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Inequalities pp 191-200 | Cite as

A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy

  • Eric A. Carlen
  • Elliott H. Lieb

Abstract

We consider the following trace function on n-tuples of positive operators:
$${\Phi _P}({A_1},{A_2},...,{A_n}) = Tr({(\sum\limits_{j = 1}^n {A_j^P} )^{1/P}})$$
and prove that it is jointly concave for 0 < p ≤ 1 and convex for p = 2. We then derive from this a Minkowski type inequality for operators on a tensor product of three Hilbert spaces, and show how this implies the strong subadditivity of quantum mechanical entropy. For p > 2, Фp is neither convex nor concave. We conjecture that Фp is convex for 1 < p < 2, but our methods do not show this.

Keywords

Positive Operator Operator Analog Selfadjoint Operator Partial Trace National Science Foundation Grant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Eric A. Carlen
    • 1
  • Elliott H. Lieb
    • 2
  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA

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