# Understanding the optimal approximate qubit cloning method

1. Feb 16, 2016

### Strilanc

The paper Optimal Cloning of Pure States by R. F. Werner describes a method for approximately expanding an unknown state $\rho$ containing n copies of a qubit, so $\rho = (\alpha \left| 0 \right\rangle + \beta \left| 1 \right\rangle)^{\otimes n}$, into a larger state with d more qubits approximating $(\alpha \left| 0 \right\rangle + \beta \left| 1 \right\rangle)^{\otimes n+d}$.

Specifically, the paper says that the optimal cloning map is to tensor $d$ randomized qubits onto the state then project onto the symmetric subspace and normalize. Symbolically: $\widehat{T}(\rho) = \frac{n+1}{n+d+1} s_{n+d} (\rho \otimes I_2^{\otimes d}) s_{n+d}$.

The thing I don't understand is what is meant operationally by "project onto the symmetric subspace".

Does it mean that we're doing a projective measurement where the symmetric subspace has a unique eigenvalue, and we give up if the measurement result is some other eigenvalue? For example we could measure the observable $S = \sum_k^{n+d} \widehat{\left| n+d \atop k \right\rangle} \widehat{\left\langle n+d \atop k \right|}$, which would return 1 if we're in the symmetric subspace and 0 otherwise. But then we'd only expect to succeed $\frac{1}{2^d}$ of the time, which is worse than what's reported in the paper.

Does it mean we're tensor-factoring the space into a symmetric part and a non-symmetric part, and then discarding the parts of the state corresponding to the non-symmetric part (i.e. we trace over it)? I'm not sure how to compute that.

Does it mean something else entirely?

2. Feb 21, 2016