Understanding the optimal approximate qubit cloning method

[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?

 

[eljose79]

I can understand your confusion about the operational meaning of "projecting onto the symmetric subspace" in the context of cloning pure states. Let me try to provide some clarification on this matter.

Firstly, the symmetric subspace refers to the subspace of the total Hilbert space that is spanned by all the symmetric states, i.e. states that are invariant under the action of the symmetric group. In the case of cloning pure states, the symmetric subspace corresponds to the subspace spanned by all the states that are invariant under the action of the symmetric group on the qubits. This means that the states in this subspace are symmetric with respect to the exchange of any two qubits.

Now, when we talk about "projecting onto the symmetric subspace", we are referring to the projection operator that projects any state onto this subspace. This projection operator can be constructed using the projection operators for each of the symmetric irreducible representations of the symmetric group. These projection operators can be expressed in terms of the symmetric group characters, which are mathematical objects that describe the properties of the symmetric group.

In the specific case of the optimal cloning map described in the paper, the authors have used the projection operator onto the symmetric subspace as part of the cloning process. This means that the resulting state after applying the cloning map will be in the symmetric subspace, and hence will be symmetric with respect to the exchange of any two qubits. This is an important property for cloned states, as it ensures that the cloned states are indistinguishable from each other, which is the desired outcome of cloning.

To summarize, "projecting onto the symmetric subspace" in the context of cloning pure states refers to the projection operator that projects any state onto the subspace spanned by all the symmetric states. This is an important step in the cloning process as it ensures that the cloned states are indistinguishable from each other. I hope this helps to clarify your understanding of the concept.