Proof of Monotonicity and Invariance Under Local Unitary Operations for Summed Measures

Question

Suppose we have a set of measures $A_1, A_2, ..., A_n$ that individually satisfy the following properties:

1-Monotonicity under Local Operations and Classical Communication (LOCC)

2-Invariance under Local Unitary Operations

I want a prove that their sum also satisfies the two properties.

Answer 1

Perhaps I am missing something as it is not clear what exactly $A_i$ is. But suppose I have a bunch of quantities $A_i(\rho)$ that depend on my choice of state somehow.

1. I have some transformation $T_1$ on my state such that $A_i(\rho) \leq A_i(T_1(\rho))$ for all $i$. Well then $$ \sum_i A_i(\rho) \leq \sum_i A_i(T_1(\rho)) $$ as each term in the sum satisfies the inequality so the sum does too.

2. I have a transformation $T_2$ on my state such that $A_i(T_2(\rho)) = A_i(\rho)$ for all $i$. Well then $$ \sum_i A_i(\rho) = \sum_i A_i(T_2(\rho)) $$ as the equality holds for every term.