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

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].

### MSC:

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