# Approximate cloning of a qubit, given multiple starting copies

Suppose I'm given several clones of a qubit in a pure unentangled state. That is to say, I'm given the state $(a \left|0\right\rangle + b \left|1\right\rangle)^{\otimes n}$. My goal is to make $d$ clones of the qubit; to expand the input state so that it approximates the target state $(a \left|0\right\rangle + b \left|1\right\rangle)^{\otimes n+d}$.

This can't be done exactly because of the No Cloning theorem. On the other hand, if $n$ is very large, then we can use tomography to get a pretty good approximation of $a$ and $b$. Once we have that, we can start churning out as many copies as we want.

I'm looking for papers/websites/answers giving bounds and on how accurately and efficiently this can be done, as well as explicit techniques for doing it and how well they perform.

For example, suppose we know that $a$ and $b$ are constrained such that $a = \sin \theta$, $b = \cos \theta$, and $0 \leq \theta \leq \pi/2$. Then a very simple $n$-to-$n+d$ cloning procedure is to count the given qubits that are ON in the computational basis and estimate $\theta$ from that (note that we're conditioning operations on the count, not measuring the count). So, given $\psi(t) = (\cos t) \left|0\right\rangle + (\sin t) \left|1\right\rangle$, we get something like:

$\psi_{\text{target}} = \psi(\theta)^{\otimes n+d} = \sum_{l=0}^{n+d} \left| n+d \atop l \right\rangle \cdot \cos^{n+d-l} \theta \cdot \sin^l \theta$

$\psi_{\text{made}} = \sum_{k=0}^{n} \left| n \atop k \right\rangle (\cos^{n-k} \theta) \cdot (\sin^k \theta) \cdot \psi(f_n(k))^{\otimes d}$

$\psi_{\text{target}}^* \cdot \psi_{\text{made}} = \sum_{k=0}^{n} \left( n \atop k \right) \cdot (\cos^{2n-2k} \theta) \cdot (\sin^{2k} \theta) (\cos^d \left( \theta - f_n(k) \right))$

Which, for $n=d=10$ gives this plot w.r.t. $\theta$:

Since the inner product stays above 0.8, the resulting state is basically at most ~35 degrees off of the true state (which is pretty good in a $2^{20}$-dimensional complex vector space).

But presumably there are processes that perform better than this, and without restricting the state space as much.

Abstract: We construct the unique optimal quantum device for turning a finite number of $d$-level quantum systems in the same unknown pure state $\sigma$ into $M$ systems of the same kind, in an approximation of the $M$-fold tensor product of the state $\sigma$.