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.


1 Answer 1


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.


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).

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).

  • $\begingroup$ Thanks for this answer. Since we know that it is true for $X = I$, is it possible to determine conditions on X such that the inequality is true? $\endgroup$ Apr 23, 2014 at 4:14

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