisa Fukuda

In this paper, we present a class of maps for which the multiplicativity of the maximal output p-norm holds for p=2 and p≥4. The class includes all positive trace-preserving maps from B(C3) to B(C2). In this sense, the result is a generalization of the corresponding result in the work of King and Koldan [“New multiplicativity results for qubit maps,” J. Math. Phys. 47, 042106 (2006)], where the multiplicativity was proved for all positive trace-preserving maps from B(C2) to B(C2) with p=2 and p≥4. Interestingly, by contrast, the multiplicativity of p-norm was investigated in the context of quantum information theory and shown not to hold, in general, for high dimensional quantum channels [Hayden, P. and Winter, A., “Counterexamples to the maximal p-norm multiplicativity conjecture for all p>1,” Commun. Math. Phys. 284, 263 (2008)]. Moreover, the Werner–Holevo channel, which is a map from B(C3) to B(C3), is a counterexample for p>4.79 [Werner and Holevo, J. Math. Phys. 43, 4353 (2002).].

arXiv:0906.3355•Journal of Mathematical Physics•Feb 1, 2010•0 citations

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

Multiplicativity properties of entrywise positive maps

Christopher King, Michael Nathanson, Mary Beth Ruskai

Multiplicativity of certain maximal p -> q norms of a tensor product of linear maps on matrix algebras is proved in situations in which the condition of complete positivity (CP) is either augmented by, or replaced by, the requirement that the entries of a matrix representative of the map are non-negative (EP). In particular, for integer t, multiplicativity holds for the maximal 2 -> 2t norm of a product of two maps, whenever one of the pair is EP; for the maximal 1 -> t norm for pairs of CP maps when one of them is also EP; and for the maximal 1 -> 2t norm for the product of an EP and a 2-positive map. Similar results are shown in the infinite-dimensional setting of convolution operators on L^2(R), with the pointwise positivity of an integral kernel replacing entrywise positivity of a matrix. These results apply in particular to Gaussian bosonic channels.

arXiv:quant-ph/0409181•Sep 26, 2004•0 citations

Comments on multiplicativity of maximal p-norms when p=2

Christopher King, Mary Beth Ruskai

We consider the maximal p-norm associated with a completely positive map and the question of its multiplicativity under tensor products. We give a condition under which this multiplicativity holds when p = 2, and we describe some maps which satisfy our condition. This class includes maps for which multiplicativity is known to fail for large p. Our work raises some questions of independent interest in matrix theory; these are discussed in two appendices.

Quantum Information and Computation•Dec 1, 2004•27 citations

Comments on multiplicativity of maximal p-norms when p = 2

Christopher King, Mary Beth Ruskai

We consider the maximal p-norm associated with a completely positive map and the question of its multiplicativity under tensor products. We give a condition under which this multiplicativity holds when p = 2, and we describe some maps which satisfy our condition. This class includes maps for which multiplicativity is known to fail for large p. Our work raises some questions of independent interest in matrix theory; these are discussed in two appendices.

arXiv:quant-ph/0401026•Jan 7, 2004•7 citations
In this paper, we present a class of maps for which the multiplicativity of the maximal output p-norm holds for p=2 and p≥4. The class includes all positive trace-preserving maps from B(C3) to B(C2). In this sense, the result is a generalization of the corresponding result in the work of King and Koldan [“New multiplicativity results for qubit maps,” J. Math. Phys. 47, 042106 (2006)], where the multiplicativity was proved for all positive trace-preserving maps from B(C2) to B(C2) with p=2 and p≥4. Interestingly, by contrast, the multiplicativity of p-norm was investigated in the context of quantum information theory and shown not to hold, in general, for high dimensional quantum channels [Hayden, P. and Winter, A., “Counterexamples to the maximal p-norm multiplicativity conjecture for all p>1,” Commun. Math. Phys. 284, 263 (2008)]. Moreover, the Werner–Holevo channel, which is a map from B(C3) to B(C3), is a counterexample for p>4.79 [Werner and Holevo, J. Math. Phys. 43, 4353 (2002).].
Multiplicativity of certain maximal p -> q norms of a tensor product of linear maps on matrix algebras is proved in situations in which the condition of complete positivity (CP) is either augmented by, or replaced by, the requirement that the entries of a matrix representative of the map are non-negative (EP). In particular, for integer t, multiplicativity holds for the maximal 2 -> 2t norm of a product of two maps, whenever one of the pair is EP; for the maximal 1 -> t norm for pairs of CP maps when one of them is also EP; and for the maximal 1 -> 2t norm for the product of an EP and a 2-positive map. Similar results are shown in the infinite-dimensional setting of convolution operators on L^2(R), with the pointwise positivity of an integral kernel replacing entrywise positivity of a matrix. These results apply in particular to Gaussian bosonic channels.