aqora

Noncommutative analogues of the Cramér-Rao inequality in the quantum measurement theory

Alexander Semenovich Holevo

Lecture notes in mathematicsJan 1, 197610 citations

On trace inequalities and their applications to noncommutative communication theory

Kenjiro Yanagi, Shigeru Furuichi, Ken Kuriyama

arXiv:quant-ph/0403187Linear Algebra and its ApplicationsOct 21, 20044 citations

On trace inequalities and their applications to noncommutative communication theory

研二郎柳, Shigeru Furuichi, 憲栗山

Jan 15, 20050 citations

Quantum information inequalities

Liang Cai, Frank Hansen

Mar 10, 20080 citations

Multivariate trace inequalities, p-fidelity, and universal recovery beyond tracial settings

Marius Junge, Nicholas D. LaRacuente

Trace inequalities are general techniques with many applications in quantum information theory, often replacing the classical functional calculus in noncommutative settings. The physics of quantum field theory and holography, however, motivates entropy inequalities in type III von Neumann algebras that lack a semifinite trace. The Haagerup and Kosaki Lp spaces enable re-expressing trace inequalities in non-tracial von Neumann algebras. In particular, we show this for the generalized Araki–Lieb–Thirring and Golden–Thompson inequalities from the work of Sutter et al. [Commun. Math. Phys. 352(1), 37 (2017)]. Then, using the Haagerup approximation method, we prove a general von Neumann algebra version of universal recovery map corrections to the data processing inequality for relative entropy. We also show subharmonicity of a logarithmic p-fidelity of recovery. Furthermore, we prove that the non-decrease of relative entropy is equivalent to the existence of an L1-isometry implementing the channel on both input states.Trace inequalities are general techniques with many applications in quantum information theory, often replacing the classical functional calculus in noncommutative settings. The physics of quantum field theory and holography, however, motivates entropy inequalities in type III von Neumann algebras that lack a semifinite trace. The Haagerup and Kosaki Lp spaces enable re-expressing trace inequalities in non-tracial von Neumann algebras. In particular, we show this for the generalized Araki–Lieb–Thirring and Golden–Thompson inequalities from the work of Sutter et al. [Commun. Math. Phys. 352(1), 37 (2017)]. Then, using the Haagerup approximation method, we prove a general von Neumann algebra version of universal recovery map corrections to the data processing inequality for relative entropy. We also show subharmonicity of a logarithmic p-fidelity of recovery. Furthermore, we prove that the non-decrease of relative entropy is equivalent to the existence of an L1-isometry implementing the channel on both input states.

arXiv:2009.11866Journal of Mathematical PhysicsDec 1, 20229 citations

Convergent Relaxations of Polynomial Optimization Problems with Noncommuting Variables

S. Pironio, Miguel Navascués, A. Acín

We consider optimization problems with polynomial inequality constraints in noncommuting variables. These noncommuting variables are viewed as bounded operators on a Hilbert space whose dimension is not fixed and the associated polynomial inequalities as semidefinite positivity constraints. Such problems arise naturally in quantum theory and quantum information science. To solve them, we introduce a hierarchy of semidefinite programming relaxations which generates a monotone sequence of lower bounds that converges to the optimal solution. We also introduce a criterion to detect whether the global optimum is reached at a given relaxation step and show how to extract a global optimizer from the solution of the corresponding semidefinite programming problem.We consider optimization problems with polynomial inequality constraints in noncommuting variables. These noncommuting variables are viewed as bounded operators on a Hilbert space whose dimension is not fixed and the associated polynomial inequalities as semidefinite positivity constraints. Such problems arise naturally in quantum theory and quantum information science. To solve them, we introduce a hierarchy of semidefinite programming relaxations which generates a monotone sequence of lower bounds that converges to the optimal solution. We also introduce a criterion to detect whether the global optimum is reached at a given relaxation step and show how to extract a global optimizer from the solution of the corresponding semidefinite programming problem.

arXiv:0903.4368SIAM Journal on OptimizationJan 1, 2010194 citations

Noncommutative versions of inequalities in quantum information theory

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