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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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