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arXiv:2502.18264 · Quantum

Indefinite Time Directed Quantum Metrology

Gaurang Agrawal, Pritam Halder, Aditi SenDe

Published Jul 3, 2025• 5 citations
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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/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.

DOI
10.22331/q-2025-07-03-1785

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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.
In quantum metrology, adaptive and causal-superposition strategies are proven to be beneficial over parallel schemes for a finite number of channel uses, but their advantages disappear in the asymptotic limit. We show that quantum operations with indefinite time direction, specifically, time-flip (TF)-assisted strategies, referred to as indefinite time-directed metrology (ITDM), can overcome this asymptotic equivalence. Using semidefinite programming, we rigorously demonstrate that TF-assisted protocols can achieve quantum Fisher information (QFI) strictly exceeding the maximum value attainable by parallel, adaptive, and causal-superposition strategies, both for finite and asymptotically many channel uses. Moreover, we identify a class of Pauli noise channels for which ITDM achieves Heisenberg scaling, while all parallel, adaptive, and causal-superposition strategies remain restricted to standard scaling. We call this phenomenon as metrological activation. Interestingly, this activation can be used to exhibit that the quantum time-flip and transposition supermaps cannot be simulated by conventional quantum circuits or causal-superposition strategies using any finite number of channel queries, thereby establishing indefinite time direction as a genuine resource for quantum metrology.
The precision limit in quantum metrology, quantified by the root-mean-square error of parameter estimation, is conventionally expected to improve at most linearly with the total interrogation time T and with the number N of queried quantum gates. Although several metrological schemes have been shown to achieve precision scaling faster than linear in T and N, they typically rely on unbounded probe-side information resources, usually qualified by an increasingly large variance of the parameter generator. This requirement complicates the interpretation of the resulting scaling advantage and poses substantial technical challenges. In this work, we employ an indefinite-time-direction encoding process to achieve a nonlinear-scaling enhancement of the precision limit. Rather than relying on increasingly informative probe states, our method converts controllable noncommuting encoding operations into metrological gain. Experimentally, we implement this protocol for angular-rotation measurement in a quantum optical system and demonstrate a nonlinear-scaling improvement in practical precision without using probe-side information resources. These results establish a practical framework for surpassing conventional linear-scaling precision limits in quantum metrology and provide new insights into precision enhancement in realistic quantum metrological and sensing applications.
Quantum metrology with indefinite causal order (ICO) has attracted intense interest due to its potential to surpass the limitations of conventional fixed-order strategies. A key open question is whether ICO can fundamentally enhance asymptotic precision scaling. In this work, we bridge this gap for the estimation of a single parameter encoded in NNN identical uses of a finite-dimensional quantum channel. We first establish a universal Heisenberg-scaling upper bound for the full general ICO process-matrix class and show that for unitary channels its optimal quantum Fisher information (QFI) coincides exactly with that of parallel strategies. For noisy channels, a structurally refined bound shows that channels restricted to the standard quantum limit (SQL) under parallel strategies remain SQL-limited under general ICO strategies. Most significantly, an asymptotically tight (AT) bound is derived to close the remaining possibility of an asymptotic ICO advantage by showing that general ICO and optimal parallel strategies have exactly the same leading QFI coefficient in both the SQL and Heisenberg regimes. For the operationally motivated class of quantum circuits with quantum control of causal order, we further obtain an iterative constraint on finite-query precision whose asymptotic limit agrees with that of the AT bound. Our results clarify the ultimate role of indefinite causality as a metrological resource for quantum channel estimation.
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.
Quantum metrology achieves Heisenberg-limited precision in single-parameter estimation, but its multiparameter extension is fundamentally constrained by both parameter-encoding and measurement incompatibility. Noncommuting signal generators may cause incompatible parameter-encoding, preventing the quantum Fisher information matrix from simultaneously achieving the Heisenberg scale for all parameters. Due to incompatible optimal measurements, the classical Fisher information matrix represents the practical attainable precision. Here, we introduce a multiparameter metrology framework based on indefinite evolution (IE), in which different control operations and signal reversal are placed in a coherent superposition. For a single-qubit probe with mutually orthogonal signal generators, IE enables compatible parameter encoding and optimal measurement without the signal reversal. For parallel generators, where only signal reversal realized by its generator is available, IE can achieve the same performance. We further extend this mechanism to noisy, many-body, and high-dimensional probes, and establish general conditions for achieving the simultaneous Heisenberg-limit. In contrast, definite evolution cannot achieve the same performance under compatible optimal measurements, even when signal reversal is available. Our results identify IE as an operational resource for overcoming multiparameter incompatibility and open a route toward attainable Heisenberg-limited sensing in interferometric platforms.
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 N2{N}^{2}N2 with respect to the probe size NNN. 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.