Inequalities pp 191-200 | Cite as

A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy

  • Eric A. Carlen
  • Elliott H. Lieb


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.


Positive Operator Operator Analog Selfadjoint Operator Partial Trace National Science Foundation Grant 
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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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