Citation: Kretschmann, Dennis, Dirk Schlingemann, and Reinhard F. Werner. "A continuity theorem for Stinespring's dilation." Journal of Functional Analysis 255.8 (2008): 1889-1904.
Web: https://arxiv.org/abs/0710.2495
Tags: Mathematical, Approximate-structure
In this lovely paper, the authors demonstrate a continuity theorem for Stinespring's dilation. That is - they show that two completely positive maps have nearby dilations in operator norm if and only if they are close in cb norm. For cptp maps between finite-dimensional matrix algebras (i.e. quantum channels) then this theorem was already resolved the year earlier:
> D. Kretschmann, D. Schlingemann, R. F. Werner: The Information-Disturbance Tradeoff and the Continuity of Stinespring's Representation, 2006; quant-ph/0605009
The present paper is tackling the infinite-dimensional case. As part of this endeavor, the authors introduce the notion of "Bures distance" to the quantum information community. The name "Bures" here is a reference to a much earlier paper:
> D. Bures: An Extension of Kakutani's Theorem on Infinite Product Measures to the Tensor Product of Semifinite w-algebras Trans. Amer. Math. Soc. 135 (1969)
In this earlier paper, Bures introduced a notion of distance for positive functionals. The notion of Bures distance for positive maps between C*-algebras generalizes this classical notion. The Bures distance here is defined as the smallest possible operator norm of a difference between two dilations, when the codomain of the cp map is the space of bounded operators on a Hilbert space (since it is in this context that the notion of dilation makes the most sense). In the case of a map for which the codomain is not the space of all bounded operators on a Hilbert space, the Bures distance is defined by a different kinda funky formula. Whenever the codomain is an "injective" C*-algebra then the main theorem holds.
In the context of quantum information the Bures distance is typically defined in terms of fidelity, now.