Deterministic Superreplication of One-Parameter Unitary Transformations

We show that one can deterministically generate, out of $N$ copies of an unknown unitary operation, up to $N^2$ almost perfect copies. The result holds for all operations generated by a Hamiltonian with an unknown interaction strength. This generalizes a similar result in the context of phase-covariant cloning where, however, superreplication comes at the price of an exponentially reduced probability of success. We also show that multiple copies of unitary operations can be emulated by operations acting on a much smaller space, e.g., a magnetic field acting on a single $n$-level system allows one to emulate the action of the field on $n^2$ qubits.

Figure 1

Illustration of the overall procedure to obtain $M$ approximate copies from $N$ applications of an (unknown) unitary operation $U(\vartheta )$. By applying the unitary basis change $A^{†}$, $A$, before and after $U(\vartheta {)}^{\otimes N}$, we can obtain the operation $\tilde{V}(\vartheta )$ of Eq. (5) which is a good approximation of $U(\vartheta {)}^{\otimes M}$, i.e., $A^{†}({\mathbb{1}}^{\otimes M}\otimes U^{\otimes N})A(|\psi ⟩\otimes |0{⟩}^{\otimes N})\approx U^{\otimes M}|\psi ⟩\otimes |0{⟩}^{\otimes N}$. Notice that $N$ auxiliary systems are used that are not affected by the transformation.

Figure 2

Illustration of the unitary mapping $A$. Eigenstates $|\mathit{k}⟩$ of $U^{\otimes M}$ with corresponding degeneracies are depicted on the right. Eigenstates $|\mathit{j}⟩$ of $U(\vartheta {)}^{\otimes N}$ are depicted on the left, where the degeneracy is achieved by adding $M$ ancillary systems on which an identity operation acts. The relevant part of the spectrum of $U(\vartheta {)}^{\otimes M}$ is mapped to the appropriate eigenstates of $U^{\otimes N}\otimes I^{\otimes M}$.