In order to make a copy, we take a system B with an identical Hilbert space and initial state $|e\rangle_B$ (which must be independent of $|\psi\rangle_A$, of which we have no prior knowledge). The coefficients ''a'' and ''b'' are unknown to us. Suppose the state of a quantum system A, which we wish to copy. The linearity of quantum mechanics, however, prevents us from cloning arbitrary, unknown states ( photoQopiers don't exist). This would allow Alice and Bob to communicate across space-like separations (if the two components of the entangled state are space-like separated), potentially violating causality. Bob will know that Alice has transmitted a $0$ if all his measurements will produce the same result otherwise, his measurements will be split evenly between $+1/2$ and $-1/2$.
Bob creates many copies of his electron's state, and measures the spin of each copy in the $z$ direction. If Alice wishes to transmit a $0$, she measures the spin of her electron in the $z$ direction (she could have chosen another direction if she had wished to transmit a $1$), collapsing Bob's state to either $|z+\rangle_B$ or $|z-\rangle_B$. Alice could send bits to Bob in the following way: Consider the EPR thought experiment, and suppose quantum states could be cloned. The no-cloning theorem prevents superluminal communication via quantum entanglement. Hardy, "Quantum theory from five reasonable axioms," 2001, arXiv:quant-ph/0101012. Aaronson, "Is Quantum Mechanics An Island In Theoryspace?," 2004, arXiv:quant-ph/0401062. Does even the teensiest bit of nonlinearity in QM bring causality to its knees, or can the damage be limited in some sense?ĭoes all of this have any implications for quantum gravity - e.g., does it help to explain why it's hard to make a theory of quantum gravity, since it's not obvious that quantum gravity can be unitary and linear? When it comes to mechanical waves, we're used to thinking of a linear wave equation as an approximation that is always violated at some level. It seems strange to me that a principle so fundamental and important can be violated simply by having some nonlinearity. Can anyone offer an argument with crayons for why this should be so?
Aaronson makes the statement, which was new to me, that nonlinearity in QM leads to superluminal signaling (as well as the solvability of hard problems in computer science by a nonlinear quantum computer). I recently came across two nice papers on the foundations of quantum mechancis, Aaronson 2004 and Hardy 2001.