Abstract <p>We consider random linear operators <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8343_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="103" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Omega\to\mathcal{L}(\mathcal{T}_{p},\mathcal{T}_{p})\)</EquationSource>  </InlineEquation> acting in a <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8343_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(p\)</EquationSource>  </InlineEquation>th Schatten class <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8343_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{T}_{p}\)</EquationSource>  </InlineEquation> in a separable Hilbert space <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8343_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{H}\)</EquationSource>  </InlineEquation> for some <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8343_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">\(1\leqslant p<\infty\)</EquationSource>  </InlineEquation>. 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 <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8343_Article_IEq6.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(e^{A_{i}t/n}\)</EquationSource>  </InlineEquation> in the Banach space <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8343_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{T}_{p}\)</EquationSource>  </InlineEquation>. The law of large numbers is known in the case <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8343_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(p=2\)</EquationSource>  </InlineEquation> 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 <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8343_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">\(1\leqslant p\leqslant 2\)</EquationSource>  </InlineEquation>.</p>

The Law of Large Numbers for Discrete Generalized Quantum Channels

期刊论文
发表日期：

2025年6月1日

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S. V. Dzhenzher,
V. Zh. Sakbaev

摘要

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 \(1\leqslant p\leqslant 2\) .