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arXiv:quant-ph/0001105

Quantum Anti-Cloning

Song, David D., Lucién Hardy

Published Jan 28, 2000• 2 citations
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Abstract

We derive the transformation for the optimal universal quantum anti-cloner which produces two anti-parallel outputs for a single input state. The fidelity is shown to be 2/3 which is same as the measurement fidelity. We consider a probabilistic quantum anti-cloner and show quantum states can be anti-cloned exactly with non-zero probability and its efficiency is higher than the efficiency of distinguishing between the two states.

DOI
10.48550/arxiv.quant-ph/0001105

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We derive the transformation for the optimal universal quantum anti-cloner which produces two anti-parallel outputs for a single input state. The fidelity is shown to be 2/3 which is same as the measurement fidelity. We consider a probabilistic quantum anti-cloner and show quantum states can be anti-cloned exactly with non-zero probability and its efficiency is higher than the efficiency of distinguishing between the two states.
We establish the best possible approximation to a perfect quantum cloning machine that produces two clones out of a single input. We analyze both universal and state-dependent cloners. The maximal fidelity of cloning is shown to be 5/6 for universal cloners. It can be achieved either by a special unitary evolution or by a teleportation scheme. We construct the optimal state-dependent cloners operating on any prescribed two nonorthogonal states and discuss their fidelities and the use of auxiliary physical resources in the process of cloning. The optimal universal cloners permit us to derive an upper bound on the quantum capacity of the depolarizing quantum channel.
We investigate the probabilistic cloning and purification of quantum states. The performance of these probabilistic operations is quantified by the average fidelity between the ideal and actual output states. We provide a simple formula for the maximal achievable average fidelity and we explicitly show how to construct a probabilistic operation that achieves this fidelity. We illustrate our method on several examples such as the phase covariant cloning of qubits, cloning of coherent states, and purification of qubits transmitted via depolarizing channel and amplitude damping channel. Our examples reveal that the probabilistic cloner may yield higher fidelity than the best deterministic cloner even when the states that should be cloned are linearly dependent and are drawn from a continuous set.
Due to the no-cloning theorem, the unknown quantum state can only be cloned approximately or exactly with some probability. There are two types of cloners: universal and state-dependent cloner. The optimal universal cloner has been found and can be viewed as a special state-dependent quantum cloner that has no information about the states. In this paper, we investigate the state-dependent cloning when the state set contains more than two states. We get some bounds of the global fidelity for these processes. This method is not dependent on the number of the states contained in the state set. It is also independent of the numbers of copying.
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.
Perfect cloning of a known set of states with arbitrary prior probabilities is possible if we allow the cloner to sometimes fail completely. In the optimal case the probability of failure is at its minimum allowed by the laws of quantum mechanics. Here we show that it is possible to lower the failure rate below that of the perfect probabilistic cloner but the price to pay is that the clones are not perfect; the global fidelity is less than one. We determine the optimal fidelity of a cloner with a Fixed Failure Rate (FFR cloner) in the case of a pair of known states. Optimality is shown to be attainable by a measure-and-prepare protocol in the limit of infinitely many clones. The optimal protocol consists of discrimination with a fixed rate of inconclusive outcome followed by preparation of the appropriate clones. The convergence shows a symmetry-breaking second-order phase transition in the fidelity of the approximate infinite clones.