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arXiv:0908.2475 · Journal of Physics A Mathematical and Theoretical

Fixed points of commutative Lüders operations

Weihua Liu, Junde Wu

Published Aug 26, 2010• 22 citations
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Abstract

This paper verifies a conjecture posed in a pair of papers on the fixed point sets for a class of quantum operations. Specifically, it is proved that if a quantum operation has mutually commuting operation elements that are effects forming a resolution of the identity, then the fixed point set of the quantum operation is exactly the commutant of the operation elements.

DOI
10.1088/1751-8113/43/39/395206

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This paper verifies a conjecture posed in a pair of papers on the fixed point sets for a class of quantum operations. Specifically, it is proved that if a quantum operation has mutually commuting operation elements that are effects forming a resolution of the identity, then the fixed point set of the quantum operation is exactly the commutant of the operation elements.
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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.
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 provide a construction of sets of (d/2+1) mutually unbiased bases (MUBs) in dimensions d=4,8 using maximal commuting classes of Pauli operators. We show that these incomplete sets cannot be extended further using the operators of the Pauli group. However, specific examples of sets of MUBs obtained using our construction are shown to be strongly unextendible; that is, there does not exist another vector that is unbiased with respect to the elements in the set. We conjecture the existence of such unextendible sets in higher dimensions (d=2^{n}, n>3) as well. Furthermore, we note an interesting connection between these unextendible sets and state-independent proofs of the Kochen-Specker Theorem for two-qubit systems. Our construction also leads to a proof of the tightness of a H_{2} entropic uncertainty relation for any set of three MUBs constructed from Pauli classes in d=4.