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arXiv:1806.02355 · Physical Review A

Quantum-limited Euler angle measurements using anticoherent states

Aaron Z. Goldberg, Daniel F. V. James

Published Sep 20, 2018• 52 citations
arXivHTMLPDF

Abstract

Many protocols require precise rotation measurement. Here we present a general class of states that surpass the shot-noise limit for measuring rotation around arbitrary axes. We then derive a quantum Cram\'er-Rao bound for simultaneously estimating all three parameters of a rotation (e.g., the Euler angles) and discuss states that achieve Heisenberg-limited sensitivities for all parameters; the bound is saturated by ``anticoherent'' states [Zimba, Electron. J. Theor. Phys. 3, 143 (2006)] (we are reluctant to use ``anticoherent'' to describe the states, but the name has become commonplace over the last decade). Anticoherent states have garnered much attention in recent years, and we elucidate a geometrical technique for finding new examples of such states. Finally, we discuss the potential for divergences in multiparameter estimation due to singularities in spherical coordinate systems. Our results are useful for a variety of quantum metrology and quantum communication applications.

DOI
10.1103/physreva.98.032113

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Many protocols require precise rotation measurement. Here we present a general class of states that surpass the shot-noise limit for measuring rotation around arbitrary axes. We then derive a quantum Cram'er-Rao bound for simultaneously estimating all three parameters of a rotation (e.g., the Euler angles) and discuss states that achieve Heisenberg-limited sensitivities for all parameters; the bound is saturated by anticoherent'' states [Zimba, Electron. J. Theor. Phys. 3, 143 (2006)] (we are reluctant to use anticoherent'' to describe the states, but the name has become commonplace over the last decade). Anticoherent states have garnered much attention in recent years, and we elucidate a geometrical technique for finding new examples of such states. Finally, we discuss the potential for divergences in multiparameter estimation due to singularities in spherical coordinate systems. Our results are useful for a variety of quantum metrology and quantum communication applications.
The efficiency of a quantum metrology protocol can be significantly diminished by the interaction of the system with its environment, leading to a loss of purity and, as a result, a mixed state for the probing system. An example is the measurement of a magnetic field through the rotation of a spin that is subject to decoherence due to its coupling to a surrounding spin or bosonic bath. In this work we define mixed optimal quantum rotosensors (OQRs) as mixed spin-jjj states that achieve maximum sensitivity to estimate infinitesimal rotations, when the rotation axis is unknown. We study two scenarios, where the probe states saturate the averaged fidelity or the averaged quantum Cram\'er-Rao bound, the latter giving the ultimate sensitivity. We find that mixed OQRs can achieve sensitivity equal to that of pure states and are obtained by mixing states from linear subspaces of anticoherent states. We present several examples of mixed OQRs and their associated anticoherent subspaces. We also show that OQRs maximize entanglement in a specific sense, preserving the known relation between entanglement and optimal rotation sensitivity for pure states, even in the context of mixed states. Our results highlight the interconnection between quantum metrology of rotations, anticoherence, and entanglement in mixed spin states.
Abstract Quantum metrology aims at delivering new quantum-mechanical improvement to technologies of parameter estimations with precision bounded by the quantum Cramér-Rao bound. The currently used quantum Cramér-Rao bound was established with measurements of observables restricted to be Hermitian. This constrains the bound and limits the precision of parameter estimation. In this paper, we lift the constraint and derive a previously unknown quantum Cramér-Rao bound. We find that the new bound can reach arbitrary small value with mixed states and it breaks the Heisenberg limit in some cases. We construct a setup to measure non-Hermitian operators and discuss the saturation of the present bound. Two examples—the phase estimation with Greenberger-Horne-Zeilinger states of trapped ions and the adiabatic quantum parameter estimation with the nuclear magnetic resonance—are employed to demonstrate the theory. The present study might open a new research direction—non-Hermitian quantum metrology.
We identify precision limits for the simultaneous estimation of multiple parameters in multimode interferometers. Quantum strategies to enhance the multiparameter sensitivity are based on entanglement among particles, modes, or combining both. The maximum attainable sensitivity of particle-separable states defines the multiparameter shot-noise limit, which can be surpassed without mode entanglement. Further enhancements up to the multiparameter Heisenberg limit are possible by adding mode entanglement. Optimal strategies that saturate the precision bounds are provided.
We present a novel strategy for obtaining optimal probe states and measurement schemes in a class of noiseless multiparameter estimation problems with symmetry among the generators. The key to the framework is the introduction of a set of quantum metrology conditions, analogous to the quantum error correction conditions of Knill and Laflamme, which are utilized to identify probe states that saturate the multiparameter quantum Cramér-Rao bound. Similar to finding two-dimensional irreps for encoding a logical qubit in error correction, we identify trivial irreps of finite groups that guarantee the satisfaction of the quantum metrology conditions. To demonstrate our framework, we analyze the SU(2) estimation with symmetric states in which three parameters define a global rotation of an ensemble of NNN qubits. For even NNN, we find that tetrahedral symmetry and, with fine-tuning, S3S_{3}S3​ symmetry, are minimal symmetry groups providing optimal probe states for SU(2) estimation, but that the quantum metrology conditions can also be satisfied in an entanglement-assisted setting by using a maximally entangled state of two spin-N/2N/2N/2 representations for any NNN. By extending the multiparameter method of moments to non-commuting observables, we use the quantum metrology conditions to construct a measurement scheme that saturates the multiparameter quantum Cramér-Rao bound for small rotation angles.
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