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arXiv · 0901.4547

On the Contractivity of Hilbert-Schmidt distance under open system dynamics

Xiaoting Wang·Sonia Schirmer
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Abstract

We show that the Hilbert-Schmidt distance, unlike the trace distance, between quantum states is generally not monotonic for open quantum systems subject to Lindblad semigroup dynamics. Sufficient conditions for contractivity of the Hilbert-Schmidt norm in terms of the dissipation generators are given. Although these conditions are not necessary, simulations suggest that non-contractivity is the typical case, i.e., that systems for which the Hilbert-Schmidt distance between quantum states is monotonically decreasing form only a small set of all possible dissipative systems for N>2, in contrast to the case N=2 where the Hilbert-Schmidt distance is always monotonically decreasing.

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Xiaoting Wang, Sonia Schirmer. 2009-02-28. On the Contractivity of Hilbert-Schmidt distance under open system dynamics. https://doi.org/10.1103/physreva.79.052326

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