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arXiv:2404.12649 · Physical Review A

Qubit-assisted quantum metrology under a time-reversal strategy

Peng Chen, Jun Jing

Published Dec 30, 2024• 6 citations
arXivHTMLPDF

Abstract

We propose a quantum metrology protocol based on a two-step joint evolution of the probe system and an ancillary qubit and quantum measurement. With a proper initial state of the ancillary qubit and an optimized evolution time, the quantum Fisher information (QFI) about the phase parameter encoded in the probe system is found to be determined by the expectation value of the square of a phase generator, irrespective of the probe initial state. Consequently, even if the probe is prepared as a finite-temperature state, faraway from the so-called resource state, e.g., the squeezed spin state or the Greenberger-Horne-Zeilinger state in atomic systems, the QFI in our protocol can approach the Heisenberg scaling ${N}^{2}$ with respect to the probe size $N$. This quadratic scaling behavior shows robustness against the imperfections about the initial state of the ancillary qubit and the optimized evolution time of the whole system. Also it is not sensitive to the deviation in system parameters and the qubit decoherence. Using the time-reversal strategy and a single-shot projective measurement, the classical Fisher information (CFI) in our metrology protocol is saturated with its quantum counterpart. Our work thus provides an economical method to reach the Heisenberg limit in metrology precision with no input of entanglement or squeezing.

DOI
10.1103/physreva.110.062425

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The Heisenberg scaling is an ultimate precision limit of parameter estimation allowed by the principles of quantum mechanics, with no counterpart in the classical realm, and has been a long-pursued goal in quantum metrology. It has been known that interactions between the probes can help reach the Heisenberg scaling without entanglement. In this paper, we show that interactions between the probes and the additional dimensions of an ancillary system may also increase the precision of parameter estimation to surpass the standard quantum limit and attain the Heisenberg scaling without entanglement, if the measurement scheme is properly designed. The quantum Fisher information exhibits periodic patterns over the evolution time, implying the existence of optimal time points for measurements that can maximize the quantum Fisher information. By implementing optimizations over the Hamiltonian, the initial states of the probes and the ancillary system, the interaction strength, and the time points for measurements, our protocol achieves the Heisenberg scaling for the parameter of the probe Hamiltonian, in terms of both evolution time and probe number. Our protocol features two aspects: (i) the Heisenberg scaling can be achieved by a product state of the probes and (ii) mere local measurement on the ancilla is sufficient, both of which reduce the quantum resources and the implementation complexity to achieve the Heisenberg scaling. The paper is concluded by the investigation of the effects of noise on this protocol.
We explore the performance of the metrology scheme by employing a quantum time flip during encoding, a specific case of processes with indefinite time direction, which we refer to as indefinite time directed metrology ( ITDM ). In the case of single parameter estimation of a unitary, we demonstrate that our protocol can achieve Heisenberg scaling (1/N) with product probe states, surpassing the standard quantum limit (1/N) , where N is the number of particles in the probe. We establish this by computing the quantum Fisher information ( QFI ) which is a lower bound on the root mean square error occurred during parameter estimation. Although we analytically prove the optimality of the symmetric product probe state in ITDM , entangled probe states produce a higher QFI than optimal product probes without enhancing scaling, highlighting the non-essentiality of entanglement. For phase estimation, we propose a single-qubit measurement on the control qubit that accomplishes near-optimal Fisher information and eventually reaches Heisenberg scaling. Our findings reveal the best orientation of product probe states in every pertinent situation, emphasizing its independence from the parameter to be estimated in the limiting case. Furthermore, we illustrate the benefits of ITDM in noisy metrology, outperforming existing techniques in some situations.