Tensor product matrices order relation

[Jufa]

TL;DR Summary

Having that $A \geq \pm B$ how can one prove that indeed $A^{\otimes n} \geq \pm B^{\otimes n}$

We mainly have to prove that this quantity


$\bra{\varphi} A^{\otimes n } \ket{\varphi} \pm \bra{\varphi} B^{\otimes n } \ket{\varphi}$

is greater or equal than zero for all $\ket{\varphi}$.

Being $\ket{\varphi}$ a product state it is straightforward to demonstrate such inequality. I am struggling though to demonstrate it for a general, perhaps entangled $\ket{\varphi}$, because of the cross products that show up.

Can someone please help me?

Many thanks in advance! :)

 

[jambaugh]

Can you show that if the inequality is true for a product state that its true for a sum of product states? Specifically for a sum of orthogonal product states?