×

Contractivity of positive and trace-preserving maps under \(L_{p}\) norms. (English) Zbl 1112.15007

Summary: We provide a complete picture of contractivity of trace preserving positive maps with respect to \(p\)-norms. We show that for \(p > 1\) contractivity holds in general if and only if the map is unital. When the domain is restricted to the traceless subspace of Hermitian matrices, then contractivity is shown to hold in the case of qubits for arbitrary \(p \geqslant 1\) and in the case of qutrits if and only if \(p=1, \infty\). In all noncontractive cases best possible bounds on the \(p\)-norms are derived.

MSC:

15A04 Linear transformations, semilinear transformations
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] DOI: 10.1007/s002200050756 · Zbl 0945.46053
[2] DOI: 10.1088/0305-4470/35/41/105 · Zbl 1050.81042
[3] DOI: 10.1103/PhysRevLett.78.2275 · Zbl 0944.81011
[4] DOI: 10.1016/S0375-9601(00)00171-7 · Zbl 0948.81515
[5] DOI: 10.1007/978-3-642-66451-9
[6] Reed M., Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness (1975) · Zbl 0308.47002
[7] Paulsen V., Completely Bounded Maps and Operator Algebras (2002) · Zbl 1029.47003
[8] DOI: 10.1007/s00220-006-0035-z · Zbl 1107.81014
[9] DOI: 10.1007/s00220-006-0034-0 · Zbl 1118.46057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.