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}##.(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Understanding the optimal approximate qubit cloning method

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