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Journal of Mathematical Physics

Characterizations of fixed points of quantum operations

Yuan Li

Published May 1, 2011• 18 citations
Publisher

Abstract

Let \documentclass[12pt]{minimal}\begin{document}$\phi _{\mathcal {A}}$\end{document}φA be a general quantum operation. An operator B is said to be a fixed point of \documentclass[12pt]{minimal}\begin{document}$\phi _{\mathcal {A}},$\end{document}φA, if \documentclass[12pt]{minimal}\begin{document}$\phi _{\mathcal {A}}(B)=B.$\end{document}φA(B)=B. In this note, we shall show conditions under which B, a fixed point \documentclass[12pt]{minimal}\begin{document}$\phi _{\mathcal {A}},$\end{document}φA, implies that B is compatible with the operation element of \documentclass[12pt]{minimal}\begin{document}$\phi _{\mathcal {A}}.$\end{document}φA. In particular, we offer an extension of the generalized Lüders theorem.

DOI
10.1063/1.3583541

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Let \documentclass[12pt]{minimal}\begin{document}ϕA\phi _{\mathcal {A}}ϕA​\end{document}φA be a general quantum operation. An operator B is said to be a fixed point of \documentclass[12pt]{minimal}\begin{document}ϕA,\phi _{\mathcal {A}},ϕA​,\end{document}φA, if \documentclass[12pt]{minimal}\begin{document}ϕA(B)=B.\phi _{\mathcal {A}}(B)=B.ϕA​(B)=B.\end{document}φA(B)=B. In this note, we shall show conditions under which B, a fixed point \documentclass[12pt]{minimal}\begin{document}ϕA,\phi _{\mathcal {A}},ϕA​,\end{document}φA, implies that B is compatible with the operation element of \documentclass[12pt]{minimal}\begin{document}ϕA.\phi _{\mathcal {A}}.ϕA​.\end{document}φA. In particular, we offer an extension of the generalized Lüders theorem.
Quantum operations frequently occur in quantum measurement theory, quantum probability, quantum computation, and quantum information theory. If an operator A is invariant under a quantum operation φ, we call A a φ-fixed point. Physically, the φ-fixed points are the operators that are not disturbed by the action of φ. Our main purpose is to answer the following question. If A is a φ-fixed point, is A compatible with the operation elements of φ? We shall show in general that the answer is no and we shall give some sufficient conditions under which the answer is yes. Our results will follow from some general theorems concerning completely positive maps and injectivity of operator systems and von Neumann algebras.
We first show that for any quantum states ρ on \documentclass[12pt]{minimal}\begin{document}H\mathcal {H}H\end{document}H and σ on \documentclass[12pt]{minimal}\begin{document}K\mathcal {K}K\end{document}K there exists a quantum channel Φ such that Φ(ρ) = σ, where \documentclass[12pt]{minimal}\begin{document}H\mathcal {H}H\end{document}H and \documentclass[12pt]{minimal}\begin{document}K\mathcal {K}K\end{document}K are finite or infinite dimensional Hilbert spaces. Then we consider some conclusions for the quantum channel Φ such that Φ(ρ) = σ and \documentclass[12pt]{minimal}\begin{document}Φ(IH)\Phi (I_{\mathcal {H}})Φ(IH​)\end{document}Φ(IH) exists or \documentclass[12pt]{minimal}\begin{document}Φ(IH)=IK.\Phi (I_{\mathcal {H}})=I_{\mathcal {K}}.Φ(IH​)=IK​.\end{document}Φ(IH)=IK.
In this paper we study fixed points of commutative super-operators. First, under some circumstances, we discuss the question of necessary and sufficient conditions for the set of fixed points of a super-operator to be non-trivial, then we establish a structure lemma to completely characterize this set. Using this lemma, we show that if a commutative super-operator is also a generalized Lüders operator, then its fixed points are precisely those operators intertwining the two sequences of operation elements associated with it. Specifically, the main results of Weihua and Junde ( 2010 J. Phys. A: Math. Theor. 43 395206 ) are special cases of our results.