Correction to: Trace Functions with Applications in Quantum Physics

1 Correction to: J Stat Phys (2014) 154:807–818 https://doi.org/10.1007/s10955-013-0890-x

The original version of this article unfortunately contained an error and it has been corrected with this erratum. We consider in [5, p. 810 and Theorem 2.2] the following results which we here put together.

Theorem 1

The functions

$$\begin{aligned} f(t)=(t^p+1)^{1/p}\qquad t>0 \end{aligned}$$

are operator monotone (thus operator concave) for $0< p\le 1$ and operator convex for $1\le p\le 2.$

The first part $(0< p\le 1)$ is proved on page 810. Note that the statement is different from Ando [1, Corollary 4.3]. The second part $(1\le p\le 2)$ is our [5, Theorem 2.2]. The perspective of an operator concave (convex) function is again an operator concave (convex) function of two variables, cf. [3, Theorem 1.1]. By applying [4, Theorem 1.1] these results entail that for arbitrary K the trace functions

$$\begin{aligned} (A,B)\rightarrow \mathrm{Tr}~K^* \big (L_A^p+R_B^p\bigr )^{1/p}(K), \end{aligned}$$

(1)

are concave for $0< p\le 1$ and convex for $1\le p\le 2.$ We then try to use these results to recover Carlen-Lieb’s theorems in [2].

Theorem 2

(Carlen-Lieb) The trace functions

$$\begin{aligned} (A,B)\rightarrow \mathrm{Tr}~(A^p+B^p)^{1/p} \end{aligned}$$

(2)

are concave for $0<p\le 1$ and convex for $1\le p\le 2.$

We do this by claiming the identity

$$\begin{aligned} \mathrm{Tr}~\big (L_A^p+R_B^p\bigr )^{1/p}(I)=\mathrm{Tr}~(A^p+B^p)^{1/p}, \end{aligned}$$

(3)

where the left hand side is obtained by setting $K=I$ (the identity operator) in (1). By replacing A with $A^{1/p}$ and B with $B^{1/p}$ this would then entail that

$$\begin{aligned} \mathrm{Tr}~\big (L_A+R_B\bigr )^{1/p}(I)=\mathrm{Tr}~(A+B)^{1/p} \end{aligned}$$

for positive definite matrices. This however is wrong. Victoria Chayes informed me in a private communication that for $p=1/4$ the difference

$$\begin{aligned} \mathrm{Tr}~\big (L_A^p+R_B^p\bigr )^{1/p}(I)- \mathrm{Tr}~(A^p+B^p)^{1/p}=\Vert AB-BA\Vert ^2_\text {HS}\ge 0, \end{aligned}$$

and the difference therefore vanishes if and only if A and B commute.

The failed identity in (3) is of no consequence here since we already know that Carlen-Lieb’s theorem is true.

Later in the paper [5, Theorem 3.1] we prove that the functions of two variables

$$\begin{aligned} g(t,s)=\frac{t-s}{t^p-s^p}\qquad t,s>0 \end{aligned}$$

are operator concave for $0<p\le 1,$ and this implies that the trace functions

$$\begin{aligned} (A,B)\rightarrow \mathrm{Tr}~K^*\frac{L_A-R_B}{L_A^p-R_B^p}(K)\qquad 0\le p\le 1 \end{aligned}$$

are concave for arbitrary K. We then put $K=I$ and try to use an identity similar to (3) to obtain [5, Theorem 3.2] which claims that the trace functions

$$\begin{aligned} (A,B)\rightarrow \mathrm{Tr}~\frac{A-B}{A^p-B^p}\qquad 0<p\le 1 \end{aligned}$$

(4)

are concave in positive definite matrices. This claim is no longer verified. In fact, numerical calculations indicate that the trace functions in (4) are concave only for $p=1/2$ and not concave for $0<p<1/2$ and $1/2<p<1.$