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 $N$ 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.
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7 pages, 2 figures; Supplemental Material included (23 pages, 2 figures)
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 N 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.
Nature imposes fundamental limits on how precisely we can probe the physical world. Quantum metrology is the study of achieving higher precision and accuracy in estimating the physical parameters by harnessing quantum resources. It is considered one of the most promising quantum technologies, offering significant advantages over classical ones in both near‑term and long‑term applications. However, existing theoretical research in quantum metrology often falls short of providing a unified and rigorous understanding of the ultimate precision under realistic experimental limitations on available quantum resources. In this thesis, we conduct a systematic investigation of the optimal precision limits in quantum metrology and identify the metrological strategies that achieve optimality under specified resource constraints. We primarily focus on the single-parameter estimation of quantum processes, a cornerstone task in quantum metrology. We first establish a general framework of strictly optimal quantum metrology using semidefinite programming within the formalism of higher-order quantum operations. This framework reveals a strict hierarchy for the performance of different families of strategies---parallel, sequential, and indefinite-causal-order ones---each subject to specific causal constraints. Our results rigorously demonstrate a quantum metrological advantage of indefinite causal order in finite-dimensional systems, which has recently aroused rapidly growing interest. In realistic scenarios, a fully optimized quantum metrology protocol may be experimentally demanding due to the complexity of state preparation and control. Tailored to practical constraints on the allowed ancilla and control operations, we propose a highly efficient tensor network algorithm to optimize the probe state and a long sequence of control for quantum metrology. When the number of control steps tends to infinity, we develop an asymptotic theory of quantum metrology for achieving the ultimate precision limit---Heisenberg limit---without ancilla. This contrasts with the traditional approach using quantum error correction for metrology that typically requires noiseless and controllable ancilla. Our results pave the way for exploiting the full potential of quantum resources in metrology. In the near term, our methods enable high-precision estimation with limited interaction with the unknown process using experimentally feasible control techniques. In the long term, they lay the foundation for scalable and fault-tolerant quantum metrology.
Quantum resources can, in principle, enable Heisenberg-limited (HL) sensing, yet no-go theorems imply that HL scaling is generically unattainable in realistic noisy devices. While quantum error correction (QEC) can suppress noise, its use in quantum sensing is constrained by stringent requirements, including prior noise characterization, restrictive signal-noise compatibility conditions, and measurement-based syndrome extraction with global control. Here we introduce an ICO-based QEC protocol, providing the first application of indefinite causal order (ICO) to QEC. By coherently placing auxiliary controls and noisy evolution in an indefinite causal order, the resulting noncommutative interference enables an auxiliary system to herald and correct errors in real time, thereby circumventing the limitations of conventional QEC and restoring HL scaling. We rigorously establish the protocol for single- and multi-noise scenarios and demonstrate its performance in single-qubit, many-body, and continuous-variable platforms. We further identify regimes in which error correction can be implemented entirely by unitary control, without measurements. Our results reveal ICO as a powerful resource for metrological QEC and provide a broadly applicable framework for noise-resilient quantum information processing.
The quantum switch, the canonical example of a process with indefinite causal order, has been claimed to provide various advantages over processes with definite causal orders for some particular tasks in the field of quantum metrology. In this work, we argue that some of these advantages in fact do not hold if a fairer comparison is made. To this end, we consider a framework that allows for a proper comparison between the performance, quantified by the quantum Fisher information, of different classes of indefinite causal order processes and that of causal strategies on a given metrological task. More generally, by considering the recently proposed classes of circuits with classical or quantum control of the causal order, we come up with different examples where processes with indefinite causal order offer (or not) an advantage over processes with definite causal order, qualifying the interest of indefinite causal order regarding quantum metrology. As it turns out, for a range of examples, the class of quantum circuits with quantum control of causal order, which are known to be physically realizable, is shown to provide a strict advantage over causally ordered quantum circuits as well as over the class of quantum circuits with causal superposition. Thus, the consideration of this class provides new evidence that indefinite causal order strategies can strictly outperform definite causal order strategies in quantum metrology.
Indefinite causal order is known to offer enhancements in quantum metrology, notably including an unbounded advantage in the measurement of a geometric phase of a harmonic oscillator. This advantage, however, is specific to the infinite dimensional setting, and its finite dimensional analogue remains elusive, with recent findings suggesting that advantages in finite dimensions may be fundamentally limited to bounded constant factors. Here we show that, in fact, arbitrarily large advantages arise for finite dimensional systems in the finite sample regime. Specifically, we establish an unbounded separation between definite and indefinite causal order in the estimation of a geometric phase associated to two sets of N displacements generated by discrete position and momentum operators on a d-dimensional quantum system: for any given constant R, there exist values of N and d=Ω(N2) such that a strategy with indefinite order uses an initial probe with R times less energy than the probe required by every strategy with definite order achieving the same mean squared error, whenever the number of measurement shots ν is bounded as ν=O(exp(πd/16)/poly(d)). In other words, indefinite order offers an energy saving that grows arbitrarily large with the parameters of the problem. To prove this result, we establish an approximate Weyl relation for discrete Gaussian wavepackets, which is of independent technical interest.
We investigate the advantage of coherent superposition of two different coded channels in quantum metrology. In a continuous variable system, we show that the Heisenberg limit 1/ N can be beaten by the coherent superposition without the help of indefinite causal order. And in parameter estimation, we demonstrate that the strategy with the coherent superposition can perform better than the strategy with quantum switch which can generate indefinite causal order. We analytically obtain the general form of estimation precision in terms of the quantum Fisher information and further prove that the nonlinear Hamiltonian can improve the estimation precision and make the measurement uncertainty scale as 1/ N m for m ≥ 2. Our results can help to construct a high-precision measurement equipment, which can be applied to the detection of coupling strength and the test of time dilation and the modification of the canonical commutation relation.
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.