Optimal Universal Quantum Cloning and State Estimation

D. Bruss

Dagmar Bruß, ARTUR K. EKERT, Chiara Macchiavello

We derive a tight upper bound for the fidelity of a universal $N\ensuremath{\rightarrow}M$ qubit cloner, valid for any $M\ensuremath{\ge}N$, where the output of the cloner is required to be supported on the symmetric subspace. Our proof is based on the concatenation of two cloners and the connection between quantum cloning and quantum state estimation. We generalize the operation of a quantum cloner to mixed and/or entangled input qubits described by a density matrix supported on the symmetric subspace of the constituent qubits. We also extend the validity of optimal state estimation methods to inputs of this kind.

Giulio Chiribella, Giacomo Mauro D’Ariano, Paolo Perinotti

Chiara Macchiavello

We describe in a unified way universal operations for quantum systems with arbitrary finite dimension and show some links among them. In particular, we discuss a connection between optimal universal cloning and optimal state estimation. We then describe the operation of the depolarising quantum channel, its connection to quantum cloning and ways to contrast the undesired effects of its action.

A. Hayashi, T. Hashimoto, M. Horibe

A direct derivation is given for the optimal mean fidelity of quantum state estimation of a d-dimensional unknown pure state with its N copies given as input, which was first obtained by Bruss and Macchiavello by establishing a connection to optimal quantum cloning. An explicit condition for POVM measurement operators for optimal estimators is obtained, by which we show optimal estimators with finite POVM exist but not generally universal, where universality means the fidelity is independent of input states. However any optimal estimator for M(>N) copies is universal if it is used for N copies as input.

Vladimír Bužek, Mark S. Hillery