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arXiv:quant-ph/0512185 · Journal of Mathematical Physics

New multiplicativity results for qubit maps

Christopher King, Nilufer Koldan

Published Apr 1, 2006• 6 citations
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Abstract

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.

DOI
10.1063/1.2191787

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Quantum Information and Computation•Dec 1, 2004•27 citations
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.
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.
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).].
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 qqq-norm, and is denoted by νqν_qνq​. The multiplicativity problem is the question whether two channels used in parallel have a combined νqν_qνq​ that is the product of the νqν_qνq​ 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>1q>1q>1. Nevertheless, these counterexamples require that the dimension of these channels increases with 1−q1-q1−q and therefore do not rule out multiplicativity for qqq in intervals [1,q0)[1,q_0)[1,q0​) with q0q_0q0​ 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.