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arXiv:1309.5325 · Physical Review A

Drawbacks of the use of fidelity to assess quantum resources

Matteo Bina, Antonio Mandarino, Stefano Olivares, Matteo G. A. Paris

Published Jan 6, 2014• 46 citations
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Abstract

Fidelity is a figure of merit widely employed in quantum technology in order to quantify similarity between quantum states and, in turn, to assess quantum resources or reconstruction techniques. Fidelities higher than, say, 0.9 or 0.99, are usually considered as a piece of evidence to say that two states are very close in the Hilbert space. On the other hand, on the basis of several examples for qubits and continuous variable systems, we show that such high fidelities may be achieved by pairs of states with considerably different physical properties, including separable and entangled states or classical and nonclassical ones. We conclude that fidelity as a tool to assess quantum resources should be employed with caution, possibly combined with additional constraints restricting the pool of achievable states, or only as a mere summary of a full tomographic reconstruction.

DOI
10.1103/physreva.89.012305

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Fidelity is a figure of merit widely employed in quantum technology in order to quantify similarity between quantum states and, in turn, to assess quantum resources or reconstruction techniques. Fidelities higher than, say, 0.9 or 0.99, are usually considered as a piece of evidence to say that two states are very close in the Hilbert space. On the other hand, on the basis of several examples for qubits and continuous variable systems, we show that such high fidelities may be achieved by pairs of states with considerably different physical properties, including separable and entangled states or classical and nonclassical ones. We conclude that fidelity as a tool to assess quantum resources should be employed with caution, possibly combined with additional constraints restricting the pool of achievable states, or only as a mere summary of a full tomographic reconstruction.
We present examples of continuous variable (CV) states having high fidelity to a given target, say F > 0.9 or F > 0.99, and still showing striking differences in their physical properties, including classical and quantum states within the set, separable and entangled ones, or nearly Gaussian and strongly non-Gaussian ones. We also show that the phenomenon persists also when one imposes additional constraints on the energy or the squeezing fraction of the states, thus generally questioning the use of fidelity to assess properties of CV systems without a tomographic set of additional constraints.
We experimentally address the significance of fidelity as a figure of merit in quantum state reconstruction of discrete (DV) and continuous-variable (CV) quantum optical systems. In particular, we analyze the use of fidelity in quantum homodyne tomography of CV states and maximum-likelihood polarization tomography of DV ones, focusing attention on nonclassicality, entanglement, and quantum discord as a function of fidelity to a target state. Our findings show that high values of fidelity, despite well quantifying geometrical proximity in the Hilbert space, may be obtained for states displaying opposite physical properties, e.g., quantum or semiclassical features. In particular, we analyze in detail the quantum-to-classical transition for squeezed thermal states of a single-mode optical system and for Werner states of a two-photon polarization qubit system.
We present examples of continuous variable (CV) states having high fidelity to a given target, say F>0.9F > 0.9F>0.9 or F>0.99F > 0.99F>0.99, and still showing striking differences in their physical properties, including classical and quantum states within the set, separable and entangled ones, or nearly Gaussian and strongly non-Gaussian ones. We also show that the phenomenon persists also when one imposes additional constraints on the energy or the squeezing fraction of the states, thus generally questioning the use of fidelity to assess properties of CV systems.
The size of the Hilbert space for a multiqubit state scales exponentially with the number of constituent qubits. Often this leads to a similar exponential scaling of the experimental resources required to characterize the state. Contrary to this, we propose a physically motivated method for experimentally assessing the fidelity of an important class of entangled states known as cluster states. The proposed method always yields a lower bound of the fidelity with a number of measurement settings scaling only linearly with the system size, and is tailored to correctly account for the errors most likely to occur in experiments. For one-dimensional cluster states, the constructed fidelity measure is tight to lowest order in the error probability for experimentally realistic noise sources and thus closely matches the true fidelity. Furthermore, it is tight for the majority of higher-order errors, except for a small subset of certain nonlocal multiqubit errors irrelevant in typical experimental situations. The scheme also performs very well for higher-dimensional cluster states, assessing correctly the majority of experimentally relevant errors.
The thread of this thesis is an attempt to clarify some assumptions that sound reasonable but whose correctness has never been proved or demonstrated formally. The first question I consider is: Do two quantum states near in the Hilbert space have the same, or almost the same, physical properties? First of all we have to clarify with which kind of measure we would like to measure the distance between two states. I do consider a measure, called Fidelity, that acts from the Hilbert space of a bipartite system to [0; 1]. Fidelity is null when the two states are orthogonal while if it is 1 the two states coincides. I show that high fidelities may be achieved by pairs of states with considerably different physical properties, including separable and entangled states or classical and nonclassical ones. Therefore, fidelity alone cannot be used to asses the very quantum properties of two states,but a tomographic reconstruction of the state is required. Secondly I question: Does a good quantum thermometer could be build with a system showing phase transition? I employ a critical magnetic system, known as Lipkin-Meshkov-Glick model, I study it in contact with a reservoir with it has thermalized but whose temperature is an unknown temperature to be estimated. I show that the best precision for temperature estimation it is not at critical point as someone could suspect but nevertheless criticality is an unmatchable resource even in the quantum realm. Last question is: when two different probes spatially separated interact with a common or a separate bath? It is quite a common opinion that systems, two objects located far away one from each others, feel the effect of different environments. Dissipation and decoherence are the two main processes which a quantum system undergo when it is not isolated, but when it interacts with another system much bigger. I explore how two different systems put in contact with a common bath show decoherence with typical quantum interference leading to a non obvious definition of separate environments.
Applications of quantum technology often require fidelities to quantify performance. These provide a fundamental yardstick for the comparison of two quantum states. While this is straightforward in the case of pure states, it is much more subtle for the more general case of mixed quantum states often found in practice. A large number of different proposals exist. In this review, we summarize the required properties of a quantum fidelity measure, and compare them, to determine which properties each of the different measures has. We show that there are large classes of measures that satisfy all the required properties of a fidelity measure, just as there are many norms of Hilbert space operators, and many measures of entropy. We compare these fidelities, with detailed proofs of their properties. We also summarize briefly the applications of these measures in teleportation, quantum memories and quantum computers, quantum communications, and quantum phase-space simulations.