A Minkowski type trace inequality and strong subadditivity of quantum entropy. (English) Zbl 0933.47014

Carlen, Eric A.; Lieb, Elliott H.

Buslaev, V. (ed.) et al., Differential operators and spectral theory. M. Sh. Birman’s 70th anniversary collection. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 189(41), 59-68 (1999).

Let \(P_H\) be the set of all positive semidefinite operators on a finite-dimensional Hilbert space \(H\) with inner product \(\langle\cdot,\cdot\rangle\).
For any natural number \(n\), finite \(p> 0\), \(\forall A_i\in P_H\), \(1\leq i\leq n\), denote \[ \Phi_p(A_1,A_2,\dots, A_n)= \text{Tr}\Biggl(\Biggl(\sum^n_{j= 1} A^p_j\Biggr)^{1/p}\Biggr). \] The main result of this article is
Theorem 1. For \(0\leq p\leq 1\), \(\Phi_p\) is a jointly concave function of its arguments. For \(p= 2\), \(\Phi_p\) is jointly convex. For \(p>2\), \(\Phi_p\) is neither convex nor concave.
Theorem 1 is used to obtain other two theorems.
Theorem 2. Let \(A\) be a positive operator on the tensor product of two Hilbert spaces \(H_1\otimes H_2\). Then for all \(p\geq 1\) \[ (\text{Tr}_2(\text{Tr}_1A)^p)^{1/p}\leq \text{Tr}_1((\text{Tr}_2 A^p)^{1/p}). \] The last inequality reverses for \(0<p\leq 1\).
Theorem 3. Let \(A\) be a positive operator on the tensor product of three Hilbert spaces \(H_1\otimes H_2\otimes H_3\). Then \[ \text{Tr}_3(\text{Tr}_2( \text{Tr}_1 A)^p)^{1/p}\leq \text{Tr}_{1,3}((\text{Tr}_2 A^p)^{1/p}). \] For \(p= 2\) and, trivially, \(p=1\), while the reverse inequality holds for \(0<p\leq 1\).
For the entire collection see [Zbl 0911.00011].

Reviewer: A.Kondrat’yev (Pensacola)

MSC:

47A63	Linear operator inequalities
15A90	Applications of matrix theory to physics (MSC2000)

Keywords:

trace function; inequality; positive semidefinite operators; finite-dimensional Hilbert space; inner product; jointly concave function; jointly convex; tensor product