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arXiv:1103.5709 · Physical Review A

Cloning of a quantum measurement

Alessandro Bisio, Giacomo Mauro D’Ariano, Paolo Perinotti, Michal Sedlák

Published Oct 18, 2011• 8 citations
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Abstract

We analyze quantum algorithms for cloning of a quantum measurement. Our aim is to mimic two uses of a device performing an unknown von Neumann measurement with a single use of the device. When the unknown device has to be used before the bipartite state to be measured is available we talk about $1\ensuremath{\rightarrow}2$ learning of the measurement, otherwise the task is called $1\ensuremath{\rightarrow}2$ cloning of a measurement. We perform the optimization for both learning and cloning for arbitrary dimension $d$ of the Hilbert space. For $1\ensuremath{\rightarrow}2$ cloning we also propose a simple quantum network that achieves the optimal fidelity. The optimal fidelity for $1\ensuremath{\rightarrow}2$ learning just slightly outperforms the estimate and prepare strategy in which one first estimates the unknown measurement and depending on the result suitably prepares the duplicate.

DOI
10.1103/physreva.84.042330

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We analyze quantum algorithms for cloning of a quantum measurement. Our aim is to mimic two uses of a device performing an unknown von Neumann measurement with a single use of the device. When the unknown device has to be used before the bipartite state to be measured is available we talk about 1\ensuremath→21\ensuremath{\rightarrow}21\ensuremath→2 learning of the measurement, otherwise the task is called 1\ensuremath→21\ensuremath{\rightarrow}21\ensuremath→2 cloning of a measurement. We perform the optimization for both learning and cloning for arbitrary dimension ddd of the Hilbert space. For 1\ensuremath→21\ensuremath{\rightarrow}21\ensuremath→2 cloning we also propose a simple quantum network that achieves the optimal fidelity. The optimal fidelity for 1\ensuremath→21\ensuremath{\rightarrow}21\ensuremath→2 learning just slightly outperforms the estimate and prepare strategy in which one first estimates the unknown measurement and depending on the result suitably prepares the duplicate.
We investigate the asymptotic relationship between quantum cloning and quantum estimation from the global point of view where all the copies produced by the cloner are considered jointly. For an N-to-M cloner, we consider the overall fidelity between the state of the M output systems and the state of M ideal copies, and we ask whether the optimal fidelity is attained by a measure and-prepare protocol in the limit M -> \infty. In order to gain intuition into the general problem, we analyze two concrete examples: i) cloning qubit states on the equator of the Bloch sphere and ii) cloning two-qubit maximally entangled states. In the first case, we show that the optimal measure-and-prepare fidelity converges to the fidelity of the optimal cloner in the limit M -> \infty. In the second case, we restrict our attention to economical covariant cloners, and again, we exhibit a measure-and-prepare protocol that achieves asymptotically the optimal fidelity. Quite counterintuitively, in both cases the optimal states that have to be prepared in order to maximize the overall fidelity are not product states corresponding to M identical copies, but instead suitable M-partite entangled states.
After introducing the no-cloning theorem and the most common forms of approximate quantum cloning, universal quantum cloning is considered in detail. The connections it has with universal NOT-gate, quantum cryptography and state estimation are presented and briefly discussed. The state estimation connection is used to show that the amount of extractable classical information and total Bloch vector length are conserved in universal quantum cloning. The 1  2 qubit cloner is also shown to obey a complementarity relation between local and nonlocal information. These are interpreted to be a consequence of the conservation of total information in cloning. Finally, the performance of the 1  M cloning network discovered by Bužek, Hillery and Knight is studied in the presence of decoherence using the Barenco et al. approach where random phase fluctuations are attached to 2-qubit gates. The expression for average fidelity is calculated for three cases and it is found to depend on the optimal fidelity and the average of the phase fluctuations in a specific way. It is conjectured to be the form of the average fidelity in the general case. While the cloning network is found to be rather robust, it is nevertheless argued that the scalability of the quantum network implementation is poor by studying the effect of decoherence during the preparation of the initial state of the cloning machine in the 1 ! 2 case and observing that the loss in average fidelity can be large. This affirms the result by Maruyama and Knight, who reached the same conclusion in a slightly different manner.
We address the problem of learning an unknown unitary transformation from a finite number of examples. The problem consists in finding the learning machine that optimally emulates the examples, thus reproducing the unknown unitary with maximum fidelity. Learning a unitary is equivalent to storing it in the state of a quantum memory (the memory of the learning machine) and subsequently retrieving it. We prove that, whenever the unknown unitary is drawn from a group, the optimal strategy consists in a parallel call of the available uses followed by a ``measure-and-rotate'' retrieving. Differing from the case of quantum cloning, where the incoherent ``measure-and-prepare'' strategies are typically suboptimal, in the case of learning the ``measure-and-rotate'' strategy is optimal even when the learning machine is asked to reproduce a single copy of the unknown unitary. We finally address the problem of the optimal inversion of an unknown unitary evolution, showing also in this case the optimality of the ``measure-and-rotate'' strategies and applying our result to the optimal approximate realignment of reference frames for quantum communication.
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 present the universal cloning transformation of states in arbitrary-dimensional Hilbert spaces. This unitary transformation attains the optimal fidelity of cloning as specified by Werner [Phys. Rev. A 58, 1827 (1998)]. With this cloning transformation, pure as well as impure states can be optimally copied, and the quality of the copies does not depend on the state being copied. We discuss the properties of quantum clones. In particular, we show that in the limit of high dimension the fidelity of clones does not converge to zero but attains the limit 1/21/21/2. We also show that our cloning transformation is most suitable for cloning of entanglement.
The problem of optimally estimating an unknown unitary quantum operation with the aid of entanglement is addressed. The idea is to prepare an entangled pair, apply the unknown unitary to one of the two parts, and then measure the joint output state. This measurement could be an entangled one or it could be separable (e.g., measurements which can be implemented with local operations and classical communication or LOCC). A comparison is made between these possibilities and it is shown that by using nonseparable measurements one can improve the accuracy of the estimation by a factor of 2(d+1)/d2(d+1)/d2(d+1)/d where d is the dimension of the Hilbert space on which U acts.