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arXiv:1604.00313 · Physical Review A

Assessing the significance of fidelity as a figure of merit in quantum state reconstruction of discrete and continuous-variable systems

Antonio Mandarino, Matteo Bina, Carmen Porto, S. Cialdi, Stefano Olivares, Matteo G. A. Paris

Published Jun 20, 2016• 34 citations
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Abstract

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.

DOI
10.1103/physreva.93.062118

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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.
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 address the experimental determination of entropic quantum discord for systems made of a pair of polarization qubits. We compare results from full and partial tomography and found that the two determinations are statistically compatible, with partial tomography leading to a smaller value of discord for depolarized states. Despite the fact that our states are well described, in terms of fidelity, by families of depolarized or phase-damped states, their entropic discord may be largely different from that predicted for these classes of states, such that no reliable estimation procedure beyond tomography may be effectively implemented. Our results, together with the lack of an analytic formula for the entropic discord of a generic two-qubit state, demonstrate that the estimation of quantum discord is an intrinsically noisy procedure. Besides, we question the use of fidelity as a figure of merit to assess quantum correlations.
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 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 quantum-to-classical transition of a quantum state is a topic of great interest in fundamental and practical aspects. A coarse-graining in quantum measurement has recently been suggested as its possible account in addition to the usual decoherence model. We here investigate the reconstruction of a Gaussian state (single mode and two modes) by coarse-grained homodyne measurements. To this aim, we employ two methods, the direct reconstruction of the covariance matrix and the maximum likelihood estimation (MLE), respectively, and examine the reconstructed state under each scheme compared to the state interacting with a Gaussian (squeezed thermal) reservoir. We clearly demonstrate that the coarse-graining model, though applied equally to all quadrature amplitudes, is not compatible with the decoherence model by a thermal (phase-insensitive) reservoir. Furthermore, we compare the performance of the direct reconstruction and the MLE methods by investigating the fidelity and the nonclassicality of the reconstructed states and show that the MLE method can generally yield a more reliable reconstruction, particularly without information on a reference frame (phase of input state).