Indefinite Time Directed Quantum Metrology

Gaurang Agrawal; Pritam Halder; Aditi SenDe

doi:10.22331/q-2025-07-03-1785

Indefinite Time Directed Quantum Metrology

Gaurang Agrawal^1,2, Pritam Halder², and Aditi SenDe²

¹Indian Institute of Science Education and Research, Homi Bhabha Rd, Pashan, Pune 411 008, India
²Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute, Chhatnag Road, Jhunsi, Allahabad - 211019, India

Published:	2025-07-03, volume 9, page 1785
Editor:	Angelo Carollo
Eprint:	arXiv:2502.18264v3
Doi:	https://doi.org/10.22331/q-2025-07-03-1785
Citation:	Quantum 9, 1785 (2025).

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Updated version: The authors have uploaded version v4 of this work to the arXiv which may contain updates or corrections not contained in the published version v3. The authors left the following comment on the arXiv:

17+11 pages, 4+4 figures

Abstract

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 $(1/{\sqrt{N}})$, 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.

Featured image: Indefinite time ordered metrology (ITDM) scheme — the encoding of the parameter to be estimated is performed through time-flip operation (as illustrated in the right side of the image) on input states. Without an initial entangled state, ITDM can attain Heisenberg scaling in quantum metrology with product probe states.

Popular summary

Quantum metrology exploits quantum phenomena to measure physical parameters with exceptional accuracy. In particular, entanglement has long been considered an indispensable resource for achieving higher precision than that can be achieved without entanglement. However, generating multipartite entangled states for a large number of qubits poses significant experimental challenges. Recently, there has been a surge of interest in finding other quantum resources that can surpass classical limits in estimation theory.

Quantum theory, in contrast to classical ones, allows objects to be in a superposition, e.g., a superposition of causal orders, which exhibits benefits across various domains of quantum information processing and thermodynamic tasks. Recent studies have revealed that physical processes with arbitrary superpositions of the forward and backward time directions, named indefinite time-directed processes, offer quantum advantage in certain quantum games and communication scenarios, along with their experimental validations.

In this work, we exhibit that harnessing indefinite time direction, initial product probe states with all the particles aligned in the same direction, can provide quantum advantage in metrology by achieving Heisenberg scaling, thereby showing entanglement to be unnecessary. We also present simple schemes, involving single-qubit decoding measurements in a fixed basis to accomplish the same scaling. Additionally, our protocol also outperforms existing methods in realistic noisy settings, highlighting its practical potential for next-generation quantum sensing technologies.

► BibTeX data

@article{Agrawal2025indefinitetime,
  doi = {10.22331/q-2025-07-03-1785},
  url = {https://doi.org/10.22331/q-2025-07-03-1785},
  title = {Indefinite {T}ime {D}irected {Q}uantum {M}etrology},
  author = {Agrawal, Gaurang and Halder, Pritam and SenDe, Aditi},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1785},
  month = jul,
  year = {2025}
}

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[4] Hang Xu, Xiaoyang Deng, Ze Zheng, Tailong Xiao, and Guihua Zeng, "Noise-Resilient Heisenberg-Limited Quantum Sensing via Indefinite-Causal-Order Error Correction", Physical Review Letters 137 2, 020801 (2026).

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