Lieb ’ s simple proof of concavity of ( A , B ) 7 → Tr A p K † B 1 − p K and remarks on related inequalities

A simple, self-contained proof is presented for the concavity of the map (A,B) 7→ TrApK†B1−pK. The author makes no claim to originality; this note gives Lieb’s original argument in its simplest, rather than its most general, form. A sketch of the chain of implications from this result to concavity of A 7→ Tr eK+logA is then presented. An independent elementary proof is given for the joint convexity of the map (A,B,X) 7→ Tr ∫∞ 0 X † 1 A+uIX 1 B+uI du which plays a key role in entropy inequalities.