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How well can an arbitrary unknown (quantum) state $\rvert \psi \rangle = \alpha\rvert 0 \rangle + \beta \rvert 1 \rangle$, be imperfectly/approximately cloned?

Given an unknown state ${\rvert \psi \rangle}$, say one can clone ${\rvert \phi \rangle}$ , such that $${\langle \phi }{\rvert \psi \rangle}= \delta$$

If $\rvert \phi \rangle$ and $\rvert \psi \rangle$ are identical and same, $\delta = 1$, however we know, due the no cloning theorem, that $\delta<1$.

My question is what is the 'highest' value of $\delta$ one can achieve using polynomial time 'resources', i.e. how well can an UNKNOWN quantum state be imperfectly cloned?

(Let's agree we $\rvert \psi \rangle$, both $\alpha, \beta$ are non-zero, i.e. not basis states)

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This problem has been studied in great detail, not just for the case of imperfectly cloning 1 qubit to get 2 copies, but more general problems of how to get m copies of a state given n copies, etc. I don't work in the area, but I'll try to give you some answers. Others may be able to provide a better answer.

For the problem you describe, 1 qubit to 2 qubit cloning, the optimal cloner is due to

Bužek, Vladimir, and Mark Hillery. "Quantum copying: Beyond the no-cloning theorem." Physical Review A 54, no. 3 (1996): 1844.

They show how to take an input state $|\psi\rangle$ and output two qubits such that the reduced density matrix of either output qubit (let's call it $\rho$) satisfies $\langle\psi|\rho|\psi\rangle=5/6$.

This was later shown to be optimal by several authors:

Bruß, Dagmar, David P. DiVincenzo, Artur Ekert, Christopher A. Fuchs, Chiara Macchiavello, and John A. Smolin. "Optimal universal and state-dependent quantum cloning." Physical Review A 57, no. 4 (1998): 2368.

Gisin, Nicolas, and Serge Massar. "Optimal quantum cloning machines." Physical review letters 79, no. 11 (1997): 2153.

Gisin, Nicolas. "Quantum cloning without signaling." Physics Letters A, Volume 242, Issues 1–2, 1998, Pages 1-3

A survey on the topic that covers this and much more:

Scarani, Valerio, Sofyan Iblisdir, Nicolas Gisin, and Antonio Acin. "Quantum cloning." Reviews of Modern Physics 77, no. 4 (2005): 1225.

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