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.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.

quant-pharXiv:2609.05355Sep 7, 20260 citations

Probing the quantum depolarizing channel with mixed Indefinite causal order

Michael R. Frey

Indefinite causal ordering (ICO) refers to quantum channels arranged so that the path through them is a super- position of different possible paths. The ICO is called pure when the path superposition is a pure quantum state. Channel probing, in which probes in prepared quantum states are passed through copies of the channel to estimate one or more channel parameters, is aided by ICO. Specifically, pure ICO is known to increase the quantum Fisher information (QFI) in the processed probe state about the unknown parameter(s). We consider the case complementary to pure ICO in which the path state is maximally mixed for a given indefiniteness. Deriving the QFI under this condition by a new approach, we find that mixed ICO-assisted probing, while not as beneficial as pure ICO probing, still yields greater QFI than does the comparable scheme with definite causal order. The d-dimensional quantum depolarizing channel is used as the channel model for this study. While mixed ICO-assisted probing, like pure ICO-assisted probing, is generally advantageous for probing the depolarizing channel, mixed ICO's relative effectiveness decreases with probe dimension, just as for pure ICO-assisted probing, each being most effective for qubit probes.Indefinite causal ordering (ICO) refers to quantum channels arranged so that the path through them is a super- position of different possible paths. The ICO is called pure when the path superposition is a pure quantum state. Channel probing, in which probes in prepared quantum states are passed through copies of the channel to estimate one or more channel parameters, is aided by ICO. Specifically, pure ICO is known to increase the quantum Fisher information (QFI) in the processed probe state about the unknown parameter(s). We consider the case complementary to pure ICO in which the path state is maximally mixed for a given indefiniteness. Deriving the QFI under this condition by a new approach, we find that mixed ICO-assisted probing, while not as beneficial as pure ICO probing, still yields greater QFI than does the comparable scheme with definite causal order. The d-dimensional quantum depolarizing channel is used as the channel model for this study. While mixed ICO-assisted probing, like pure ICO-assisted probing, is generally advantageous for probing the depolarizing channel, mixed ICO's relative effectiveness decreases with probe dimension, just as for pure ICO-assisted probing, each being most effective for qubit probes.

May 13, 20192 citations

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.