A Tensor product matrices order relation

Jufa
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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! :)
 
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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?
 

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