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arXiv:1412.3361 · Physical Review A

No-signaling bounds for quantum cloning and metrology

Pavel Sekatski, Michalis Skotiniotis, Wolfgang Dür

Published Aug 28, 2015• 6 citations
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Abstract

The impossibility of superluminal communication is a fundamental principle of physics. Here we show that this principle underpins the performance of several fundamental tasks in quantum information processing and quantum metrology. In particular, we derive tight no-signaling bounds for probabilistic cloning and super-replication that coincide with the corresponding optimal achievable fidelities and rates known. In the context of quantum metrology, we derive the Heisenberg limit from the no-signaling principle for certain scenarios including reference frame alignment and maximum likelihood state estimation. We elaborate on the equivalence of assymptotic phase-covariant cloning and phase estimation for different figures of merit.

DOI
10.1103/physreva.92.022355

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The impossibility of superluminal communication is a fundamental principle of physics. Here we show that this principle underpins the performance of several fundamental tasks in quantum information processing and quantum metrology. In particular, we derive tight no-signaling bounds for probabilistic cloning and super-replication that coincide with the corresponding optimal achievable fidelities and rates known. In the context of quantum metrology, we derive the Heisenberg limit from the no-signaling principle for certain scenarios including reference frame alignment and maximum likelihood state estimation. We elaborate on the equivalence of assymptotic phase-covariant cloning and phase estimation for different figures of merit.
The no-cloning theorem is a cornerstone of quantum cryptography. Here we generalize and rederive in a unified framework various upper bounds on the maximum achievable fidelity of probabilistic and deterministic cloning machines. Building on ideas by Gisin [Phys. Lett. A 242, 1 (1998)], our result follows from the fact that remote state preparation is possible and the no-signaling principle holds. We apply our general theorem to several subsets of states that are of interest in quantum cryptography.
We consider the problem of deterministically cloning quantum channels with respect to the best attainable rate and the highest quality, so-called optimal cloning. We demonstrate that cloning quantum states is, in-fact, equivalent to cloning the trash-and-replace channel and therefore the former is a special case of the more general problem. By appealing to higher-order quantum operations (quantum processes) we construct a unified framework to deal with the most general cloning tasks and establish necessary conditions for a family of channels to exhibit super-replication -- a quadratic cloning rate with vanishing error. We find that noisy phase-gate channels satisfy these conditions, and we construct the explicit super-replicating process for the task. Conversely, we find that the criteria are not met by the full set of noisy unitary gates; classical noise channels; or amplitude damping channels, whose respective cloning rates are at most linear. In this paradigm, we not only derive new results, but also refigure known ones. We derive a strong converse for state cloning, and for unitary channels we construct an alternative super-replication process to that of Dür et al. [PRL 114, 120503 (2015)] and Chiribella et al. [PRL 114, 120504 (2015)] based on a measure-and-prepare process, which allows us to establish a direct connection between optimal channel cloning and Bayesian channel estimation. Finally we give an SDP algorithm to search for optimal cloning processes and study the advantage of coherent vs measure-and-prepare protocols on concrete examples.
It has recently been shown that probabilistic protocols based on postselection boost the performances of the replication of quantum clocks and phase estimation. Here we demonstrate that the improvements in these two tasks have to match exactly in the macroscopic limit where the number of clones grows to infinity, preserving the equivalence between asymptotic cloning and state estimation for arbitrary values of the success probability. Remarkably, the cloning fidelity depends critically on the number of rationally independent eigenvalues of the clock Hamiltonian. We also prove that probabilistic metrology can simulate cloning in the macroscopic limit for arbitrary sets of states when the performance of the simulation is measured by testing small groups of clones.
We investigate the class of quantum cloning machines that equally duplicate all real states in a Hilbert space of arbitrary dimension. By using the no-signaling condition, namely, that cloning cannot make superluminal communication possible, we derive an upper bound on the fidelity of this class of quantum cloning machines. Then, for each dimension d, we construct an optimal symmetric cloner whose fidelity saturates this bound. Similar calculations can also be performed in order to recover the fidelity of the optimal universal cloner in d dimensions.