aqora

Proof of the strong subadditivity of quantum-mechanical entropy

Élliott H. Lieb, Mary Beth Ruskai

Springer eBooksJan 1, 20024 citations

A simple proof of the strong subadditivity inequality

Michael A. Nielsen, D�nes Petz

Arguably the deepest fact known about the von~Neumann entropy, the strong subadditivity inequality is a potent hammer in the quantum information theorist's toolkit. This short tutorial describes a simple proof of strong subadditivity due to Petz [Rep. on Math. Phys. \textbf{23} (1), 57--65 (1986)]. It assumes only knowledge of elementary linear algebra and quantum mechanics.

arXiv:quant-ph/0408130Quantum Information and ComputationSep 1, 200565 citations

Stronger subadditivity of entropy

Élliott H. Lieb, Robert Seiringer

The strong subadditivity of entropy plays a key role in several areas of physics and mathematics. It states that the entropy $S[\ensuremath{\varrho}]=\ensuremath{-}\mathrm{Tr}(\ensuremath{\varrho}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varrho})$ of a density matrix ${\ensuremath{\varrho}}_{123}$ on the product of three Hilbert spaces satisfies $S[{\ensuremath{\varrho}}_{123}]\ensuremath{-}S[{\ensuremath{\varrho}}_{12}]\ensuremath{\leqslant}S[{\ensuremath{\varrho}}_{23}]\ensuremath{-}S[{\ensuremath{\varrho}}_{2}]$. We strengthen this to $S[{\ensuremath{\varrho}}_{123}]\ensuremath{-}S[{\ensuremath{\varrho}}_{12}]\ensuremath{\leqslant}{\ensuremath{\sum}}_{\ensuremath{\alpha}}{n}^{\ensuremath{\alpha}}(S[{\ensuremath{\varrho}}_{23}^{\ensuremath{\alpha}}]\ensuremath{-}S[{\ensuremath{\varrho}}_{2}^{\ensuremath{\alpha}}])$, where the ${n}^{\ensuremath{\alpha}}$ are weights and the ${\ensuremath{\varrho}}_{23}^{\ensuremath{\alpha}}$ are partitions of ${\ensuremath{\varrho}}_{23}$. Correspondingly, there is a strengthening of the theorem that the map $A\ensuremath{\mapsto}\mathrm{Tr}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[L+\mathrm{ln}\phantom{\rule{0.2em}{0ex}}A]$ is concave. As applications we prove some monotonicity and convexity properties of the Wehrl coherent state entropy and entropy inequalities for quantum gases.The strong subadditivity of entropy plays a key role in several areas of physics and mathematics. It states that the entropy $S[\ensuremath{\varrho}]=\ensuremath{-}\mathrm{Tr}(\ensuremath{\varrho}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varrho})$ of a density matrix ${\ensuremath{\varrho}}_{123}$ on the product of three Hilbert spaces satisfies $S[{\ensuremath{\varrho}}_{123}]\ensuremath{-}S[{\ensuremath{\varrho}}_{12}]\ensuremath{\leqslant}S[{\ensuremath{\varrho}}_{23}]\ensuremath{-}S[{\ensuremath{\varrho}}_{2}]$. We strengthen this to $S[{\ensuremath{\varrho}}_{123}]\ensuremath{-}S[{\ensuremath{\varrho}}_{12}]\ensuremath{\leqslant}{\ensuremath{\sum}}_{\ensuremath{\alpha}}{n}^{\ensuremath{\alpha}}(S[{\ensuremath{\varrho}}_{23}^{\ensuremath{\alpha}}]\ensuremath{-}S[{\ensuremath{\varrho}}_{2}^{\ensuremath{\alpha}}])$, where the ${n}^{\ensuremath{\alpha}}$ are weights and the ${\ensuremath{\varrho}}_{23}^{\ensuremath{\alpha}}$ are partitions of ${\ensuremath{\varrho}}_{23}$. Correspondingly, there is a strengthening of the theorem that the map $A\ensuremath{\mapsto}\mathrm{Tr}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[L+\mathrm{ln}\phantom{\rule{0.2em}{0ex}}A]$ is concave. As applications we prove some monotonicity and convexity properties of the Wehrl coherent state entropy and entropy inequalities for quantum gases.

arXiv:math-ph/0412009Physical Review AJun 23, 200527 citations

A simple proof of the strong subadditivity inequality

A NielsenMichael, PetzDénes

Arguably the deepest fact known about the von Neumann entropy, the strong subadditivity inequality is a potent hammer in the quantum information theorist's toolkit. This short tutorial describes a ...

Quantum Information and ComputationSep 1, 20051 citation

Operator extension of strong subadditivity of entropy

Isaac H. Kim

We prove an operator inequality that extends strong subadditivity of entropy: after taking a trace, the operator inequality becomes the strong subadditivity of entropy.

arXiv:1210.5190Journal of Mathematical PhysicsNov 30, 201228 citations

Another short and elementary proof of strong subadditivity of quantum entropy

Mary Beth Ruskai

arXiv:quant-ph/0604206Reports on Mathematical PhysicsAug 1, 200732 citations

Proof of the strong subadditivity of quantum-mechanical entropy

Abstract

Related papers

Proof of the strong subadditivity of quantum-mechanical entropy

Abstract

Related papers