Minkowski型迹不等式与量子熵II的强次可加性：凸与凹
A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity
摘要：
我们重新讨论并证明了在第一部分中讨论的迹函数的一些凸性不等式。所考虑的主要泛函是Φ_{p，q}（A_1，A_2，.，A_m）=（迹（（\sum_{j=1}^m A_j^p）^{q/p}））^{1/q}.在第一部分中，我们只考虑了q=1的情形，证明了0 < p \leq 1时Φ_{p，1}的凸性和p=2时的凸性.我们证明了Φ_{p，1}的凸性，其中1< p < 2。这里我们不仅解决了1 \leq p \leq 2的联合凸性的未解决的情况，我们还能够包括参数q \geq 1并且仍然保持凸性。除此之外，这导致了当1 \leq p \leq 2时算子的L^q（L^p）范数的定义，以及三个希尔伯特空间的张量积上的算子的闵可夫斯基不等式-这导致了熵的强次可加性的另一个证明。我们还证明了其他一些相关泛函的凸性/凸性。
英文摘要：
We revisit and prove some convexity inequalities for trace functions conjectured in the earlier part I. The main functional considered is Φ_{p,q}(A_1,A_2,...,A_m) = (trace((\sum_{j=1}^m A_j^p)^{q/p}))^{1/q} for m positive definite operators A_j. In part I we only considered the case q=1 and proved the concavity of Φ_{p,1} for 0 < p \leq 1 and the convexity for p=2. We conjectured the convexity of Φ_{p,1} for 1< p < 2. Here we not only settle the unresolved case of joint convexity for 1 \leq p \leq 2, we are also able to include the parameter q\geq 1 and still retain the convexity. Among other things this leads to a definition of an L^q(L^p) norm for operators when 1 \leq p \leq 2 and a Minkowski inequality for operators on a tensor product of three Hilbert spaces -- which leads to another proof of strong subadditivity of entropy. We also prove convexity/concavity properties of some other, related functionals.
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