A special case of the well known Lieb concavity theorem states that the following function is concave on positive operators A and B:


$$
(A,B) \to \text{Tr} \{A^s X B^{1-s} X^\dagger \}
$$


for $s \in [0,1]$ and an arbitrary operator $X$. I am wondering if it is known whether the following function is concave as well:


$$
(A,B) \to \text{Tr} \{A^s B^{(1-s)/2} X X^\dagger B^{(1-s)/2} \}
$$


If not, do you have a counterexample? Apologies if this question has already been posted somewhere else in math overflow...


I have a similar question as the above one for the Ando convexity theorem.



variation of the Lieb concavity theorem
The conjectured inequality is **false**. 


Using cyclicity of the trace, let's first write it in a slightly nicer form
\begin{equation\*}
 f(A,B) := \operatorname{tr}(X^\*B^{(1-s)/2} A^sB^{(1-s)/2}X).
\end{equation\*}


For joint-concavity to hold, we'd like to show
\begin{equation\*}
 f\left(\tfrac{A+B}{2}, \tfrac{U+V}{2} \right) \ge \tfrac12f(A,U)+\tfrac12f(B,V).\qquad (\*)
\end{equation\*}


But picking $A,B,U,V,X$ as follows, we have an immediate counterexample (I used $s=0.5$):
\begin{equation\*}
 A = \begin{pmatrix} 41 & 25\\ 25 & 26\end{pmatrix},\ 
 B = \begin{pmatrix} 2 & 7\\ 7 & 25\end{pmatrix},\ 
 U = \begin{pmatrix} 25 & 22\\ 22 & 20\end{pmatrix},\ 
 V = \begin{pmatrix} 5 & 5\\ 5 & 10\end{pmatrix},\\ 
 X = \begin{pmatrix} 1 & 0\\ 7 & -1\end{pmatrix}.
\end{equation\*}


With this choice of variables, we obtain 5.964618e+03 < 6.529930e+03 in $(\*)$ above.




---


EDIT
----


In case you are interested or for others who may interested in concavity / convexity results similar in flavor to the one asked above, I'd like to mention the following nice paper: [*Concavity of certain matrix trace and norm functions*, F. Hiai (2013).](http://arxiv.org/pdf/1210.7524)


In that paper, Hiai studies (among others), joint concavity and convexity of maps of the form
\begin{equation\*}
 (A,B) \mapsto \mbox{tr}(\Phi(A^p)^{1/2}\Psi(B^q)\Phi(A^p)^{1/2})^s,
\end{equation\*}
for suitable choices of reals $p,q,s$, and positive linear maps $\Phi$ and $\Psi$ (e.g, with $\Phi(A) = X^\*AX$, $\Psi=\mbox{Id}$, and $s=1$, we recover Lieb's concavity).


