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arXiv:1811.10554 · Physical Review A

Ancilla-assisted schemes are beneficial for Gaussian state phase estimation

Zixin Huang, Chiara Macchiavello, Lorenzo Maccone, Pieter Kok

Published Jan 29, 2020• 5 citations
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Abstract

We study interferometry with Gaussian states and show that an ancilla-assisted scheme outperforms coherent state interferometry for all levels of loss. We also compare the ancilla-assisted scheme to other interferometric schemes involving squeezing, and show that it has the most advantage in the high-loss, high photon-number regime. In fact, in the presence of high loss, it outperforms many other strategies proposed to date. We find the optimal measurement observable for each scheme discussed. We also find that, with the appropriate measurement, the achievable precision of the proposal by Caves [Phys. Rev. D 23, 1693 (1981)] can be improved upon, and is less vulnerable to losses than previously thought.

DOI
10.1103/physreva.101.012124

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We study interferometry with Gaussian states and show that an ancilla-assisted scheme outperforms coherent state interferometry for all levels of loss. We also compare the ancilla-assisted scheme to other interferometric schemes involving squeezing, and show that it has the most advantage in the high-loss, high photon-number regime. In fact, in the presence of high loss, it outperforms many other strategies proposed to date. We find the optimal measurement observable for each scheme discussed. We also find that, with the appropriate measurement, the achievable precision of the proposal by Caves [Phys. Rev. D 23, 1693 (1981)] can be improved upon, and is less vulnerable to losses than previously thought.
We address high-precision measurements by active and passive interferometric schemes based on Gaussian states and operations. In particular, we look for the best states to be injected into their ports according to the quantum Cramer–Rao bound, i.e., maximizing the quantum Fisher information over all the involved parameters, given a constraint on the overall mean number of photons entering into the interferometer. We found that for passive interferometers involving only beam splitters, the optimal input leading to Heisenberg scaling is a pair of identical squeezed-coherent states with at most one-third of the total energy employed in squeezing. For active interferometers involving optical amplifiers, input coherent signals are enough to achieve Heisenberg scaling, given an optimal value of the amplification gain. For passive schemes our results clarify the role of squeezing in improving both the reference phase and the signal phase of an interferometer.
We give a detailed discussion of optimal quantum states for optical two-mode interferometry in the presence of photon losses. We derive analytical formulae for the precision of phase estimation obtainable using quantum states of light with a definite photon number and prove that maximization of the precision is a convex optimization problem. The corresponding optimal precision, i.e., the lowest possible uncertainty, is shown to beat the standard quantum limit thus outperforming classical interferometry. Furthermore, we discuss more general inputs: states with indefinite photon number and states with photons distributed between distinguishable time bins. We prove that neither of these is helpful in improving phase estimation precision.
We study the precise phase estimation using squeezed states in the presence of photon loss. Our exact quantum Fisher information calculation shows significant enhancement even though the loss is very large and sets a benchmark for experimental realization. We show that if we blindly use the existing parity measurement scheme [Anisimov et al., Phys. Rev. Lett. 104, 103602 (2010)] for the ideal case, the result will be even worse than the classical case given very small loss. Using our formulas, we can optimize the measurement result by making loss-dependent phase shift and choosing appropriate squeezed states (average photon number of squeezed states).
This thesis explores various protocols which can be applied to quantum metrology, such that precision of the measurement can beat the best classical approaches. We first demonstrate that ancilla-assisted schemes can improve phase estimation in the presence of noise, and we propose an experiment to demonstrate these effects. Given that the use of ancillae helps, it is no surprise that there are noisy channels for which the optimal strategy may depend on the noise level. We show that there is a non-trivial crossover between the ancilla-assisted strategy and the parallel-entangled strategy for Pauli noise channels and the amplitude damping channel. We see that under certain circumstances, the environment cannot access the parameter of interest. We then integrated techniques from quantum communication, to derive a general framework that renders quantum metrology protocols into cryptographic ones. Here the protocol is unconditionally secure if the trusted parties wish only to achieve the standard quantum limit. If Heisenberg-limited precision is to be achieved, they sacrifice some security, where adversaries can access information, but only at the risk of getting caught at it (cheat-sensitivity). In many scenarios, the quantum Fisher information is achieved for only one value of the phase. In a particular phase estimation scheme that measures the parity at the output of a Mach-Zehnder interferometer where the input is a two-mode squeezed vacuum, this problem is particularly imminent. We apply an adaptive technique to resolve this issue, showing that one can achieve precisions substantially below the shot noise limit. The key to a quadratically enhanced improvement in quantum metrology is entanglement. Finally, we demonstrate in a photonic system how high-dimensional entanglement can be certified, and provide an alternate interpretation of entanglement from an information-theoretic perspective.
For phase estimation using qubits, we show that for some noise channels, the optimal entanglement-assisted strategy depends on the noise level. We note that there is a nontrivial crossover between the parallel-entangled strategy and the ancilla-assisted strategy: in the former the probes are all entangled; in the latter the probes are entangled with a noiseless ancilla but not among themselves. The transition can be explained by the fact that separable states are more robust against noise and therefore are optimal in the high-noise limit, but they are in turn outperformed by ancilla-assisted ones.
The sensitivity in optical interferometry is strongly affected by losses during the signal propagation or at the detection stage. The optimal quantum states of the probing signals in the presence of loss were recently found. However, in many cases of practical interest, their associated accuracy is worse than the one obtainable without employing quantum resources (e.g., entanglement and squeezing) but neglecting the detector's loss. Here, we detail an experiment that can reach the latter even in the presence of imperfect detectors: it employs a phase-sensitive amplification of the signals after the phase sensing, before the detection. We experimentally demonstrated the feasibility of a phase estimation experiment able to reach its optimal working regime. Since our method uses coherent states as input signals, it is a practical technique that can be used for high-sensitivity interferometry and, in contrast to the optimal strategies, does not require one to have an exact characterization of the loss beforehand.