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arXiv:2412.12085 · Physical Review A

Qubit-channel metrology with highly noisy initial states and additional undesirable noisy evolution

David Collins, Taylor Larrechea

Published Nov 20, 2025• 0 citations
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Abstract

We consider protocols for estimating the parameter in a single-parameter unital qubit channel, assuming that the available initial states are highly mixed with very low purity. We compare two protocols, each querying the channel once, via their quantum Fisher informations. One uses $n$ qubits prepared in a particular correlated input state and subsequently queries the channel on one qubit. The other uses a single qubit. We extend the results of Collins [Phys. Rev. A 99, 012123 (2019)] by allowing for additional noisy evolution on the spectator qubits in the $n$-qubit protocol. We provide simple algebraic expressions that will determine when the $n$-qubit protocol is superior. We provide a technique that can alleviate certain types of noise. We show that for certain types of noisy evolution the $n$-qubit protocol will be inferior but for others it will be superior.

DOI
10.1103/f179-s1yp

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We consider protocols for estimating the parameter in a single-parameter unital qubit channel, assuming that the available initial states are highly mixed with very low purity. We compare two protocols, each querying the channel once, via their quantum Fisher informations. One uses nnn qubits prepared in a particular correlated input state and subsequently queries the channel on one qubit. The other uses a single qubit. We extend the results of Collins [Phys. Rev. A 99, 012123 (2019)] by allowing for additional noisy evolution on the spectator qubits in the nnn-qubit protocol. We provide simple algebraic expressions that will determine when the nnn-qubit protocol is superior. We provide a technique that can alleviate certain types of noise. We show that for certain types of noisy evolution the nnn-qubit protocol will be inferior but for others it will be superior.
We consider an arbitrary qubit channel depending on a single parameter, which is to be estimated by a physical process. Using the quantum Fisher information per channel invocation to quantify the estimation accuracy, we consider various estimation protocols when the available initial states are mixed with very low purity rrr. We compare a protocol using a single channel invocation on one out of nnn qubits prepared in a particular correlated input state to the optimal protocol using uncorrelated input states, with the same initial-state purity. We show that, to lowest order in initial-state purity, for a unital channel this correlated-state protocol enhances the estimation accuracy by a factor between $n\ensuremath{-}1$ and nnn, provided that $n{r}^{2}\ensuremath{\ll}1$. We also show that to lowest order in initial-state purity, a broad class of nonunital channels yields no gain regardless of the input state.
We consider protocols for estimating the parameter which characterizes a single qubit depolarizing channel, with the goal of attaining the most accurate estimate per channel use. The accuracy of any quantum estimation protocol will be quantified via the quantum Fisher information (QFI) since the Cramer-Rao bound implies thata larger QFI yields a smaller lower bound on the possible variance in any estimate of the parameter. Within this framework, the choice of input state prior to channel invocation affects estimation accuracy. The known optimal estimation scheme uses pairs of maximally entangled pure input states, giving a gain over any protocol using unentangled input states. Entangling more than two qubits initially in apure state provides no further gain in the QFI per channel use. We ask if,when the available input states are not pure,these gains persist and if correlating more than two qubits is advantageous. We present a protocol using input states correlated over any number of qubits and compare this to an independent channel useprotocol using uncorrelated mixed states. We show that the correlated state protocol yields gains in the QFI per channel use for certain physically reasonable parameter ranges. We show that, unlike the pure state case, using more than two correlated qubits can be advantageous and we show that, as the initial qubit states become highly mixed, adding additional correlated qubits can provide substantial gains in estimation accuracy. We show that for two qubits, such gainsare attained even when the state prior to channel invocationare separable. Thedepolarizing channelmaps a qubit as
We consider estimating the parameter associated with the qubit depolarizing channel when the available initial states that might be employed are mixed. We use quantum Fisher information as a measure of the accuracy of estimation to compare protocols which use collections of qubits in product states to one in which the qubits are in a correlated state. We show that, for certain parameter values and initial states, the correlated-state protocol can yield a greater accuracy per channel invocation than the product-state protocols. We show that, for some parameters and initial states, using more than two qubits and channel invocations is advantageous. These results stand in contrast to the known optimal case that uses pure initial states and a single channel invocation on a pair of entangled qubits.
Quantum metrology overcomes standard precision limits and plays a central role in science and technology. Practically, it is vulnerable to imperfections such as decoherence. Here we demonstrate quantum metrology for noisy channels such that entanglement with ancillary qubits enhances the quantum Fisher information for phase estimation but not otherwise. Our photonic experiment covers a range of noise for various types of channels, including for two randomly alternating channels such that assisted entanglement fails for each noisy channel individually. We simulate noisy channels by implementing space-multiplexed dual interferometers with quantum photonic inputs. We demonstrate the advantage of entanglement-assisted protocols in a phase estimation experiment run with either a single-probe or multiprobe approach. These results establish that entanglement with ancillae is a valuable approach for delivering quantum-enhanced metrology. Our approach to entanglement-assisted quantum metrology via a simple linear-optical interferometric network with easy-to-prepare photonic inputs provides a path towards practical quantum metrology.
Quantum metrology is superior to classical metrology in parameter estimation. This is because quantum states have coherence that, however, does not exist in classical states. Previous works were limited to the study of the relationship between quantum Fisher information (QFI) and coherence of the initial probe state only for a specific single-qubit parameter estimation process (e.g., a process described by a single-qubit spin angular momentum operator Jz{J}_{z}Jz​). Here, we consider a more general case and study the relationship between QFI and coherence of the initial probe state for a single-qubit general parameter estimation process, which is described by a single-qubit spin angular momentum operator ${J}_{\stackrel{P\vec}{n}}$ with a general unit vector $\stackrel{P\vec}{n}$. We find that the QFI is proportional to the square of coherence of the initial probe state, when the single-qubit bases are eigenstates of the spin angular momentum operator ${J}_{\stackrel{P\vec}{n}}$ describing the parametrization process. We also design and conduct a linear optical experiment to support our theory. The experimental results are in good agreement with the theoretical ones. This work provides an important guideline for enhancing the precision of parameter estimation.
We consider frequency estimation in a noisy environment with noisy probes. This builds on previous studies, most of which assume that the initial probe state is pure, while the encoding process is noisy, or that the initial probe state is mixed, while the encoding process is noiseless. Our work is more representative of reality, where noise is unavoidable in both the initial state of the probe and the estimation process itself. We prepare the probe in a GHZ diagonal state, starting from n + 1 qubits in an arbitrary uncorrelated mixed state, and subject it to parameter encoding under dephasing noise. For this scheme, we derive a simple formula for the (quantum and classical) Fisher information, and show that quantum enhancements do not depend on the initial mixedness of the qubits. That is, we show that the so-called 'Zeno' scaling is attainable when the noise present in the encoding process is time inhomogeneous. This scaling does not depend on the mixedness of the initial probe state, and it is retained even for highly mixed states that can never be entangled. We then show that the sensitivity of the probe in our protocol is invariant under permutations of qubits, and monotonic in purity of the initial state of the probe. Finally, we discuss two limiting cases, where purity is either distributed evenly among the probes or concentrated in a single probe.