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A world with identical micro-physical structure, but with no qualia may be conceived of.
I can conceive of another world in which there is a person with identical micro-physical states, but lacks all qualia whatsoever, has no subjective experience, possesses no phenomenal states of consciousness. This is called the "philosophical zombie" thought experiment of Chalmers.
Conceivability entails possibility.
If it is possible that my mental properties are independent of physical properties, then it follows that mental properties and physical properties are independent.
Mental properties have no causal efficacy, but are only the effects or counterparts of physical states, and only constitute the knowledge of what physical states are like, or seem to us.
Given that the sentence "colourless green ideas sleep furiously" is grammatically correct (and arguably conceivable) does not mean it is possible, or holds any meaning. Similarly philosophical zombie is a seemingly logical, but inherently paradoxical and impossible hypothesis.
Conceivability does not entail metaphysical possibility. I can imagine that I can break the laws of nature, but this does not make it possible evidently, therefore the conclusion does not follow.
I care only about nomological possibilities, not metaphysical possibilities that cannot be supported by any evidence. Thus, the conclusion does not follow.
All neuroscience research strongly supports neuro-physical identity theory, ruling out such a conclusion. Maybe, if we were cavemen we could believe in Chalmer's annoyingly superstitious ideas.
if my mind has non-physical properties, how can I know anything about them in the physical world?
The perceived "I" exists solely in the mind, and the mind has only access to non-physical phenomenological realm of mental properties, whereas physical properties are inaccessible as directly to the mind except through their corresponding mental states.