Could a non-unitary time evolution violate the no-cloning theorem?

Question

It is written in some places that the unitarity of time evolution is what prevents quantum cloning. However, consider the typical definition of a cloning operator $A$. For all $\left|\psi\right>$ and a standard state $\left|0\right>$, $$ A[\left|\psi\right>\otimes \left|0\right>] = \left|\psi\right>\otimes \left|\psi\right> $$ Without using the unitarity of $A$, I can follow the proof in these notes to demonstrate no-cloning. With a superposition state $\left|\chi\right> = a\left|\psi\right>+b\left|\phi\right>$, $A$ can be appled to find, $$ A[\left|\chi\right>\otimes \left|0\right>] = a(\left|\psi\right>\otimes \left|\psi\right>)+b(\left|\phi\right>\otimes\left|\phi\right>) $$ $A$ could also be applied to find, $$ A[\left|\chi\right>\otimes \left|0\right>] =\left|\chi\right>\otimes\left|\chi\right> = (a\left|\psi\right>+b\left|\phi\right>)\otimes(a\left|\psi\right>+b\left|\phi\right>) $$

These expressions are not equal, so there is a contradiction. This appears to be a proof that no-cloning is impossible for any linear time evolution, even in an alternate universe where quantum time evolution does not have to be unitary. Is this reasoning correct?

Answer 1

In fact, this is one of the two popular kinds of proofs by contradiction of the no-cloning theorem which claims the nonexistence of quantum operation that can duplicate arbitrary unknown quantum state. It may be helpful to clearly give two kinds of proofs here:

(i) proof by contradiction based on the linearity: just as what you have done, assume that there is a quantum operation $U_{\mathrm{clone}}$ which can duplicate arbitrary unknown quantum state. Then for arbitrary state $|\Psi\rangle=\alpha|\phi_1\rangle+\beta|\phi_2\rangle$, $$U_{\mathrm{clone}}|\Psi\rangle|0\rangle=|\Psi\rangle|\Psi\rangle.$$ However, by the assumption of linearity of this quantum cloning operation, we have $$U_{\mathrm{clone}}|\Psi\rangle|0\rangle=\alpha U_{\mathrm{clone}}|\phi_1\rangle|0\rangle+\beta U_{\mathrm{clone}}|\phi_2\rangle|0\rangle =\alpha |\phi_1\rangle|\phi_1\rangle+\beta |\phi_2\rangle|\phi_2\rangle.$$ We thus arrive at a contradiction. Here I would like to point out that this proof is actually the initial proof of no-cloning theorem used by Wotters and Zurek in their paper in 1982 and also by Dieks in his 1982 paper where he also indicated that the linearity of quantum mechanics can be used to prove the impossibility of superluminal communication.

(ii) proof by contradiction based on the unitarity of the quantum operation: we first assume that the quantum clone operation $U_{\mathrm{clone}}$ is unitary, then for arbitrary states $|\Psi\rangle$ and $|\Phi\rangle$, $U_{\mathrm{clone}}|\Psi\rangle|0\rangle=|\Psi\rangle|\Psi\rangle$ and $U_{\mathrm{clone}}|\Phi\rangle|0\rangle=|\Phi\rangle|\Phi\rangle$. Taking the inner product of two sides of the above two equations and using the unitarity assumption of $U_{\mathrm{clone}}$, we arrive at $\langle \Psi|\Phi\rangle=(\langle\Psi|\Phi\rangle)^2$ which is the case only when $|\Psi\rangle$ and $|\Phi\rangle$ are orthogonal. This proof is first proposed by Yuen in his paper in 1986. Now this proof is more popular in quantum information books, e.g., Peres' book Quantum Theory: Concepts and Methods, Nielsen and Chuang's book Quantum Computation and Quantum Information.