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.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.

Journal of Mathematical PhysicsNov 25, 2002129 citations

Some Properties of Generalized Quantum Operations

Yuan Li, Xiu-Hong Sun, Yan-Ni Dou

International Journal of Theoretical PhysicsJan 11, 20133 citations

Fixed Points of Completely Positive Trace-Preserving Maps in Infinite Dimension

Roderich Tumulka, Jonte Weixler

Completely positive trace-preserving maps $S$ , also known as quantum channels, arise in quantum physics as a description of how the density operator $ρ$ 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 $\mathscr{H}$ that $ρ$ acts on is finite-dimensional, then every $S$ must have a fixed point, i.e., a density operator $ρ_0$ with $S(ρ_0)=ρ_0$ . In infinite dimension, $S$ 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 $S$ 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.Completely positive trace-preserving maps $S$, also known as quantum channels, arise in quantum physics as a description of how the density operator $ρ$ 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 $\mathscr{H}$ that $ρ$ acts on is finite-dimensional, then every $S$ must have a fixed point, i.e., a density operator $ρ_0$ with $S(ρ_0)=ρ_0$. In infinite dimension, $S$ 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 $S$ 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.

arXiv:2411.14800Nov 22, 20240 citations

Completely positive trace-preserving maps

S

, also known as quantum channels, arise in quantum physics as a description of how the density operator

ρ

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

\mathscr{H}

that

ρ

acts on is finite-dimensional, then every

S

must have a fixed point, i.e., a density operator

ρ_0

with

S(ρ_0)=ρ_0

. In infinite dimension,

S

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

S

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.