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Remarks on Kim's strong subadditivity matrix inequality: Extensions and equality conditions

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### Abstract

We describe recent work of Kim [J. Math. Phys.53, 122204 (2012)] to show that operator convex functions associated with quasi-entropies can be used to prove a large class of new matrix inequalities in the tri-partite and bi-partite setting by taking a judiciously chosen partial trace over all but one of the spaces. We give some additional examples in both settings. Furthermore, we observe that the equality conditions for all the new inequalities are essentially the same as those for strong subadditivity.

© 2013 AIP Publishing LLC

Received 03 December 2012
Accepted 05 September 2013
Published online 16 October 2013

Acknowledgments:
It is a pleasure to thank Jon Tyson for drawing my attention to Ref. 19 and for subsequent stimulating discussions. This work was partially supported by NSF Grant No. CFF-1018401 which is administered by Tufts University.

Article outline:

I. BACKGROUND
II. GENERAL THEORY
A. Some basics
B. Main result
C. Adjoint and conjugate forms
III. SPECIFIC INEQUALITIES
A. Subadditive type
B. Relative entropy
C. WYD inequalities
D. But Cauchy-Schwarz matrix inequalities are not new
E. More examples of new inequalities
IV. EQUALITY CONDITIONS
V. HISTORICAL REMARKS