New multiplicativity results for qubit maps

Christopher King, Nilufer Koldan

Published Dec 21, 2005• 9 citations

Abstract

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.

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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/0512185Journal of Mathematical PhysicsApr 1, 20066 citations

Multiplicativity properties of entrywise positive maps

Christopher King, Michael Nathanson, Mary Beth Ruskai

arXiv:quant-ph/0409181Sep 26, 20040 citations

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

Motohisa Fukuda

arXiv:0906.3355Journal of Mathematical PhysicsFeb 1, 20100 citations

Notes on multiplicativity of maximal output purity for completely positive qubit maps

K M R Audenaert

arXiv:0709.0855Journal of Physics Conference SeriesMar 1, 20080 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/0401026Jan 7, 20047 citations

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

Christopher King, Mary Beth Ruskai

Quantum Information and ComputationDec 1, 200427 citations

New multiplicativity results for qubit maps

Christopher King, Nilufer Koldan

Published Dec 21, 2005• 9 citations

OpenAlex

Abstract

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.

Related papers

New multiplicativity results for qubit maps

Christopher King, Nilufer Koldan

arXiv:quant-ph/0512185Journal of Mathematical PhysicsApr 1, 20066 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.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/0409181Sep 26, 20040 citations

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).].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:0906.3355Journal of Mathematical PhysicsFeb 1, 20100 citations

Notes on multiplicativity of maximal output purity for completely positive qubit maps

K M R Audenaert

A problem in quantum information theory that has received considerable attention in recent years is the question of multiplicativity of the so-called maximal output purity (MOP) of a quantum channel. This quantity is defined as the maximum value of the purity one can get at the output of a channel by varying over all physical input states, when purity is measured by the Schatten $q$ -norm, and is denoted by $ν_q$ . The multiplicativity problem is the question whether two channels used in parallel have a combined $ν_q$ that is the product of the $ν_q$ of the two channels. A positive answer would imply a number of other additivity results in QIT. Very recently, P. Hayden has found counterexamples for every value of $q>1$ . Nevertheless, these counterexamples require that the dimension of these channels increases with $1-q$ and therefore do not rule out multiplicativity for $q$ in intervals $[1,q_0)$ with $q_0$ depending on the channel dimension. I argue that this would be enough to prove additivity of entanglement of formation and of the classical capacity of quantum channels. More importantly, no counterexamples have as yet been found in the important special case where one of the channels is a qubit-channel, i.e. its input states are 2-dimensional. In this paper I focus attention to this qubit case and I rephrase the multiplicativity conjecture in the language of block matrices and prove the conjecture in a number of special cases.A problem in quantum information theory that has received considerable attention in recent years is the question of multiplicativity of the so-called maximal output purity (MOP) of a quantum channel. This quantity is defined as the maximum value of the purity one can get at the output of a channel by varying over all physical input states, when purity is measured by the Schatten $q$-norm, and is denoted by $ν_q$. The multiplicativity problem is the question whether two channels used in parallel have a combined $ν_q$ that is the product of the $ν_q$ of the two channels. A positive answer would imply a number of other additivity results in QIT. Very recently, P. Hayden has found counterexamples for every value of $q>1$. Nevertheless, these counterexamples require that the dimension of these channels increases with $1-q$ and therefore do not rule out multiplicativity for $q$ in intervals $[1,q_0)$ with $q_0$ depending on the channel dimension. I argue that this would be enough to prove additivity of entanglement of formation and of the classical capacity of quantum channels. More importantly, no counterexamples have as yet been found in the important special case where one of the channels is a qubit-channel, i.e. its input states are 2-dimensional. In this paper I focus attention to this qubit case and I rephrase the multiplicativity conjecture in the language of block matrices and prove the conjecture in a number of special cases.

arXiv:0709.0855Journal of Physics Conference SeriesMar 1, 20080 citations

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

Christopher King, Mary Beth Ruskai

arXiv:quant-ph/0401026Jan 7, 20047 citations

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

Christopher King, Mary Beth Ruskai

Quantum Information and ComputationDec 1, 200427 citations

q

-norm, and is denoted by

ν_q

. The multiplicativity problem is the question whether two channels used in parallel have a combined

ν_q

that is the product of the

ν_q

of the two channels. A positive answer would imply a number of other additivity results in QIT. Very recently, P. Hayden has found counterexamples for every value of

q>1

. Nevertheless, these counterexamples require that the dimension of these channels increases with

1-q

and therefore do not rule out multiplicativity for

q

in intervals

[1,q_0)

with

q_0

depending on the channel dimension. I argue that this would be enough to prove additivity of entanglement of formation and of the classical capacity of quantum channels. More importantly, no counterexamples have as yet been found in the important special case where one of the channels is a qubit-channel, i.e. its input states are 2-dimensional. In this paper I focus attention to this qubit case and I rephrase the multiplicativity conjecture in the language of block matrices and prove the conjecture in a number of special cases.