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arXiv:1509.04330 · New Journal of Physics

Building versatile bipartite probes for quantum metrology

Alessandro Farace, Antonella De Pasquale, Gerardo Adesso, Vittorio Giovannetti

Published Jan 25, 2016• 18 citations
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Abstract

We consider bipartite systems as versatile probes for the estimation of transformations acting locally on one of the subsystems. We investigate what resources are required for the probes to offer a guaranteed level of metrological performance, when the latter is averaged over specific sets of local transformations. We quantify such a performance via the average skew information (AvSk), a convex quantity which we compute in closed form for bipartite states of arbitrary dimensions, and which is shown to be strongly dependent on the degree of local purity of the probes. Our analysis contrasts and complements the recent series of studies focused on the minimum, rather than the average, performance of bipartite probes in local estimation tasks, which was instead determined by quantum correlations other than entanglement. We provide explicit prescriptions to characterize the most reliable states maximizing the AvSk, and elucidate the role of state purity, separability and correlations in the classification of optimal probes. Our results can help in the identification of useful resources for sensing, estimation and discrimination applications when complete knowledge of the interaction mechanism realizing the local transformation is unavailable, and access to pure entangled probes is technologically limited.

DOI
10.1088/1367-2630/18/1/013049

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We consider bipartite systems as versatile probes for the estimation of transformations acting locally on one of the subsystems. We investigate what resources are required for the probes to offer a guaranteed level of metrological performance, when the latter is averaged over specific sets of local transformations. We quantify such a performance via the average skew information (AvSk), a convex quantity which we compute in closed form for bipartite states of arbitrary dimensions, and which is shown to be strongly dependent on the degree of local purity of the probes. Our analysis contrasts and complements the recent series of studies focused on the minimum, rather than the average, performance of bipartite probes in local estimation tasks, which was instead determined by quantum correlations other than entanglement. We provide explicit prescriptions to characterize the most reliable states maximizing the AvSk, and elucidate the role of state purity, separability and correlations in the classification of optimal probes. Our results can help in the identification of useful resources for sensing, estimation and discrimination applications when complete knowledge of the interaction mechanism realizing the local transformation is unavailable, and access to pure entangled probes is technologically limited.
Quantum mechanics predicts that measurements of incompatible observables carry a minimum uncertainty which is independent of technical deficiencies of the measurement apparatus or incomplete knowledge of the state of the system. Nothing yet seems to prevent a single physical quantity, such as one spin component, from being measured with arbitrary precision. Here, we show that an intrinsic quantum uncertainty on a single observable is ineludible in a number of physical situations. When revealed on local observables of a bipartite system, such uncertainty defines an entire class of bona fide measures of nonclassical correlations. For the case of 2 × d systems, we find that a unique measure is defined, which we evaluate in closed form. We then discuss the role that these correlations, which are of the "discord" type, can play in the context of quantum metrology. We show in particular that the amount of discord present in a bipartite mixed probe state guarantees a minimum precision, as quantified by the quantum Fisher information, in the optimal phase estimation protocol.
For pure two-qudit probes used in a single fixed setting to estimate a common SU(d) transformation, maximal total Fisher sensitivity and full local identifiability are mutually exclusive. We derive exact identities that determine the trace of the quantum Fisher information matrix from two measurable probe properties: exchange symmetry and collective polarization. Every maximizer is exchange symmetric and maximally entangled. Its Fisher matrix has equal sensitivity in every visible direction, while d(d−1)/2d(d-1)/2d(d−1)/2 generator directions are blind. More generally, weak compatibility, defined by vanishing mean generator commutators, forces at least ⌊d/2⌋\lfloor d/2\rfloor⌊d/2⌋ blind directions for every pure bipartite probe. A probe-defined antiunitary invariant classifies the complete Fisher spectrum at maximal entanglement, and approaches to the optimum that preserve exchange symmetry have a divergent Holevo cost. The bipartite obstruction is sharp in particle number: for N≥3N\geq 3N≥3, generalized GHZ probes can satisfy weak compatibility, attain the corresponding NNN-partite Fisher trace bound, and retain full local identifiability. Hence, Fisher sensitivity, weak compatibility, identifiability, and attainable precision are distinct resources in quantum metrology with noncommuting generators.
Quantum metrology exploits quantum mechanical laws to improve the precision in estimating technologically relevant parameters such as phase, frequency, or magnetic fields.Probe states are usually tailored to the particular dynamics whose parameters are being estimated.Here we consider a novel framework where quantum estimation is performed in an interferometric configuration, using bipartite probe states prepared when only the spectrum of the generating Hamiltonian is known.We introduce a figure of merit for the scheme, given by the worst-case precision over all suitable Hamiltonians, and prove that it amounts exactly to a computable measure of discord-type quantum correlations for the input probe.We complement our theoretical results with a metrology experiment, realized in a highly controllable room-temperature nuclear magnetic resonance setup, which provides a proof-of-concept demonstration for the usefulness of discord in sensing applications.Discordant probes are shown to guarantee a nonzero phase sensitivity for all the chosen generating Hamiltonians, while classically correlated probes are unable to accomplish the estimation in a worst-case setting.This work establishes a rigorous and direct operational interpretation for general quantum correlations, shedding light on their potential for quantum technology.
In the context of quantifying entanglement we study those functions of a multipartite state which do not increase under the set of local transformations. A mathematical characterization of these monotone magnitudes is presented. They are then related to optimal strategies of conversion of shared states. More detailed results are presented for pure states of bipartite systems. It is shown that more than one measure is required simultaneously in order to quantify completely the non-local resources contained in a bipartite pure state, while examining how this fact does not hold in the so-called asymptotic limit. Finally, monotonicity under local transformations is proposed as the only natural requirement for measures of entanglement.
Quantum metrology offers a quadratic advantage over classical approaches to parameter estimation problems by utilising entanglement and nonclassicality. However, the hurdle of actually implementing the necessary quantum probe states and measurements, which vary drastically for different metrological scenarios, is usually not taken into account. We show that for a wide range of tasks in metrology, 2D cluster states (a particular family of states useful for measurement-based quantum computation) can serve as flexible resources that allow one to efficiently prepare any required state for sensing, and perform appropriate (entangled) measurements using only single qubit operations. Crucially, the overhead in the number of qubits is less than quadratic, thus preserving the quantum scaling advantage. This is ensured by using a compression to a logarithmically sized space that contains all relevant information for sensing. We specifically demonstrate how our method can be used to obtain optimal scaling for phase and frequency estimation in local estimation problems, as well as for the Bayesian equivalents with Gaussian priors of varying widths. Furthermore, we show that in the paradigmatic case of local phase estimation 1D cluster states are sufficient for optimal state preparation and measurement.
Quantum metrology relies on observables that maximize information about the parameter of interest, but are often hard to access. We develop a protocol that combines Pauli moments estimated from a fixed ensemble of randomized local measurements into an optimized method-of-moments estimator. For local Pauli classical shadows, we derive the shadow information which determines estimator variance and sample-complexity guarantees. Identifying the separable-state sensitivity limit, we show that entanglement can lead to a constant-factor sensitivity enhancement, as illustrated with one-axis-twisted probes. This protocol allows for efficient circuit-based metrology without choosing an observable.