In this project, quantum cloning machines are analyzed that take in N quantum systems in the same unknown pure state and output M quantum systems with M > N, such that the output best resembles the ideal, but impossible output of an M-fold tensor product of the pure input state. The proof of Keyl and Werner is reviewed in which a unique optimal solution is constructed, both in the case where the quality of the cloning is determined by the entire output, and in the case where the quality is determined by measurements on a single clone. Furthermore, it is shown that previous work on qubit cloning machines is a special case of the presented optimal solution.
We consider quantum devices for turning a finite number N of d-level quantum systems in the same unknown pure state σ into M>N systems of the same kind, in an approximation of the M-fold tensor product of the state σ. In a previous paper it was shown that this problem has a unique optimal solution, when the quality of the output is judged by arbitrary measurements, involving also the correlations between the clones. We show in this paper, that if the quality judgment is based solely on measurements of single output clones, there is again a unique optimal cloning device, which coincides with the one found previously.
We construct a unique optimal quantum device for turning a finite number of
d-level quantum systems in the same unknown pure state \ensuremath{\sigma} into
M systems of the same kind, in an approximation of the
M-fold tensor product of the state \ensuremath{\sigma}.
We study the problem of universal quantum cloning – taking several identical copies of a pure but unknown quantum state and producing further copies. While it is well known that it is impossible to perfectly reproduce the state, how well the copies can be cloned can be quantified using the fidelity. We examine how individual fidelities can be traded against each other, and how different fidelity measures can be incorporated. The broadly applicable formalism into which we transform the cloning problem is described as a series of quadratic constraints which are amenable to mathematical and computational scrutiny. As such, we reproduce all known results on optimal universal cloning, and push the recent results on asymmetric cloning much further, giving new trade-off relations between fidelities for broad classes of optimal cloning machines. We also provide substantial evidence that motivates why other parameter ranges (number of input copies) have not, and will not, yield to similar analysis.
We consider the problem of approximate cloning of quantum states: given
n copies of an unknown state
ρ∈Cd×d, prepare an
(n+k)-copy state with high fidelity to
ρ⊗(n+k). Werner's pure state cloner is the optimal channel for the pure state case, and shows that
n=Θ(kd/ε) copies are necessary and sufficient to clone
k additional copies of an unknown pure state to fidelity
1−ε. The random purification channel gives a straightforward extension of Werner's cloner to mixed state inputs: given
n copies of a mixed state, randomly purify your input, apply Werner's channel in the larger Hilbert space, and then trace out the auxiliary registers. This gives a mixed state cloner using
n=O(krd/ε) copies to clone rank-
r states. Can one do any better? We show that the answer is no: one must use
n=Ω(krd/ε) copies. We prove our lower bound by studying the special case of projector cloning, in which the input state
ρ is promised to be of the form
P/r, where
P is a rank-
r orthogonal projector.
As a further application of our techniques, we consider the closely related problem of approximate transposition of quantum states, where one seeks to convert
ρ⊗n to a
k-copy state with high fidelity to
(ρT)⊗k. Here, we again show
n=Θ(krd/ε) copies are necessary and sufficient for this task.
The impossibility of perfectly copying (or cloning) an unknown quantum state is one of the basic rules governing the physics of quantum systems. The processes that perform the optimal approximate cloning have been found in many cases. These ``quantum cloning machines'' are important tools for studying a wide variety of tasks, e.g., state estimation and eavesdropping on quantum cryptography. This paper provides a comprehensive review of quantum cloning machines both for discrete-dimensional and for continuous-variable quantum systems. In addition, it presents the role of cloning in quantum cryptography, the link between optimal cloning and light amplification via stimulated emission, and the experimental demonstrations of optimal quantum cloning.