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arXiv:1206.2821 · Physical Review A

Broadcasting quantum Fisher information

Xiao-Ming Lu, Zhe Sun, Xiaoguang Wang, Shunlong Luo, C. H. Oh

Published May 16, 2013• 44 citations
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Abstract

It is well known that classical information can be cloned, but nonorthogonal quantum states cannot be cloned, and noncommuting quantum states cannot be broadcast. We conceive a scenario in which the object we want to broadcast is the statistical distinguishability, as quantified by quantum Fisher information, about a signal parameter encoded in quantum states. We show that quantum Fisher information cannot be cloned, while it might be broadcast even when the input states are noncommuting. This situation interpolates between cloning of classical information and no broadcasting of quantum information, and indicates a hybrid way of information broadcasting which is of particular significance from both practical and theoretical perspectives.

DOI
10.1103/physreva.87.050302

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It is well known that classical information can be cloned, but nonorthogonal quantum states cannot be cloned, and noncommuting quantum states cannot be broadcast. We conceive a scenario in which the object we want to broadcast is the statistical distinguishability, as quantified by quantum Fisher information, about a signal parameter encoded in quantum states. We show that quantum Fisher information cannot be cloned, while it might be broadcast even when the input states are noncommuting. This situation interpolates between cloning of classical information and no broadcasting of quantum information, and indicates a hybrid way of information broadcasting which is of particular significance from both practical and theoretical perspectives.
Perfect cloning of an unknown quantum state is impossible. Approximate cloning, which is optimal in various senses, has been found in many cases. Paradigmatic examples are Wootters-Zurek cloning and universal cloning. These cloning machines aim at optimal cloning of the full quantum states. However, in practice, what is important and relevant may only involve partial information in quantum states, rather than quantum states themselves. For example, signals are often encoded as parameters in quantum states, whose information content is well synthesized by quantum Fisher information. This raises the basic issue of evaluating the information transferring capability (e.g., distributing quantum Fisher information) of quantum cloning. We assess and compare Wootters-Zurek cloning and universal cloning from this perspective and show that, on average, Wootters-Zurek cloning performs better than universal cloning for the phase (as well as amplitude) parameter, although they are incomparable individually, and universal cloning has many advantages over Wootters-Zurek cloning in other contexts. Physical insights and related issues are further discussed.
An unknown quantum state cannot be copied and broadcast freely due to the no-cloning theorem. Approximate cloning schemes have been proposed to achieve the optimal cloning characterized by the maximal fidelity between the original and its copies. Here, from the perspective of quantum Fisher information (QFI), we investigate the distribution of QFI in asymmetric cloning machines which produce two nonidentical copies. As one might expect, improving the QFI of one copy results in decreasing the QFI of the other copy. It is perhaps also unsurprising that asymmetric phase-covariant cloning outperforms universal cloning in distributing QFI since a priori information of the input state has been utilized. However, interesting results appear when we compare the distributabilities of fidelity (which quantifies the full information of quantum states), and QFI (which only captures the information of relevant parameters) in asymmetric cloning machines. Unlike the results of fidelity, where the distributability of symmetric cloning is always optimal for any d-dimensional cloning, we find that any asymmetric cloning outperforms symmetric cloning on the distribution of QFI for d ≤ 18, whereas some but not all asymmetric cloning strategies could be worse than symmetric ones when d > 18.
We prove generic versions of the no-cloning and no-broadcasting theorems, applicable to essentially {\em any} non-classical finite-dimensional probabilistic model that satisfies a no-signaling criterion. This includes quantum theory as well as models supporting ``super-quantum'' correlations that violate the Bell inequalities to a larger extent than quantum theory. The proof of our no-broadcasting theorem is significantly more natural and more self-contained than others we have seen: we show that a set of states is broadcastable if, and only if, it is contained in a simplex whose vertices are cloneable, and therefore distinguishable by a single measurement. This necessary and sufficient condition generalizes the quantum requirement that a broadcastable set of states commute.
Quantum coherence has recently emerged as a key candidate for use as a resource in various quantum information processing tasks. Therefore, it is of utmost importance to explore the possibility of creating a greater number of coherent states from an existing coherent pair. In other words, we start with an initial incoherent pair and induce coherence via quantum cloning. More specifically, we start with a genuinely incoherent state that remains incoherent with the change of basis and make it a coherent state at the end. This process is known as broadcasting of coherence via cloning, which can either be optimal or nonoptimal. Interestingly, in this work, we are able to give a method by which we can introduce coherence in the genuinely incoherent state. We use the computational basis representation of the most general two-qubit mixed state, shared between Alice and Bob, as the input state for the universal symmetric optimal Buzek-Hillery cloner. First of all we show that it is impossible to ensure optimal broadcast of coherence. Second, in the case of nonoptimal broadcasting, we show that the coherence introduced in the output states of the cloner will always be lesser than the initial coherence of the input state. Finally, we take the examples of statistical mixture of the most coherent state and most incoherent state, and then the Bell-diagonal states, to obtain their respective ranges of nonoptimal broadcasting in terms of their input state parameters.
It is well known that quantum theory forbids the exact copying of an unknown quantum state. Therefore in broadcasting of classical information by a quantum channel an additional contribution to the error in the decoding is expected. We consider the optimal copying transformation which is adapted to classical information transmission by two linearly independent quantum states, and show that there is no additional contribution to the error. Instead the clones are correlated, and this breaks their usefulness: The entanglement increases with non-orthogonality of the states. The capacity of the corresponding quantum channel is considered also.
We consider transmission of classical information through a quantum channel from one sender to several receivers. In contrast to quantum information, classical information carried by two non-orthogonal states of a quantum system can be cloned exactly. This restricts a naive extrapolation of the standard no-cloning theorem. The optimal cloning induces entanglement between clones which increases with non-orthogonality of the states. The capacity of the corresponding quantum channel is considered. As a byproduct of our results, we obtain that for an entangled system with an unknown state there is no transformation which removes the entanglement totally without affecting the corresponding marginal density matrices.