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A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity

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Abstract

We revisit and prove some convexity inequalities for trace functions conjectured in this paper’s antecedent. The main functional considered is

$$ \Phi_{p,q} (A_1,\, A_2, \ldots, A_m) = \left({\rm Tr}\left[\left( \, {\sum\limits_{j=1}^m A_j^p } \, \right) ^{q/p} \right] \right)^{1/q} $$

for m positive definite operators A j . In our earlier paper, we only considered the case q = 1 and proved the concavity of Φ p,1 for 0 < p ≤ 1 and the convexity for p = 2. We conjectured the convexity of Φ p,1 for 1 < p < 2. Here we not only settle the unresolved case of joint convexity for 1 ≤ p ≤ 2, we are also able to include the parameter q ≥ 1 and still retain the convexity. Among other things this leads to a definition of an L q(L p) norm for operators when 1 ≤ p ≤ 2 and a Minkowski inequality for operators on a tensor product of three Hilbert spaces – which leads to another proof of strong subadditivity of entropy. We also prove convexity/concavity properties of some other, related functionals.

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Authors and Affiliations

  1. Department of Mathematics, Hill Center, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ, 08854, USA

    Eric A. Carlen

  2. Department of Mathematics, Fine Hall, Princeton, NJ, 08544, USA

    Elliott H. Lieb

  3. Departments of Mathematics and Physics, Jadwin Hall, Princeton University, P. O. Box 708, Princeton, NJ, 08544, USA

    Elliott H. Lieb

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  1. Eric A. Carlen
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  2. Elliott H. Lieb
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Correspondence to Eric A. Carlen.

Additional information

Eric A. Carlen’s work was partially supported by U.S. National Science Foundation grant DMS 06-00037.

Elliott H. Lieb’s work was partially supported by U.S. National Science Foundation grant PHY 06-52854.

© 2008 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

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Carlen, E.A., Lieb, E.H. A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity. Lett Math Phys 83, 107–126 (2008). https://doi.org/10.1007/s11005-008-0223-1

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  • DOI: https://doi.org/10.1007/s11005-008-0223-1

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