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arXiv:2411.14800

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

Roderich Tumulka, Jonte Weixler

Published Nov 22, 2024• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2411.14800

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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.
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Abstract In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi’s characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A completely positive map μ is called a quantum channel, if it is trace preserving, and μ is called a quantum subchannel if it decreases the trace of a positive operator.We give simple neccesary and sufficient condtions for μ to be a quantum subchannel.We show that μ is a quantum subchannel if and only if it hasHellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.
Abstract We establish an operator algebra generalization of Watrous’ theorem (Watrous 2009 Quantum Inf. Comput. 9 403–413) on mixing unital quantum channels (completely positive trace-preserving maps) with the completely depolarizing channel, wherein the more general objects of focus become (finite-dimensional) von Neumann algebras, the unique trace preserving conditional expectation onto the algebra, the group of unitary operators in the commutant of the algebra, and the fixed point algebra of the channel. As an application, we obtain a result on the asymptotic theory of quantum channels, showing that all unital channels are eventually mixed unitary. We also discuss the special case of the diagonal algebra in detail, and draw connections to the theory of correlation matrices and Schur product maps.
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In this paper we study quantum dynamical mappings called also quantum processes.The set of values of such mapping is a one-parameter family of completely positive trace-preserving linear operators defined on a finite-dimensional Hilbert space.In quantum information theory such operators are referred to as quantum channels.An important concept for quantum dynamical mappings is their divisibility.There are different types of this concept.The present paper deals with so-called completely positive divisible quantum processes.For two such processes, which are bijective and satisfy a commutativity condition, we construct a compound quantum process.It is shown that this compound quantum process is also completely positive divisible.Endowing a set of quantum channels with the norm topology, we consider continuous quantum processes and continuous completely positive evolutions.The latter are defined as two-parameter families of quantum channels satisfying additional properties.We prove that a continuous bijective completely positive divisible quantum process generates a continuous completely positive evolution.In order to illustrate the considered concepts and the results on them, we provide examples of quantum dynamical mappings with values in the set of qubit channels.In particular, a completely positive divisible compound quantum process is constructed for two bijective commuting quantum processes.Geometric and physical interpretations of this compound quantum process are given.
Let H\mathcal{H}H be a complex finite-dimensional or infinite-dimensional separable Hilbert space, B(H)\mathcal{B(H)}B(H) and T(H)\mathcal{T(H)}T(H) be the Banach spaces of all bounded linear operators and of all trace class operators on H,\mathcal{H},H, respectively. In this paper, we give a concrete description of the linear maps Φ:T(H)→B(H⊗H)Φ:\mathcal{T(H)}\rightarrow \mathcal{B(H\otimes H)}Φ:T(H)→B(H⊗H) that are continuous relative to the norm topology and covariance under unitary evolution (i.e., Φ(UXU∗)=(U⊗U)Φ(X)(U∗⊗U∗)Φ(UXU^*)=(U\otimes U)Φ(X)(U^*\otimes U^*)Φ(UXU∗)=(U⊗U)Φ(X)(U∗⊗U∗) for all X∈T(H)X\in\mathcal{T(H)}X∈T(H) and unitary operators U∈B(H)).U\in\mathcal{B(H)}).U∈B(H)). Using this, we obtain the equivalent conditions for this class of maps to be self-adjoint or positive. As a corollary, we get that the virtual broadcasting map Bvb:T(H)→B(H⊗H)\mathcal{B}_{vb}:\mathcal{T(H)}\rightarrow \mathcal{B(H\otimes H)}Bvb​:T(H)→B(H⊗H) with the form Bvb(X)=12[S(I⊗X)+S(X⊗I)]\mathcal{B}_{vb}(X)=\frac{ 1}{2}[S(I\otimes X)+S(X\otimes I)]Bvb​(X)=21​[S(I⊗X)+S(X⊗I)] is uniquely determined by three conditions: covariance under unitary evolution, invariance under permutation of the copies and consistency with classical broadcasting, where S∈B(H⊗H)S\in\mathcal{B(H\otimes H)}S∈B(H⊗H) is the swap operator. Moreover, the linear maps Ψ:B(H)→B(H⊗H)Ψ:\mathcal{B(H)}\rightarrow \mathcal{B(H\otimes H)}Ψ:B(H)→B(H⊗H) that are continuous relative to the W∗W^*W∗-topology and covariance under unitary evolution are also characterized.