Quantum Dichotomies and Coherent Thermodynamics beyond First-Order Asymptotics

We address the problem of exact and approximate transformation of quantum dichotomies in the asymptotic regime, i.e., the existence of a quantum channel <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:mrow><a:mrow><a:mi mathvariant="script">E</a:mi></a:mrow></a:mrow></a:math> mapping <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><e:msubsup><e:mi>ρ</e:mi><e:mn>1</e:mn><e:mrow><e:mo>⊗</e:mo><e:mi>n</e:mi></e:mrow></e:msubsup></e:math> into <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><h:msubsup><h:mi>ρ</h:mi><h:mn>2</h:mn><h:mrow><h:mo>⊗</h:mo><h:msub><h:mi>R</h:mi><h:mi>n</h:mi></h:msub><h:mi>n</h:mi></h:mrow></h:msubsup></h:math> with an error <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><k:msub><k:mi>ϵ</k:mi><k:mi>n</k:mi></k:msub></k:math> (measured by trace distance) and <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><n:msubsup><n:mi>σ</n:mi><n:mn>1</n:mn><n:mrow><n:mo>⊗</n:mo><n:mi>n</n:mi></n:mrow></n:msubsup></n:math> into <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><q:msubsup><q:mi>σ</q:mi><q:mn>2</q:mn><q:mrow><q:mo>⊗</q:mo><q:msub><q:mi>R</q:mi><q:mi>n</q:mi></q:msub><q:mi>n</q:mi></q:mrow></q:msubsup></q:math> exactly, for a large number <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><t:mi>n</t:mi></t:math>. We derive second-order asymptotic expressions for the optimal transformation rate <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><w:msub><w:mi>R</w:mi><w:mi>n</w:mi></w:msub></w:math> in the small-, moderate-, and large-deviation error regimes, as well as the zero-error regime, for an arbitrary pair <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><z:mo stretchy="false">(</z:mo><z:msub><z:mi>ρ</z:mi><z:mn>1</z:mn></z:msub><z:mo>,</z:mo><z:msub><z:mi>σ</z:mi><z:mn>1</z:mn></z:msub><z:mo stretchy="false">)</z:mo></z:math> of initial states and a commuting pair <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><eb:mo stretchy="false">(</eb:mo><eb:msub><eb:mi>ρ</eb:mi><eb:mn>2</eb:mn></eb:msub><eb:mo>,</eb:mo><eb:msub><eb:mi>σ</eb:mi><eb:mn>2</eb:mn></eb:msub><eb:mo stretchy="false">)</eb:mo></eb:math> of final states. We also prove that for <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><jb:msub><jb:mi>σ</jb:mi><jb:mn>1</jb:mn></jb:msub></jb:math> and <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mb:msub><mb:mi>σ</mb:mi><mb:mn>2</mb:mn></mb:msub></mb:math> given by thermal Gibbs states, the derived optimal transformation rates in the first three regimes can be attained by thermal operations. This allows us, for the first time, to study the second-order asymptotics of thermodynamic state interconversion with fully general initial states that may have coherence between different energy eigenspaces. Thus, we discuss the optimal performance of thermodynamic protocols with coherent inputs and describe three novel resonance phenomena allowing one to significantly reduce transformation errors induced by finite-size effects. What is more, our result on quantum dichotomies can also be used to obtain, up to second-order asymptotic terms, optimal conversion rates between pure bipartite entangled states under local operations and classical communication. Published by the American Physical Society 2024