The Crown is the head of our State; all authority derives from our Sovereign.
http://www.parl.gc.ca/About/House/compendium/web-content/c_d...
http://www.parl.gc.ca/About/House/compendium/web-content/c_d...
Yes, Canada is a monarchy, yes, the Queen has the ultimate power in theory, but the head of state is the Queen of Canada, not the Queen of the UK.
I'll repeat myself, them being both the same person is incidental, if you think the fact Canada being a monarchy means it's not independent, does that also mean the UK is not?