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arXiv:1106.3380

Complete Characterization of the Fixed Space of Completely Positive Trace Preserving Super-operators

Ansis Rosmanis

Published Jun 17, 2011• 0 citations
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Abstract

We examine the fixed space of positive trace-preserving super-operators. We describe a specific structure that this space must have and what the projection onto it must look like. We show how these results, in turn, lead to an alternative proof of the complete characterization of the fixed space of completely positive trace-preserving super-operators.

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We examine the fixed space of positive trace-preserving super-operators. We describe a specific structure that this space must have and what the projection onto it must look like. We show how these results, in turn, lead to an alternative proof of the complete characterization of the fixed space of completely positive trace-preserving super-operators.
Let Φ\PhiΦ be a super-operator, i.e., a linear mapping of the form Φ:L(F)→L(G)\Phi:\mathrm{L}(\mathcal{F})\rightarrow\mathrm{L}(\mathcal{G})Φ:L(F)→L(G) for finite dimensional Hilbert spaces F\mathcal{F}F and G\mathcal{G}G. This paper considers basic properties of the super-operator norms defined by ∥Φ∥q→p=sup⁡{∥Φ(X)∥p/∥X∥q : X≠0}\|\Phi\|_{q\rightarrow p}= \sup\{\|\Phi(X)\|_p/\|X\|_q\,:\,X\not=0\}∥Φ∥q→p​=sup{∥Φ(X)∥p​/∥X∥q​:X=0}, induced by Schatten norms for 1≤p,q≤∞1\leq p,q\leq\infty1≤p,q≤∞. These super-operator norms arise in various contexts in the study of quantum information. In this paper it is proved that if Φ\PhiΦ is completely positive, the value of the supremum in the definition of ∥Φ∥q→p\|\Phi\|_{q\rightarrow p}∥Φ∥q→p​ is achieved by a positive semidefinite operator XXX, answering a question recently posed by King and Ruskai~\cite{KingR04}. However, for any choice of p∈[1,∞]p\in [1,\infty]p∈[1,∞], there exists a super-operator Φ\PhiΦ that is the {\em difference} of two completely positive, trace-preserving super-operators such that all Hermitian XXX fail to achieve the supremum in the definition of ∥Φ∥1→p\|\Phi\|_{1\rightarrow p}∥Φ∥1→p​. Also considered are the properties of the above norms for super-operators tensored with the identity super-operator. In particular, it is proved that for all p≥2p\geq 2p≥2, q≤2q\leq 2q≤2, and arbitrary Φ\PhiΦ, the norm ∥Φ∥q→p\|\Phi \|_{q\rightarrow p}∥Φ∥q→p​ is stable under tensoring Φ\PhiΦ with the identity super-operator, meaning that ∥Φ∥q→p=∥Φ⊗I∥q→p\|\Phi \|_{q\rightarrow p} = \|\Phi \otimes I\|_{q\rightarrow p}∥Φ∥q→p​=∥Φ⊗I∥q→p​. For 1≤p<21\leq p < 21≤p<2, the norm ∥Φ∥1→p\|\Phi\|_{1\rightarrow p}∥Φ∥1→p​ may fail to be stable with respect to tensoring Φ\PhiΦ with the identity super-operator as just described, but ∥Φ⊗I∥1→p\|\Phi\otimes I\|_{1\rightarrow p}∥Φ⊗I∥1→p​ is stable in this sense for III the identity super-operator on L(H)\mathrm{L}(\mathcal{H})L(H) for $\op{dim}(\mathcal{H}) = \op{dim}(\mathcal{F})$. This generalizes and simplifies a proof due to Kitaev \cite{Kitaev97} that established this fact for the case p=1p=1p=1.
Completely positive trace-preserving maps SSS, also known as quantum channels, arise in quantum physics as a description of how the density operator ρρρ of a system changes in a given time interval, allowing not only for unitary evolution but arbitrary operations including measurements or other interaction with an environment. It is known that if the Hilbert space H\mathscr{H}H that ρρρ acts on is finite-dimensional, then every SSS must have a fixed point, i.e., a density operator ρ0ρ_0ρ0​ with S(ρ0)=ρ0S(ρ_0)=ρ_0S(ρ0​)=ρ0​. In infinite dimension, SSS need not have a fixed point in general. However, we prove here the existence of a fixed point under a certain additional assumption which is, roughly speaking, that SSS leaves invariant a certain set of density operators with bounded ``cost'' of preparation. The proof is an application of the Schauder-Tychonoff fixed point theorem. Our motivation for this question comes from a proposal of Deutsch for how to define quantum theory in a space-time with closed timelike curves; our result supports the viability of Deutsch's proposal.