arXiv:2504.10033 · Lobachevskii Journal of Mathematics

The Law of Large Numbers for Discrete Generalized Quantum Channels

Sviatoslav V. Dzhenzher, Vsevolod Zhanovich Sakbaev

Published Jun 1, 2025• 1 citation

Abstract

We consider random linear operators $$\Omega\to\mathcal{L}(\mathcal{T}_{p},\mathcal{T}_{p})$$ 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 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 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 $$1\leqslant p\leqslant 2$$ .

DOI: 10.1134/s1995080225608112

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arXiv:2504.10033 · Lobachevskii Journal of Mathematics

The Law of Large Numbers for Discrete Generalized Quantum Channels

Sviatoslav V. Dzhenzher, Vsevolod Zhanovich Sakbaev

Published Jun 1, 2025• 1 citation

arXiv HTML PDF

Abstract

We consider random linear operators

\Omega\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 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 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

DOI: 10.1134/s1995080225608112

Related papers

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Максим Евгеньевич Широков

We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra $\mathfrak{B}(\mathcal{H})$ of bounded operators on a separable Hilbert space $\mathcal{H}$ ) to the space of all operators on $\mathcal{H}$ relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation $\Phi$ can be uniquely extended to a bounded linear operator on the Banach space of all $\sqrt{G}$ -bounded operators on $\mathcal{H}$ provided that the operation $\Phi$ is $G$ -limited: the predual operation $\Phi_{*}$ maps the set of positive trace class operators $\rho$ with finite value of $\textrm{Tr}\rho G$ into itself. Assuming that $G$ has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the $G$ -limited property. Applications to the theory of Bosonic Gaussian channels are considered.We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra $$\mathfrak{B}(\mathcal{H})$$ of bounded operators on a separable Hilbert space $$\mathcal{H}$$ ) to the space of all operators on $$\mathcal{H}$$ relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation $$\Phi$$ can be uniquely extended to a bounded linear operator on the Banach space of all $$\sqrt{G}$$ -bounded operators on $$\mathcal{H}$$ provided that the operation $$\Phi$$ is $$G$$ -limited: the predual operation $$\Phi_{*}$$ maps the set of positive trace class operators $$\rho$$ with finite value of $$\textrm{Tr}\rho G$$ into itself. Assuming that $$G$$ has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the $$G$$ -limited property. Applications to the theory of Bosonic Gaussian channels are considered.

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We consider random linear operators

\Omega\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

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

We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra

\mathfrak{B}(\mathcal{H})

of bounded operators on a separable Hilbert space

\mathcal{H}

) to the space of all operators on

\mathcal{H}

relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation

\Phi

can be uniquely extended to a bounded linear operator on the Banach space of all

\sqrt{G}

-bounded operators on

\mathcal{H}

provided that the operation

\Phi

G

-limited: the predual operation

\Phi_{*}

maps the set of positive trace class operators

\rho

with finite value of

\textrm{Tr}\rho G

into itself. Assuming that

G

has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the

G

-limited property. Applications to the theory of Bosonic Gaussian channels are considered.

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.

Given a positive operator-valued measure

ν

acting on the Borel sets of a locally compact Hausdorff space

X

, with outcomes in the algebra

\mathcal B(\mathcal H)

of all bounded operators on a (possibly infinite-dimensional) Hilbert space

\mathcal H

, one can consider

ν

-integrable functions

X\rightarrow \mathcal B(\mathcal H)

that are positive quantum random variables. We define a seminorm on the span of such functions which in the quotient leads to a Banach space. We consider bistochastic operators acting on this space and majorization of quantum random variables is then defined with respect to these operators. As in classical majorization theory, we relate majorization in this context to an inequality involving all possible convex functions of a certain type. Unlike the classical setting, continuity and convergence issues arise throughout the work.