Singularizing preserving countable additivity
quantum channels on quantum measurable cardinals
May 24, 2026
We investigate the structural and dynamical properties of a class of quantum channels on von Neumann algebras induced by averaging over operator groups via the Pettis integral. Utilizing the classical Yosida–Hewitt decomposition, we focus on the interplay between the set-theoretic properties of the underlying space and the topological nature of the resulting quantum states. We establish a suffucient condition under which a channel preserves \(\sigma\)-additivity and exhibits a singularizing property, completely suppressing the normal component of any incoming state. In conjunction with the theory of Ulam real-valued measurable cardinals, this framework reveals a novel phenomenon: the existence of quantum channels that transform normal states into singular yet strictly \(\sigma\)-additive states. Furthermore, we analyze the structural constraints on the preservation of state purity imposed by the cardinality of the continuum, and extend our constructions to invariant measures on groups and their unitary representations, establishing the convergence of their Cesàro averages in the strong operator topology.
Keywords: Quantum channels, von Neumann algebras, Singular quantum states, Yosida–Hewitt decomposition, Ulam measurable cardinals, Pettis integral.
MSC2020 Subject Classification: Primary 03E55, 46L30; Secondary 81P47, 46G10.
The theory of quantum channels and the evolution of quantum states on von Neumann algebras constitute the mathematical foundation of modern quantum mechanics and quantum information theory [1]. In physical applications, primary attention is traditionally devoted to normal states, which possess the property of complete additivity and are associated with density matrices in separable Hilbert spaces. However, in infinite-dimensional systems, especially when transitioning to spaces of uncountable dimension, singular states begin to play a significant role [2], [3]. These linear functionals vanish on the ideal of compact operators [4] and lack a direct physical analogue in the form of density vectors [5], [6]. See also [7] for the dynamics of singular quantum states generated by the averaging of random shifts over some particular measures.
According to the classical Yosida–Hewitt decomposition [8], any quantum state can be uniquely represented as a convex combination of its normal and singular components. While the dynamics of normal states have been studied in detail [6], the mechanisms of generation and transformation of singular states under the action of quantum channels remain less explored. Of particular theoretical interest are channels possessing the singularizing property, i.e., those that map any initial state to a singular one [9], [10].
In the present paper, we investigate a class of quantum channels induced by averaging over groups of operators via the Pettis integral with respect to a given measure. The main focus is on the interplay between the set-theoretic properties of the underlying space (in particular, the properties of Ulam real-valued measurable cardinals [11]) and the topological properties of the resulting quantum states, building upon the framework of quantum measurable cardinals established in [12].
The main results of this paper are summarized as follows:
Preservation of \(\sigma\)-additivity: we prove that if a quantum channel is constructed as the Pettis integral over a \(\sigma\)-additive measure, then it strictly preserves the \(\sigma\)-additivity of quantum states (Theorem 2).
Criterion for singularization: we establish a sufficient condition for a channel to be singularizing. Specifically, if the \(\sigma\)-additive measure \(\mu\) vanishes on singletons, the corresponding channel completely suppresses the normal component of any state (Theorem 3). Combined with the theory of Ulam measurable cardinals, this leads to a fascinating phenomenon: the existence of channels that convert normal states into singular yet strictly \(\sigma\)-additive states.
Preservation of purity: we examine the conditions under which the considered channels can preserve the purity of vector states and establish structural limitations imposed by the cardinality of the continuum \(\mathfrak{c}\) (§4).
Dynamics on groups: we generalize the proposed construction to the case of left-invariant measures on groups and their unitary representations. We study the ergodic properties of these channels, characterize their fixed points, and prove the convergence of their Cesàro averages in the strong operator topology (§5).
The structure of the paper is organized as follows. In §2, we recall the necessary preliminaries regarding quantum states on von Neumann algebras, normality, singularity, and the Yosida–Hewitt decomposition. §3 is devoted to the construction of quantum channels on \(\ell^2(\kappa)\) spaces and the proofs of their key properties regarding \(\sigma\)-additivity and singularization. In §4, we analyze the geometry of the state space and the behavior of pure states under these channels using excision criteria. In §5, we consider analogous constructions for measures on general groups and their unitary representations, and investigate the asymptotic dynamics of quantum channel iterations.
Let \(\mathcal{H}\) be a Hilbert space. We presume that the inner product \((\cdot,\cdot)\) is linear by the second argument, as in quantum mechanics. Since we usually work with normed states, it is convenient to use the notation \(S^1(\mathcal{H})\subset\mathcal{H}\) for the unit sphere in \(\mathcal{H}\); that is, \(u \in S^1(\mathcal{H})\) if \(\left\lVert u\right\rVert=1\).
Denote by \(\mathcal{B}(\mathcal{H})\) the Banach algebra of linear bounded operators \(\mathcal{H}\to\mathcal{H}\). We denote by \(\mathcal{P}(\mathcal{H})\subset\mathcal{B}(\mathcal{H})\) the family of projectors. For a normed \(u\in S^1(\mathcal{H})\), denote by \({\mathbf{P}}_u \in \mathcal{P}(\mathcal{H})\) the projector on the subspace generated by \(u\); that is, for any \(v\in\mathcal{H}\) given by \[{\mathbf{P}}_uv = (u,v)u,\qquad (v, {\mathbf{P}}_u v) = \left\lvert(u,v)\right\rvert^2.\] The identity operator \({\mathbf{I}}\in\mathcal{B}(\mathcal{H})\) is the projector onto the whole \(\mathcal{H}\).
Let \(\mathcal{M}\subset\mathcal{B}(\mathcal{H})\) be a von Neumann algebra of observables; that is, a weakly closed \(*\)-algebra of \(\mathcal{B}(\mathcal{H})\) containing \({\mathbf{I}}\). For example, such algebras could be
the whole algebra \(\mathcal{B}(\mathcal{H})\),
the algebra of diagonal operators on \(\ell_2\),
the algebra \(\{\lambda{\mathbf{I}}: \lambda\in\mathbb{C}\}\).
Denote by \(\Sigma(\mathcal{M}) := S^{1+}(\mathcal{M}^*)\) the space of quantum states, that is, the linear functionals \(\mathcal{M}\to \mathbb{C}\), lying on the intersection of the positive cone and the unit sphere. We denote the action of a quantum state \(\rho \in \Sigma(\mathcal{M})\) on an observable \({\mathbf{A}}\in \mathcal{M}\) by \[\braket{\rho, {\mathbf{A}}}.\]
Recall that for a collection \(\{r_i:i\in I\}\) of non-negative real numbers, their sum is defined as \[\sum_{i \in I}r_i = \sup \left\{\sum_{i \in E} r_i : \text{E\subset I is finite}\right\}.\] If the sum is finite, then there is at most a countable collection of non-zero \(r_i\).
A state \(\rho\in\Sigma(\mathcal{M})\) is normal if it is \(*\)-weakly continuous. We denote the space of normal states by \(\Sigma_n(\mathcal{M})\). It is known [1], [12] that a state \(\rho\in\Sigma(\mathcal{M})\) is normal if and only if it is completely additive, which means that the equality \[\Braket{\rho, \sum_\alpha {\mathbf{P}}_\alpha} = \sum_\alpha \Braket{\rho, {\mathbf{P}}_\alpha}\] holds for any family of pairwise orthogonal projectors \({\mathbf{P}}_\alpha\in\mathcal{P}(\mathcal{H})\cap\mathcal{M}\). It is also known that a state \(\rho\in\Sigma(\mathcal{B}(\mathcal{H}))\) is normal if and only if \[\sup \Braket{\rho, {\mathbf{P}}} = 1,\] where the supremum is taken over projectors onto finite-dimensional subspaces. It is well known in the case when \(\mathcal{H}\) is separable. We present Proposition 1 below just to rid ourselves of any doubts.
A state \(\rho\in\Sigma(\mathcal{M})\) is singular [2], if there does not exist a normal state \(\rho_n\in\Sigma_n(\mathcal{M})\) such that \(\rho\geqslant\rho_n\). It is well known, see for example [2], [1], that \(\rho\) is singular, if and only if for any non-zero projector \({\mathbf{P}}\in\mathcal{M}\), there exists a projector \({\mathbf{P}}_0 \leqslant{\mathbf{P}}\) (from \(\mathcal{M}\)) such that \[\Braket{\rho, {\mathbf{P}}_0} = 0.\] We denote by \(\Sigma_s(\mathcal{M})\) the space of singular states. It is well known [4] that a state \(\rho\in\Sigma(\mathcal{B}(\mathcal{H}))\) is singular if and only if it vanishes on compact operators.
The well known result [1], which is a generalization of Yosida–Hewitt decomposition [8], states that any \(\rho\in\Sigma(\mathcal{M})\) can be represented as \[\rho = \lambda\rho_n + (1-\lambda)\rho_s\] in the unique way for some \(\lambda\in[\,0,1\,]\), and states \(\rho_n\in\Sigma_n(\mathcal{M})\) and \(\rho_s\in\Sigma_s(\mathcal{M})\).
Now we are ready to prove the promised criterion of normality.
Proposition 1. Let \(\mathcal{H}\) be a Hilbert space. A state \(\rho\in\Sigma(\mathcal{B}(\mathcal{H}))\) is normal if and only if \[\sup \Braket{\rho, {\mathbf{P}}} = 1,\] where the supremum is taken over projectors onto finite-dimensional subspaces.
Proof. Suppose \(\rho\) is normal. Then it is completely additive. Hence, taking the orthonormal basis \(\{e_\alpha\}\) in \(\mathcal{H}\), we have \[1 = \Braket{\rho, {\mathbf{I}}} = \sum_\alpha \Braket{\rho, {\mathbf{P}}_\alpha} = \sup \Braket{\rho, {\mathbf{P}}_{\text{finite}}},\] where the first equality follows from the normalization of states, the second equality follows from complete additivity, and the last equality follows from the definition of the sum.
Conversely, suppose that \[\sup\Braket{\rho, {\mathbf{P}}_{\text{finite}}} = 1.\] Take the decomposition \[\rho = \lambda\rho_n + (1-\lambda)\rho_s,\] where \(\rho_n\) is normal and \(\rho_s\) is singular. Hence \[\Braket{\rho, {\mathbf{P}}_{\text{finite}}} = \lambda\Braket{\rho_n, {\mathbf{P}}_{\text{finite}}},\] and so \[1 = \sup\Braket{\rho, {\mathbf{P}}_{\text{finite}}} = \lambda\sup\Braket{\rho_n, {\mathbf{P}}_{\text{finite}}} = \lambda.\] Thus, \(\rho=\rho_n\) is normal. ◻
A state \(\rho\in\Sigma(\mathcal{M})\) is pure if it is an extreme point of \(\Sigma(\mathcal{M})\).
For \(u\in S^1(\mathcal{H})\), a normal state \(\rho=\rho_u\in\Sigma_n(\mathcal{M})\) is called a vector state, if for all \({\mathbf{A}}\in\mathcal{M}\) we have \[\Braket{\rho, {\mathbf{A}}} = (u, {\mathbf{A}}u).\] It is pure if \(\mathcal{M}=\mathcal{B}(\mathcal{H})\), and in some other cases. See Example 1 for the case when a vector state is not pure, and Proposition 7 for details.
A state \(\rho\in\Sigma(\mathcal{M})\) is \(\sigma\)-additive (or countably additive) if \[\Braket{\rho, \sum_{n\in\mathbb{N}} {\mathbf{P}}_n} = \sum_{n\in\mathbb{N}} \Braket{\rho, {\mathbf{P}}_n}\] for any countable family of pairwise orthogonal projectors \({\mathbf{P}}_n\in\mathcal{M}\). From the above, it is clear that all normal states are \(\sigma\)-additive.
Fix any cardinal \(\kappa>\aleph_0\). For a function \(f\colon\kappa\to\mathbb{C}\), we define its support as \[\mathop{\mathrm{supp}}f := \{i \in \kappa : f(i)\neq 0\}.\] As in [10], [12], our primary focus will be on the Hilbert space \(\ell_2(\kappa)\) of functions \(f\colon \kappa\to\mathbb{C}\) with at most a countable support and the norm \[\left\lVert f\right\rVert := \sqrt{\sum_{i < \kappa}\left\lvert f(i)\right\rvert^2}.\] We denote by \(\{e_i\}\) its orthonormal basis.
Let \(\kappa\) additionally be equipped with an Abelian group structure. For example, it can be done by taking the group \(\bigoplus_{i<\kappa} \mathbb{Z}_2\) of finite functions \(\kappa\to\mathbb{Z}_2\), which is equinumerous to \(\kappa\). Or, for \(\kappa=\mathbb{R}\), it can be just the usual group \((\mathbb{R},+)\). Define the shift operator \({\mathbf{S}}_j \in \mathcal{B}(\mathcal{H})\) by the rule \[{\mathbf{S}}_j e_i := e_{i+j}.\]
By \(2^\kappa\) we denote the set of all subsets of \(\kappa\).
In this text, we consider finitely-additive measures \(\mu\colon2^\kappa\to[\,0,1\,]\) with \(\mu(\kappa)=1\). Such measures always exist: for example, they can be delta-measures. If \(\kappa\) is a Ulam real-valued measurable cardinal, then [11] there exists a \(\sigma\)-additive measure that vanishes on singletons.
Recall that a quantum channel \(\Sigma(\mathcal{M})\to\Sigma(\mathcal{M})\) is a linear completely-positive map. For a measure \(\mu\colon2^\kappa\to[\,0,1\,]\), define the quantum channel \({\mathbf{\Phi}}_\mu\colon\Sigma(\mathcal{B}(\ell_2(\kappa)))\to\Sigma(\mathcal{B}(\ell_2(\kappa)))\) given by the Pettis integral \[{\mathbf{\Phi}}_\mu\rho := \int {\mathbf{S}}_j\rho{\mathbf{S}}_j^*\,d\mu(j).\]
Theorem 2. Let \(\mu\colon2^\kappa\to[\,0,1\,]\) be a \(\sigma\)-additive measure. Then the channel \({\mathbf{\Phi}}_\mu\) preserves \(\sigma\)-additivity.
Proof. Suppose that \(\rho\in\Sigma(\mathcal{B}(\ell_2(\kappa)))\) is \(\sigma\)-additive. Then, for any pairwise orthogonal projectors \({\mathbf{P}}_n\in\mathcal{B}(\ell_2(\kappa))\), we have \[\begin{gather} \Braket{{\mathbf{\Phi}}_\mu\rho, \sum_n {\mathbf{P}}_n} = \int \Braket{\rho, \sum_n{\mathbf{S}}_j^*{\mathbf{P}}_n{\mathbf{S}}_j}\,d\mu(j) = \int \sum_n \Braket{\rho, {\mathbf{S}}_j^*{\mathbf{P}}_n{\mathbf{S}}_j}\,d\mu(j) = \\ = \sum_n \int \Braket{\rho, {\mathbf{S}}_j^*{\mathbf{P}}_n{\mathbf{S}}_j}\,d\mu(j) = \sum_n \Braket{{\mathbf{\Phi}}_\mu\rho, {\mathbf{P}}_n}, \end{gather}\] where:
the first and the last are by the definition of \({\mathbf{\Phi}}_\mu\),
the second is by \(\sigma\)-additivity of \(\rho\),
the third is by the Levi monotone convergence theorem.
This proves the \(\sigma\)-additivity of \({\mathbf{\Phi}}_\mu\rho\). ◻
We say that a quantum channel \({\mathbf{\Phi}}\) is singularizing if it maps initial states to singular ones; strictly speaking, if \({\mathbf{\Phi}}\rho\) is singular for any quantum state \(\rho\). Cf. the following theorem with the results in [9], [10], where analogous singularizing quantum channels were obtained.
Theorem 3. Let \(\mu\colon2^\kappa\to[\,0,1\,]\) be a \(\sigma\)-additive measure that vanishes on singletons. Then the channel \({\mathbf{\Phi}}_\mu\) is singularizing.
Proof. Let \(v \in S^1(\ell_2(\kappa))\) be from the unit sphere. If \(\rho=\rho_u\) is a pure vector state, then \[\Braket{{\mathbf{\Phi}}_\mu\rho, {\mathbf{P}}_v} = \int ({\mathbf{S}}_j u, {\mathbf{P}}_v {\mathbf{S}}_ju)\,d\mu(j) =0,\] where the last equality follows since \(({\mathbf{S}}_j u, {\mathbf{P}}_v {\mathbf{S}}_ju)\neq 0\) holds for at most a countable set of \(j\), and \(\mu\) is \(\sigma\)-additive.
Hence, for any normal state \(\rho\in\Sigma_n(\mathcal{B}(\ell_2(\kappa)))\), the outcome \({\mathbf{\Phi}}_\mu\rho\) vanishes on all finite-dimensional projectors, which means \({\mathbf{\Phi}}_\mu\rho\) is singular.
Finally, for a singular \(\rho\in\Sigma_s(\mathcal{B}(\ell_2(\kappa)))\), we have \[\Braket{{\mathbf{\Phi}}_\mu\rho, {\mathbf{P}}_v} = \int \Braket{\rho, {\mathbf{P}}_{{\mathbf{S}}_jv}}\,d\mu(j) =0,\] where the last equality follows since the term under the integral is zero. ◻
Theorems 2 and 3 together claim that if \(\mu\) is a \(\sigma\)-additive measure that vanishes on singletons (constructed on a Ulam real-valued measurable cardinal), then \({\mathbf{Q}}_\mu\) converts normal states to singular \(\sigma\)-additive states, which is fascinating.
Recall that [6] for any \(\rho\in\Sigma(\mathcal{B}(\ell_2(\kappa)))\), there exists a finite non-negative finitely-additive measure \(\nu\) on \(S^1(\ell_2(\kappa))\) such that \[\rho = \int \rho_u\,d\nu(u).\] So, there is another way to define a channel by the rule \[{\mathbf{\Phi}}_\mu' \rho := \int d\nu(u)\int {\mathbf{S}}_j\rho_u{\mathbf{S}}_j^*\,d\mu(j).\]
Lemma 1. The quantum channels \({\mathbf{\Phi}}_\mu\) and \({\mathbf{\Phi}}_\mu'\) coincide. In particular, the quantum channel \({\mathbf{\Phi}}_\mu'\) does not depend on the choice of \(\nu\).
Proof. Fix any \({\mathbf{A}}\in\mathcal{B}(\ell_2(\kappa))\) and consider the operator \({\mathbf{A}}_\mu\) given by the Pettis integral \[{\mathbf{A}}_\mu = \int {\mathbf{S}}_j^*{\mathbf{A}}{\mathbf{S}}_j\,d\mu(j).\] Then \[\Braket{{\mathbf{\Phi}}_\mu'\rho, {\mathbf{A}}} = \int d\nu(u)\int \Braket{\rho_u, {\mathbf{S}}_j^*{\mathbf{A}}{\mathbf{S}}_j}\,d\mu(j) = \Braket{\rho, {\mathbf{A}}_\mu} = \int \Braket{\rho, {\mathbf{S}}_j^*{\mathbf{A}}{\mathbf{S}}_j}\,d\mu(j) = \Braket{{\mathbf{\Phi}}_\mu\rho, {\mathbf{A}}}.\] ◻
If the measure \(\mu\) was not defined on all subsets of \(\kappa\), then the channels might not coincide; moreover, the channel \({\mathbf{\Phi}}_\mu\) might be undefined, as in [9]. So, the (real-valued Ulam) measurability of the cardinal is a very strong condition.
In this section, we turn our attention to the geometric and facial structure of the state space under the action of the constructed quantum channels. Specifically, we investigate the conditions under which the averaging operators preserve the purity of states, mapping pure vector states into pure quantum states. As we shall demonstrate, this property is intimately connected to the algebraic structure of the channel’s support and is heavily constrained by the cardinality of the continuum \(\mathfrak{c}\). By employing excision criteria for states on operator algebras, we establish structural limits on the existence of non-trivial purity-preserving singularizing channels.
Theorem 4. Let \(\kappa \leqslant\mathfrak{c}\). Let \(\mu\colon2^\kappa\to[\,0,1\,]\) be a \(\sigma\)-additive measure that vanishes on singletons. Then for any \(\sigma\)-additive state \(\rho\in\Sigma(\mathcal{B}(\ell_2(\kappa)))\), the outcome \({\mathbf{\Phi}}_\mu\rho\) is not pure. In particular, this holds for any normal quantum state \(\rho\in\Sigma_n(\mathcal{B}(\ell_2(\kappa)))\), and for any vector pure state.
Proof. By theorems 2 and 3, \({\mathbf{\Phi}}_\mu\) maps \(\sigma\)-additive states into singular \(\sigma\)-additive states. By [12], if there exists a singular \(\sigma\)-additive pure state, then \(\kappa > \mathfrak{c}\). ◻
Below, the object of our interest is whether \({\mathbf{\Phi}}_\mu\) preserves purity regardless of \(\kappa\) being greater than \(\mathfrak{c}\), or not.
Theorem 5. Let \(\mu\colon2^\kappa\to[\,0,1\,]\) be a measure. If \(\mu\) is not two-valued, then \({\mathbf{\Phi}}_\mu\) does not preserve the purity of any vector pure state.
Proof. Since \(\mu\) is not two-valued, there exists a decomposition \(A \sqcup B=\kappa\) such that \[0 < \mu(A),\mu(B) < 1.\] For a vector pure state \(\rho=\rho_u\), consider two states \[\rho_A := \frac{1}{\mu(A)}\int_A {\mathbf{S}}_j\rho{\mathbf{S}}_j^*\,d\mu(j) \qquad\text{and}\qquad \rho_B := \frac{1}{\mu(B)}\int_B {\mathbf{S}}_j\rho{\mathbf{S}}_j^*\,d\mu(j).\] Then \[{\mathbf{\Phi}}_\mu \rho = \mu(A)\rho_A + \mu(B)\rho_B,\] and it remains to prove that \(\rho_A\neq \rho_B\).
Let \(\mathcal{H}_A\) be a (closed) subspace generated by \({\mathbf{S}}_j u\) for \(j\in A\). Let \({\mathbf{P}}_A\) be the projector onto \(\mathcal{H}_A\). Then \[\Braket{\rho_A, {\mathbf{P}}_A} = \frac{1}{\mu(A)}\int_A \Braket{\rho_{{\mathbf{S}}_j u}, {\mathbf{P}}_A}\,d\mu(j) = \frac{1}{\mu(A)}\int_A 1\,d\mu(j) = 1.\] For \(\mu\)-almost every \(j \in B\), we have \(\Braket{\rho_{{\mathbf{S}}_j u}, {\mathbf{P}}_A}=0\). Hence, \[\Braket{\rho_B, {\mathbf{P}}_A} = \frac{1}{\mu(B)}\int_B \Braket{\rho_{{\mathbf{S}}_j u}, {\mathbf{P}}_A}\,d\mu(j) = 0.\] So, \(\Braket{\rho_A, {\mathbf{P}}_A}=1\neq 0 =\Braket{\rho_B, {\mathbf{P}}_A}\), and thus \(\rho_A\neq\rho_B\). ◻
In order to establish whether \({\mathbf{\Phi}}_\mu\) preserves purity for a two-valued measure \(\mu\), we need to develop the theory of pure states. We will follow the notations from [12].
A family \(\mathcal{F}\subset \mathcal{P}(\mathcal{H})\cap\mathcal{M}\) of projectors is called a quantum filter if the following two properties hold:
if \({\mathbf{P}}\in \mathcal{F}\) and \({\mathbf{P}}\leqslant{\mathbf{P}}'\in\mathcal{P}(\mathcal{H})\cap\mathcal{M}\), then \({\mathbf{P}}'\in\mathcal{F}\),
if \({\mathbf{P}}_1,\ldots,{\mathbf{P}}_n\in\mathcal{F}\), then \(\left\lVert{\mathbf{P}}_1\ldots{\mathbf{P}}_n\right\rVert=1\).
For example, for a state \(\rho\in\Sigma(\mathcal{M})\), the family \[\mathcal{F}_\rho := \left\{{\mathbf{P}}\in\mathcal{P}(\mathcal{H})\cap\mathcal{M}\;:\; \Braket{\rho, {\mathbf{P}}}=1\right\}\] is a quantum filter; it is not empty since, for example, it contains the identity operator. We will not use quantum filters except \(\mathcal{F}_\rho\).
Proposition 6. Let \(\rho\in\Sigma(\mathcal{M})\) be a quantum state on a von Neumann algebra \(\mathcal{M}\). Then for any \({\mathbf{P}}\in\mathcal{F}_\rho\) and \({\mathbf{A}}\in\mathcal{M}\) we have \[\Braket{\rho, {\mathbf{A}}} = \Braket{\rho, {\mathbf{P}}{\mathbf{A}}} = \Braket{\rho, {\mathbf{A}}{\mathbf{P}}}.\]
Proof. Due to the symmetry of the equation, it is sufficient to prove the first equality. Let \[{\mathbf{P}}^\perp = {\mathbf{I}}- {\mathbf{P}}\] the orthogonal projector. By linearity, \[\Braket{\rho, {\mathbf{P}}^\perp} = \Braket{\rho, {\mathbf{I}}} - \Braket{\rho, {\mathbf{P}}} = 1 - 1 = 0.\] Then, by the Cauchy–Schwarz inequality [4], \[\left\lvert\Braket{\rho, {\mathbf{P}}^\perp{\mathbf{A}}}\right\rvert^2 \leqslant\Braket{\rho, {\mathbf{P}}^\perp}\Braket{\rho, {\mathbf{A}}^*{\mathbf{A}}} = 0.\] Thus \[\Braket{\rho, {\mathbf{P}}^\perp{\mathbf{A}}} = 0,\] and so \[\Braket{\rho, {\mathbf{A}}} = \Braket{\rho, {\mathbf{P}}{\mathbf{A}}} + \Braket{\rho, {\mathbf{P}}^\perp{\mathbf{A}}} = \Braket{\rho, {\mathbf{P}}{\mathbf{A}}}.\] ◻
In [13], the criterion of purity was given in terms of excision. We give a slightly different definition of excision, as in [12]. So, let \(\rho\in\Sigma(\mathcal{M})\) be a quantum state on a von Neumann algebra \(\mathcal{M}\). For \({\mathbf{A}}\in\mathcal{M}\), we say that \({\mathbf{P}}\in\mathcal{F}_\rho\) excises \(\rho\) for \({\mathbf{A}}\), if \[{\mathbf{P}}{\mathbf{A}}{\mathbf{P}}= \Braket{\rho, {\mathbf{A}}}{\mathbf{P}}.\]
Lemma 2 (Excision Criterion). Let \(\rho\in\Sigma(\mathcal{H})\) be a quantum state on a von Neumann algebra \(\mathcal{M}\). If for any \({\mathbf{A}}\in\mathcal{M}\) there exists a projection \({\mathbf{P}}\in\mathcal{F}_\rho\) that excises \(\rho\) for \({\mathbf{A}}\), then \(\rho\) is pure. The converse holds if \(\rho\) is \(\sigma\)-additive.
Proof. The <<converse>> part is exactly [12], so we need to prove the <<if>> part. Take a decomposition \[\rho = \lambda\rho_1 + (1-\lambda)\rho_2\] for some \(0 < \lambda < 1\) and \(\rho_1\neq\rho_2\). Fix an arbitrary \({\mathbf{A}}\in\mathcal{M}\). Take the projector \({\mathbf{P}}\in\mathcal{F}_\rho\) that excises \(\rho\) for \({\mathbf{A}}\). Applying the action of \(\rho_1\) on \[{\mathbf{P}}{\mathbf{A}}{\mathbf{P}}= \Braket{\rho, {\mathbf{A}}}{\mathbf{P}},\] we obtain \[\Braket{\rho_1, {\mathbf{P}}{\mathbf{A}}{\mathbf{P}}} = \Braket{\rho, {\mathbf{A}}}\Braket{\rho_1, {\mathbf{P}}} = \Braket{\rho, {\mathbf{A}}}.\] Since \[1 = \Braket{\rho, {\mathbf{P}}} = \lambda\Braket{\rho_1, {\mathbf{P}}} + (1-\lambda)\Braket{\rho_2, {\mathbf{P}}} \leqslant \lambda + (1-\lambda) = 1,\] we obtain that both quantum filters \(\mathcal{F}_{\rho_1}\) and \(\mathcal{F}_{\rho_2}\) contain \({\mathbf{P}}\). Then by Proposition 6, \[\Braket{\rho_1, {\mathbf{A}}} = \Braket{\rho_1, {\mathbf{P}}{\mathbf{A}}{\mathbf{P}}} = \Braket{\rho, {\mathbf{A}}}.\] Due to the arbitrary choice of \({\mathbf{A}}\), we obtain that \(\rho=\rho_1\). Analogously, \(\rho=\rho_2\), which concludes the proof. ◻
Example 1. Consider the Hilbert space \(\mathcal{H}:= \mathbb{C}^2\) of qubits, and a von Neumann algebra \(\mathcal{M}\) of diagonal matrices. Let \(u = \frac{1}{\sqrt{2}}\begin{pmatrix} 1&1 \end{pmatrix}\), and \(\rho = \rho_u\). There are only four projectors in \(\mathcal{M}\), and only \({\mathbf{I}}\) of them all lies in \(\mathcal{F}_\rho\). Take \[{\mathbf{A}}= \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}.\] Then \(\Braket{\rho, {\mathbf{A}}} = \frac{1}{2}\), and \[{\mathbf{I}}{\mathbf{A}}{\mathbf{I}}= {\mathbf{A}}\neq \Braket{\rho, {\mathbf{A}}}{\mathbf{I}}.\] Hence, by Lemma 2, the vector state \(\rho_u\) is not pure.
In light of the previous example, the following proposition does not seem so trivial.
Proposition 7. Let \(u\in S^1(\mathcal{H})\). Let \(\rho = \rho_u\in\Sigma(\mathcal{M})\) be a vector state, and \({\mathbf{P}}_u \in \mathcal{M}\). Then \(\rho_u\) is pure if and only if the projector \({\mathbf{P}}_u\) on \(u\) excises \(\rho_u\) for any \({\mathbf{A}}\in\mathcal{M}\).
Proof. The <<if>> part is by Lemma 2, so we need to prove the <<only if>> part. It is clear that \({\mathbf{P}}_u \in \mathcal{F}_{\rho_u}\). Let \({\mathbf{A}}\in\mathcal{M}\). So it is sufficient to prove that for any \(v \in \mathcal{H}\), \[{\mathbf{P}}_u{\mathbf{A}}{\mathbf{P}}_u v = \Braket{\rho, {\mathbf{A}}}{\mathbf{P}}_u v.\] We have \[{\mathbf{P}}_u v = (u,v)u.\] Hence \[{\mathbf{P}}_u{\mathbf{A}}{\mathbf{P}}_u v = (u,v) (u, {\mathbf{A}}u) u.\] The equality \(\Braket{\rho_u, {\mathbf{A}}} = (u, {\mathbf{A}}u)\) implies the result. ◻
We will need more technical criteria for purity.
Lemma 3 (Approximative Excision). Let \(\rho\in\Sigma(\mathcal{M})\) be a \(\sigma\)-additive state on a von Neumann algebra \(\mathcal{M}\). Then \(\rho\) is pure if and only if for any \({\mathbf{A}}\in\mathcal{M}\) there exists a countable family of projectors \({\mathbf{P}}_n \in \mathcal{F}_\rho\) such that \[\lim_{n\to\infty} \left\lVert{\mathbf{P}}_n({\mathbf{A}}- \Braket{\rho, {\mathbf{A}}}{\mathbf{I}}){\mathbf{P}}_n\right\rVert = 0.\]
Proof. The <<only if>> part holds by Lemma 2 (Excision Criterion) since \(\rho\) is \(\sigma\)-additive, and we may take \({\mathbf{P}}_n\) being the same projector excising \(\rho\) for \({\mathbf{A}}\). So we need to prove the <<if>> part.
Fix any \({\mathbf{A}}\in\mathcal{M}\). Denote \[{\mathbf{P}}:= \bigwedge_n {\mathbf{P}}_n.\] Since \(\rho\) is \(\sigma\)-additive, by [12] \[\Braket{\rho, {\mathbf{P}}^\perp} = \Braket{\rho, \bigvee_n{\mathbf{P}}_n^\perp} = 0,\] and so \({\mathbf{P}}\in \mathcal{F}_\rho\). Thus, by Lemma 2 (Excision Criterion), it is sufficient to prove that \[{\mathbf{P}}{\mathbf{A}}{\mathbf{P}}= \Braket{\rho, {\mathbf{A}}}{\mathbf{P}}.\] Since \({\mathbf{P}}= {\mathbf{P}}_n{\mathbf{P}}= {\mathbf{P}}{\mathbf{P}}_n\), we have \[\left\lVert{\mathbf{P}}({\mathbf{A}}- \Braket{\rho, {\mathbf{A}}}{\mathbf{I}}){\mathbf{P}}\right\rVert \leqslant\left\lVert{\mathbf{P}}_n({\mathbf{A}}- \Braket{\rho, {\mathbf{A}}}{\mathbf{I}}){\mathbf{P}}_n\right\rVert \xrightarrow[n\to\infty]{} 0,\] which concludes the proof. ◻
Now recall that there is a bijection between all two-valued measures \(2^\kappa\to\{0,1\}\) and ultrafilters on \(\kappa\). This means that to each two-valued measure \(\mu\colon2^\kappa\to\{0,1\}\) corresponds an ultrafilter \(\mathcal{U}_\mu\) of sets of measure \(1\). It is clear that
\(\mu\) is \(\sigma\)-additive if and only if \(\mathcal{U}_\mu\) is \(\sigma\)-complete (which means that \(\mathcal{U}_\mu\) is closed under countable intersections), and
\(\mu\) vanishes on singletons if and only if \(\mathcal{U}_\mu\) is non-principal.
There exists an even stronger connection between such measures and ultrafilters. The following result is probably well known, but we do not know the reference for it.
Lemma 4. Let \(\mu\colon2^\kappa\to\{0,1\}\) be a two-valued measure. Then for any bounded function \(f\colon\kappa\to\mathbb{C}\), the integral over \(\mu\) and the limit along \(\mathcal{U}_\mu\) coincide, meaning \[\lim_{j\to \mathcal{U}_\mu} f(j) = \int f(j)\,d\mu(j).\]
Proof. It is sufficient to prove the case \(f\colon\kappa\to\mathbb{R}\). Since \(f\) is bounded, there exists a unique \[L := \lim_{j\to \mathcal{U}_\mu} f(j) \in \mathbb{R}.\] By definition of \(L\), for any \(\varepsilon>0\) holds \[A_\varepsilon := \{j\in\kappa:L-\varepsilon < f(j) < L+\varepsilon\} \in \mathcal{U}_\mu.\] The latter means that \(\mu(A_\varepsilon)=1\). So, \[L-\varepsilon<\int_{A_\varepsilon} f(j)\,d\mu(j) < L+\varepsilon.\] The proof is concluded by the fact that \[\int f(j)\,d\mu(j) = \int_{A_\varepsilon} f(j)\,d\mu(j) + \int_{\kappa\setminus A_\varepsilon} f(j)\,d\mu(j) = \int_{A_\varepsilon} f(j)\,d\mu(j)\] and the arbitrary choice of \(\varepsilon\). ◻
Theorem 8. Let \(\mu\colon2^\kappa\to\{0,1\}\) be a \(\sigma\)-additive two-valued measure. Then \({\mathbf{\Phi}}_\mu\) preserves the purity of pure vector states constructed on basis vectors \(e_i\).
Proof. It is sufficient to prove the theorem for a pure vector state \(\rho = \rho_{e_0}\), since for others they differ only by shift. Take an ultrafilter \(\mathcal{U}_\mu\). Then, by Lemma 4, \[{\mathbf{\Phi}}_\mu \rho = \lim_{j\to\mathcal{U}_\mu} {\mathbf{S}}_j\rho_{e_0}{\mathbf{S}}_j^* = \lim_{j\to\mathcal{U}_\mu} \rho_{e_j},\] where the ultralimit is in the ultraweak sense. By [12], it is sufficient to prove that \({\mathbf{\Phi}}_\mu \rho\) is pure on the diagonal subalgebra. Fix any diagonal \({\mathbf{A}}\in\mathcal{B}(\mathcal{H})\) and \(\varepsilon>0\). In order to apply Lemma 3 (Approximative Excision), we need to construct the excision projector. Let \[{\mathbf{A}}= \sum_{j\in\kappa} a_j{\mathbf{P}}_{e_j}.\] By definition of ultralimit, \[X_\varepsilon := \left\{j\in \kappa : \left\lvert a_j - \Braket{{\mathbf{\Phi}}_\mu\rho,{\mathbf{A}}}\right\rvert < \varepsilon\right\} \in \mathcal{U}_\mu.\] Thus, we are able to define the projector \[{\mathbf{P}}_\varepsilon := \sum_{j \in X_\varepsilon} {\mathbf{P}}_{e_j}.\] Since \(X_\varepsilon \in \mathcal{U}_\mu\), \[\Braket{{\mathbf{\Phi}}_\mu\rho, {\mathbf{P}}_\varepsilon} = \lim_{j\to\mathcal{U}_\mu} (e_j, {\mathbf{P}}_\varepsilon e_j) = \lim_{X_\varepsilon \ni j\to\mathcal{U}_\mu} (e_j, {\mathbf{P}}_\varepsilon e_j) = \lim_{X_\varepsilon \ni j\to\mathcal{U}_\mu} 1 = 1,\] and so \({\mathbf{P}}_\varepsilon \in \mathcal{F}_{{\mathbf{\Phi}}_\mu \rho}\).
Denote \[{\mathbf{D}}:= {\mathbf{A}}- \Braket{{\mathbf{\Phi}}_\mu\rho, {\mathbf{A}}}{\mathbf{I}}= \sum_{i\in\kappa} d_i{\mathbf{P}}_{e_i},\] where \[d_i := a_i-\Braket{{\mathbf{\Phi}}_\mu\rho, {\mathbf{A}}}.\] It is clear that \[{\mathbf{P}}_\varepsilon{\mathbf{D}}{\mathbf{P}}_\varepsilon = \sum_{i\in\kappa}\sum_{j\in Y_\varepsilon}\sum_{k\in X_\varepsilon} d_i{\mathbf{P}}_{e_j}{\mathbf{P}}_{e_i}{\mathbf{P}}_{e_k} = \sum_{j\in X_\varepsilon} d_j{\mathbf{P}}_{e_j}.\] Since \(\left\lvert d_j\right\rvert< \varepsilon\) for \(j\in X_\varepsilon\), so is \(\left\lVert{\mathbf{P}}_\varepsilon{\mathbf{D}}{\mathbf{P}}_\varepsilon\right\rVert\leqslant\varepsilon\). Thus, Lemma 3 (Approximative Excision) concludes the proof. ◻
Example 2. Theorem 8 does not have to hold for arbitrary pure vector state. Indeed, let \(\kappa=\mathbb{R}\), and take \(u = \frac{e_0 + e_1}{\sqrt{2}}\). Let \(+\) be the usual summation. For an ultrafilter \(\mathcal{U}\) on \(\mathbb{R}\), define \[\rho_\mathcal{U}:= \lim_{j\to\mathcal{U}_\mu} \rho_{e_j}.\] For any diagonal operator \({\mathbf{D}}\in\mathcal{B}(\ell_2(\mathbb{R}))\), \[\Braket{\rho_u, {\mathbf{D}}} = \frac{1}{2}(e_0,{\mathbf{D}}e_0) + \frac{1}{2}(e_1,{\mathbf{D}}e_1).\] Then \[{\mathbf{\Phi}}_\mu\rho_u = \frac{1}{2}\rho_{\mathcal{U}_\mu} + \frac{1}{2}\rho_{\mathcal{U}_\mu - 1}.\] It remains to notice that \(\mathcal{U}_\mu\) is not translation invariant: indeed, for example, if \[X := \bigsqcup_{n=-\infty}^\infty [\,2n, 2n+2)\] lies in \(\mathcal{U}_\mu\), then \(X+1 = \mathbb{R}\setminus X\) does not, and vise versa. So, taking \({\mathbf{D}}\) to be the projector on \(X\), we see that \[\Braket{\rho_{\mathcal{U}_\mu},{\mathbf{D}}} = 1 \quad\text{and}\quad \Braket{\rho_{\mathcal{U}_\mu-1},{\mathbf{D}}} = 0,\] or vise versa. So, \(\rho_{\mathcal{U}_\mu}\neq\rho_{\mathcal{U}_\mu-1}\), and thus \({\mathbf{\Phi}}_\mu\rho_u\) is not pure.
In the previous sections, our focus was on the group of translations \(\{{\mathbf{S}}_j\}_{j\in \kappa}\). Now our goal is to analyze the wider class of groups of unitary operators. So, let \(\mathcal{U}(\mathcal{H})\) be the group of unitary operators on a Hilbert space \(\mathcal{H}\). Let \((G,\cdot)\) be some other group; we will omit the \(\cdot\) operation. Let \(\mu\colon 2^{G}\to[\,0,1\,]\) be a finitely-additive measure on this group. Let \(\pi\colon2^G\to\mathcal{U}(\mathcal{H})\subset GL(\mathcal{H})\) be a representation of this group; below we presume that \(\pi\) is fixed in some way. Define the channel \({\mathbf{Q}}_\mu\colon \Sigma(\mathcal{B}(\mathcal{H}))\to\Sigma(\mathcal{B}(\mathcal{H}))\) by the Pettis integral \[{\mathbf{Q}}_\mu \rho := \int \pi(g)\rho\pi(g)^*\,d\mu(g).\] The channel \({\mathbf{\Phi}}_\mu\) is a particular case of the channel \({\mathbf{Q}}_\mu\) for a measure \(\mu\) concentrated on the family of shift operators. In [6], it was shown that if \(\mathcal{H}\) is separable and \(\mu\) is \(\sigma\)-additive, then \({\mathbf{Q}}_\mu\) preserves normality; see also [6] for the channel constructed by the representation over a topological group. Note that if \(\kappa\) is Ulam (real-valued) measurable, then so is \(\mathcal{U}(\ell_2(\kappa))\), since one can induce the measure from a subset to the whole set.
Theorem 9. Let \(\mathcal{H}\) be a Hilbert space. Let \(\mu\colon 2^G\to[\,0,1\,]\) be a \(\sigma\)-additive measure on a group \(G\). Then the channel \({\mathbf{Q}}_\mu\) preserves \(\sigma\)-additivity.
Proof. The proof is analogous to the proof of Theorem 2. ◻
Example 3. Here we show that the analogue of Theorem 3 may not hold for \({\mathbf{Q}}_\mu\) for some representation \(\pi\). Indeed, let \(\pi(g):={\mathbf{I}}\). Then \({\mathbf{Q}}_\mu \rho = \rho\) regardless of \(\mu\), and there is no singularization.
As shown in Example 2, the channel \({\mathbf{Q}}_\mu\) does not necessarily preserve purity. However, this quantum channel is interesting in another way.
The measure \(\mu\) on the group \(G\) is said to be left-invariant if \(\mu(E) = \mu(g E)\) for any \(g\in G\) and \(E\subset G\). See for example [14] for the introduction to Haar measures. In the case of discrete groups, the left-invariant measure is the counting measure, which is obviously \(\sigma\)-additive; this is the case when \({\mathbf{Q}}_\mu\) is just given by the Krauss representation.
We say that a state \(\rho\in\mathcal{B}(\mathcal{H})\) is a fixed point of any unitary evolution if \(\rho={\mathbf{U}}\rho{\mathbf{U}}^*\) for any \({\mathbf{U}}\in\mathcal{U}(\mathcal{H})\).
Theorem 10. Let \(\mathcal{H}\) be an infinite-dimensional Hilbert space. Let \(\mu\colon 2^G\to[\,0,1\,]\) be left-invariant. Then \({\mathbf{Q}}_\mu^2={\mathbf{Q}}_\mu\).
Proof. The theorem follows since for any \(\rho\in\Sigma(\mathcal{B}(\mathcal{H}))\), \({\mathbf{A}}\in\mathcal{B}(\mathcal{H})\), and \(g\in G\), \[\begin{gather} \Braket{\pi(g){\mathbf{Q}}_\mu\rho\pi(g)^*, {\mathbf{A}}} = \int\Braket{\rho, \pi(h)^*\pi(g)^*{\mathbf{A}}\pi(g)\pi(h) }\,d\mu(h) = \\ = \int\Braket{\rho, \pi(t)^*{\mathbf{A}}\pi(t)}\,d\mu(g^{-1}t) = \int\Braket{\rho, \pi(t)^*{\mathbf{A}}\pi(t)}\,d\mu(t) = \Braket{{\mathbf{Q}}_\mu\rho, {\mathbf{A}}}. \end{gather}\] ◻
Thus, unlike the results in [9], where a semigroup of singularizing quantum channels was obtained for \(\mu_s*\mu_t=\mu_{s+t}\), here the quantum channels \({\mathbf{Q}}_{\mu_t}\) will not form a semigroup.
We now turn our attention to the dynamics of this quantum channel.
Recall that for two finitely-additive measures \(\mu,\nu\colon2^G\to[\,0,1\,]\), their convolution \(\mu*\nu\colon2^G\to[\,0,1\,]\) is defined by \[\mu*\nu(E) := \int \nu\left(g^{-1}E\right)\,d\mu(g).\] In other words, \[\int f(t)\,d\mu*\nu(t) = \int d\mu(g) \int d\nu(h)\,f(gh)\] for any bounded \(f\colon G\to\mathbb{C}\).
Proposition 11. For any \(\mu\colon 2^G\to [\,0,1\,]\), \[{\mathbf{Q}}_\mu^n = {\mathbf{Q}}_{\mu^{*n}}.\]
Proof. This is since for any state \(\rho\), \[{\mathbf{Q}}_\mu^n\rho = \int d\mu(g_1)\int \ldots \int d\mu(g_n)\,\pi(g_1\ldots g_n)\rho\pi(g_1\ldots g_n)^* = \int \pi(g)\rho\pi(g)^*\,d\mu^{*n}(g).\] ◻
For a measure \(\mu\) on \(G\) and a some \(h\in G\), define the left shifted measure \(L_g\mu\colon 2^G\to [\,0,1\,]\) by \[L_h\mu(E) := \mu(h E).\]
Theorem 12. Let \(\mathcal{H}\) be a Hilbert space. Let \(\mu\colon 2^{\mathcal{U}(\mathcal{H})}\to [\,0,1\,]\) be a finitely-additive measure. Suppose that for any state \(\rho\in\Sigma(\mathcal{B}(\mathcal{H}))\), iterations \({\mathbf{Q}}_\mu^n\rho\) of quantum channels \(*\)-weakly converge to a state (depending on \(\rho\)), which is a fixed point of any unitary evolution. Then \[\tau\lim_{n\to\infty}\left(\mu^{*n} - L_h(\mu^{*n})\right) = 0\] for any \(h\in G\), where \(\tau\) is the weak topology induced by functions \[g \mapsto \Braket{\rho, \pi(g)^*{\mathbf{A}}\pi(g)}, \qquad g\in G, \qquad {\mathbf{A}}\in\mathcal{B}(\mathcal{H}), \quad \rho\in\Sigma(\mathcal{B}(\mathcal{H})).\]
Proof. Take any function \[f\colon g \mapsto \Braket{\rho, \pi(g)^*{\mathbf{A}}\pi(g)},\] given by some \({\mathbf{A}}\in\mathcal{B}(\mathcal{H})\) and \(\rho\in\Sigma(\mathcal{B}(\mathcal{H}))\). Let \(\rho_\infty\) be a \(*\)-weak limit of \({\mathbf{Q}}_\mu^n\rho\). Take any \(h\in G\). Then, by Proposition 11, \[\int f\,d\mu^{*n} = \Braket{{\mathbf{Q}}_\mu^n\rho, {\mathbf{A}}} \xrightarrow[n\to\infty]{} \Braket{\rho_\infty, {\mathbf{A}}}\] and \[\int f\,dL_h\mu^{*n} = \Braket{{\mathbf{Q}}_\mu^n\rho, \pi(h)^*{\mathbf{A}}\pi(h)} \xrightarrow[n\to\infty]{} \Braket{\rho_\infty, \pi(h)^*{\mathbf{A}}\pi(h)}.\] The equality \(\Braket{\rho_\infty, {\mathbf{A}}} = \Braket{\rho_\infty, \pi(h)^*{\mathbf{A}}\pi(h)}\), which holds since \(\rho_\infty\) is a fixed point of any unitary evolution, concludes the proof. ◻
As Theorem 12 shows, if \(\mu=\mu*\mu\) is not left-invariant, then there can be no convergence to states that are fixed points of any unitary evolution.
For the final touch, we consider the dynamics of quantum channels. The following results on convergence may not hold for iterations of quantum channels, so we prove them for the Cesáro averages. We will rely heavily on the following result of Yosida and Kakutani [15].
Theorem 13 (Yosida–Kakutani). Let \(T\colon B\to B\) be a bounded linear operator on a Banach space \(B\). Suppose that there exists \(C>0\) such that for any \(n\geqslant 1\) we have \(\left\lVert T^n\right\rVert \leqslant C\). Suppose that for any \(x\in B\), the sequence of the Cesáro averages \[x_n := \frac{1}{n}\left(T+\ldots + T^n\right)x\] contains a subsequence that converges weakly to some \(\overline{x}\in B\). Then \(x_n\) converges strongly to \(\overline{x}\), and the operator \[\overline{T}\colon x\mapsto \overline{x}\] is a bounded linear operator \(B\to B\) such that \[\overline{T}T = T\overline{T} = \overline{T}^2 = \overline{T}.\]
Theorem 14. Let \(\mathcal{M}\) be a von Neumann algebra on a Hilbert space \(\mathcal{H}\). Let \({\mathbf{\Phi}}\colon\Sigma_n(\mathcal{M})\to\Sigma_n(\mathcal{M})\) be a quantum channel. Then the Cesáro averages \[\frac{1}{n}\left({\mathbf{\Phi}}+ \ldots + {\mathbf{\Phi}}^n\right)\] converge in strong operator topology to a quantum channel \(\overline{{\mathbf{\Phi}}}\) such that \[\overline{{\mathbf{\Phi}}}{\mathbf{\Phi}}= {\mathbf{\Phi}}\overline{{\mathbf{\Phi}}} = \overline{{\mathbf{\Phi}}}^2 = \overline{{\mathbf{\Phi}}}.\]
Proof. Firstly, note that we may extend \({\mathbf{\Phi}}\) to the whole predual \(\mathcal{M}_*\), and consider it as an operator \(\mathcal{M}_*\to\mathcal{M}_*\).
Fix any \(\rho\in\Sigma_n(\mathcal{M})\). Since \(\left\lVert{\mathbf{\Phi}}\right\rVert\leqslant 1\), the sequence of \[\rho_n := \frac{1}{n}\left({\mathbf{\Phi}}+ \ldots + {\mathbf{\Phi}}^n\right)\rho \in \Sigma_n(\mathcal{M})\] is well defined. Since \(\Sigma_n(\mathcal{M})\) is weakly compact, the sequence \(\{\rho_n\}\) contains a subsequence that converges weakly to some \(\overline{\rho}\in \Sigma(\mathcal{M})\). Since \(\left\lVert{\mathbf{\Phi}}\right\rVert \leqslant 1\), it remains to apply Theorem 13 for \(B=\mathcal{M}_*\), \(T={\mathbf{\Phi}}\), \(x=\rho\), and \(x_n=\rho_n\). This shows that \(\rho_n\) converges to \(\overline{\rho} =: \overline{{\mathbf{\Phi}}}\rho\) strongly. ◻
By Theorem 14 and [6], for a separable Hilbert space \(\mathcal{H}\) and a \(\sigma\)-additive measure \(\mu\) on \(2^{\mathcal{U}(\mathcal{H})}\) itself (taking the representation \(\pi\colon{\mathbf{U}}\mapsto{\mathbf{U}}\)), the Cesáro averages of \({\mathbf{Q}}_\mu\colon\Sigma_n(\mathcal{B}(\mathcal{H}))\to\Sigma_n(\mathcal{B}(\mathcal{H}))\) converge in SOT to some quantum channel \(\overline{{\mathbf{\Phi}}}\colon\Sigma_n(\mathcal{B}(\mathcal{H}))\to\Sigma_n(\mathcal{B}(\mathcal{H}))\). It is interesting to find out whether \(\overline{{\mathbf{\Phi}}}\) itself can be given as the Pettis integral over some measure, which probably should be left-invariant.
In this paper, we have investigated the structural and dynamical properties of quantum channels \({\mathbf{\Phi}}_\mu\) and \({\mathbf{Q}}_\mu\) induced by averaging procedures via the Pettis integral. By establishing a bridge between operator algebra theory and the foundational properties of Ulam real-valued measurable cardinals, we demonstrated how the analytical behavior of the underlying measure directly dictates the topological nature of the resulting quantum states. Specifically, we provided a complete characterization of channels that simultaneously preserve \(\sigma\)-additivity while enforcing a strict singularization of any incoming state. Furthermore, we extended these constructions to general groups of unitary operators, characterizing their fixed points and analyzing the asymptotic behavior of their iterations.
The framework developed herein highlights the subtle geometry of singular \(\sigma\)-additive states, which naturally emerge at the intersection of non-separable Hilbert spaces and non-trivial set-theoretic assumptions. While the ergodic properties of normal states under such channels can be successfully resolved via classical mean ergodic theorems, the singularized dynamics pose significantly deeper challenges. As it was mentioned for a separable Hilbert space \(\mathcal{H}\) and a \(\sigma\)-additive measure \(\mu\colon2^{\mathcal{U}(\mathcal{H})}\to[\,0,1\,]\), the Cesáro averages of \({\mathbf{Q}}_\mu\colon\Sigma_n(\mathcal{B}(\mathcal{H}))\to\Sigma_n(\mathcal{B}(\mathcal{H}))\) converge in SOT to some quantum channel \(\Sigma_n(\mathcal{B}(\mathcal{H}))\to\Sigma_n(\mathcal{B}(\mathcal{H}))\). It is natural to examine the analogous results for singularizing quantum channels. However, it is much harder to achieve such results, since we cannot apply the Yosida–Kakutani result (Theorem 13). There are analogous results for quasi-compact operators; see, for example, [16]. Unfortunately, we have been unable to establish the quasi-compactness of singularizing quantum channels \({\mathbf{Q}}_\mu\) in any particular noteworthy cases.
S. V. Dzhenzher: sdjenjer@yandex.ru. orcid: 0009-0008-3513-4312
Moscow Institute of Physics and Technology 141701, Institutskii per. 9, Dolgoprudny, Russia↩︎