Searcharxiv⌕ Search

arXiv subjects

S. V. Dzhenzher

Publications and source records attributed to S. V. Dzhenzher.

7 recordsLinked to original sources
arXiv2026

Quantum channels preserving sigma-additivity on diagonal von Neumann algebras and Ulam measurable cardinals

This paper investigates the interplay between the properties of quantum states on the Hilbert space \(\ell_2(κ)\) and the set-theoretic nature of the cardinal $κ$. We focus on the existence of singular $σ$-additive states~--- functionals whose induced measures are $σ$-additive yet vanish on singletons. While the existence of such states is known to be equivalent to the Ulam measurability of $κ$, their structural and dynamical properties remain largely unexplored. We prove that any $σ$-additive state on the diagonal algebra is representable as a Pettis integral over a singular $σ$-additive measure, extending the classical representation theory to the non-normal sector. Furthermore, we construct a class of quantum channels using $σ$-complete ultrafilters that map normal states to singular $σ$-additive states, effectively < > information into the singular part of the state space.

quant-ph↗
arXiv2026

The Strong Law of Large Numbers for random semigroups with unbounded generators on uniformly smooth Banach spaces

We consider random linear unbounded operators on a Banach space $\mathcal{X}$. For example, such random operators may be random quantum channels. The Law of Large Numbers is known when $\mathcal{X}$ is a Hilbert space, in the form of the usual Law of Large Numbers for random operators, and in some other particular cases. Instead of the sum of i.i.d. variables, there may be considered the composition of random semigroups $e^{A_i t/n}$. We obtain the Strong Law of Large Numbers in Strong Operator Topology for random semigroups of unbounded linear operators on a uniformly smooth Banach space.

math.FA↗
arXiv2026

Banach and counting measures, and dynamics of singular quantum states generated by averaging of operator random walks

In this paper the random channels and their compositions in the space of quantum states are studied. For compositions of i.i.d. random unitary channels, the limit behaviour of probability distributions is described. The sufficient condition for convergence in probability is obtained. The generalized convergence in distribution w.r.t. weak operator topology is obtained. The analysis of transmission of pure and normal states to the set of singular states is done. The dynamics of quantum states is described in terms of the evolution of the values of quadratic forms of operators from the algebra that implements the representation of canonical commutation relations.

quant-ph↗
arXiv2025

The Strong Law of Large Numbers for random semigroups on uniformly smooth Banach spaces

We consider random linear continuous operators $Ω\to \mathcal{L}(\mathcal{X}, \mathcal{X})$ on a Banach space $\mathcal{X}$. For example, such random operators may be random quantum channels. The Law of Large Numbers is known when $\mathcal{X}$ is a Hilbert space, in the form of the usual Law of Large Numbers for random operators, and in some other particular cases. Instead of the sum of i.i.d. variables, there may be considered the composition of random semigroups $e^{A_it/n}$. We obtain the Strong Law of Large Numbers in Strong Operator Topology for random semigroups of bounded linear operators on a uniformly smooth Banach space. We also develop another approach giving the SLLN in Weak Operator Topology for all Banach spaces.

math.FA↗
arXiv2025

The Central Limit Theorem for random exponents on a Hilbert space in the Weak Operator Topology

We consider random linear continuous operators $Ω\to \mathcal{L}(\mathcal{H}, \mathcal{H})$ on a Hilbert space $\mathcal{H}$. For example, such random operators may be random quantum channels. The Central Limit Theorem is known for the sums of i.i.d. random operators. Instead of the sum, there may be considered the composition of random exponents $e^{A_i/n}$. We obtain the Central Limit Theorem in the Weak Operator Topology for centralized and normalized random exponents of i.i.d. linear continuous operators on a Hilbert space.

math.FA↗
arXiv2025

The law of large numbers for discrete generalized quantum channels

We consider random linear operators $Ω\to \mathcal{L}(\mathcal{T}_p, \mathcal{T}_p)$ acting in a $p$-th Schatten class $\mathcal{T}_p$ in a separable Hilbert space $\mathcal{H}$ for some $1 \leqslant p < \infty$. Such a superoperator is called a pre-channel since it is an extension of a quantum channel to a wider class of operators without requirements of trace-preserving and positivity. Instead of the sum of i.i.d. variables there may be considered the composition of random semigroups $e^{A_i t/n}$ in the Banach space $\mathcal{T}_p$. The law of large numbers is known in the case $p=2$ in the form of the usual law of large numbers for random operators in a Hilbert space. We obtain the law of large numbers for the case $1\leqslant p \leqslant 2$.

math.FA↗