Abstract We comprehensively study the metrological performance of perfect W (PW) state in quantum phase measurement, including local operation and two-body interacting operation as the phase generator. The analytical precision limits are obtained in both cases and the decoherence effects are also investigated. The results show that under local operation the precision limit is slightly lower than it from W state, indicating the weaker sensitivity. Specially, with increasing m the precision limit from the general 3-qubit PW state approaches 1/Sqrt{5}. In the amplitude damping channel, the precision limit decreases first and then increases to the shot noise limit, whereas it is always decreased in phase damping channel. Alternatively, we have also provided the optimal strategies for quantum phase estimation. In the two-body interacting operations, Ising model and Lipkin-Meshkov-Glick (LMG) model are employed to explore the precision limit with respect to interacting strength. It behaves similar in both cases that with increasing the precision limit surpasses the general Heisenberg limit and gradually converges with W state. However, the difference is also evident that the more qubits involved for achieving Heisenberg limit, the larger the needed \gamma in Ising model but smaller in LMG model. Interestingly, in amplitude damping channel, with increasing an expanding platform area of precision limit emerges in the few-qubit PW state (N < 6), while it gradually disappears in the large qubit case. This will be beneficial to the stable and tunable high-precision measurement, especially under the noisy circumstances.
Abstract We study the performance of N-qubit W superposition state in quantum metrology. Taking advantage of the general Ising-type Hamiltonian (including local and nonlocal operation), we analytically present the Quantum Fisher information (QFI) of N-qubit W superposition state under different situations and then investigate its phase sensitivity. The results show that the phase sensitivity under local operation displays a crossover from W state to GHZ state, where it is same as W state in few-qubit case (N ≤ 6) but asymptotically equal to GHZ state for large qubit case (N >> 1). Interestingly, the 4-qubit W superposition state is found to have the same sensitivity with 4-qubit GHZ state, but be more robust than latter. Besides, the optimal theoretical proposals are provided for ideal metrology. Under the phase/amplitude damping channel, the phase sensitivity of W superposition state (except for N=3) is ultimately decreased to standard quantum limit, while it turns worse in depolarizing channel. Finally, the tunable phase sensitivity under nonlocal operation is studied and the general Heisenberg limit is surpassed with the increasing interaction strength γ . Meanwhile, a plateau of QFI and phase sensitivity is found for all large qubit W superposition state, which is similar to the study of GHZ state and again verifies the common feature of GHZ-type states in quantum metrology.
We study the performance of an
N-qubit
W superposition state composed of a
W state and its obverse in quantum metrology. Taking advantage of the general Ising-type Hamiltonian (including noninteracting and interacting operation), we analytically present the quantum Fisher information (QFI) of an
N-qubit
W superposition state under different situations and then investigate its phase sensitivity. The results show that the phase sensitivity under noninteracting operation displays a crossover from the
W state to Greenberger-Horne-Zeilinger (GHZ) state, where it is same as
W state in the few-qubit case $(N\ensuremath{\le}6)$ but asymptotically equal to the GHZ state for large-qubit cases $(N\ensuremath{\gg}1)$. Interestingly, the 4-qubit
W superposition state is found to have the same sensitivity as the 4-qubit GHZ state. And the optimal measurement protocols are provided for ideal metrology. Under the phase-amplitude damping channel, the phase sensitivity of the
W superposition state (except for
N=3) is ultimately decreased to the standard quantum limit, while it turns worse in a depolarizing channel. Finally, the tunable phase sensitivity under interacting operation is studied, and the general Heisenberg limit is surpassed with the increasing interaction strength $\ensuremath{\gamma}$. Meanwhile, a plateau of QFI and phase sensitivity is found for all large-qubit
W superposition states, which is similar to the study of the GHZ state and again verifies the common feature of GHZ-type states in quantum metrology.
We examine the metrological performance of WW¯-like state (αWN+βW¯N) in quantum phase estimation. Based on the framework of quantum interferometry, we analytically derive the quantum Fisher information (QFI) and analyze the precision limits. In the noninteracting environment, the metrological power of WW¯-like state is same as that of the W state in few-qubit case but symmetrically enhanced (with respect to β2) in large-qubit case. In the Ising model, with increasing interaction strength, the QFI of N≤6 qubit WW¯-like state is universally enhanced and displays different and exotic trends (with respect to β2), particularly for 4- and 6-qubit cases where it respectively shows a reversible phenomenon and a reversal scenario. Regarding others (N>6), it exhibits a similar trend that the precision limit is always better than that of WW¯ state in strong interaction.
While quantum metrology enables measurement precision beyond classical limits, its performance is often susceptible to experimental imperfections. Most prior studies have focused on imperfections in quantum states and operations. Here, we investigate the effect of coarse graining in quantum measurement through both theoretical analysis and experimental demonstration. Using an interferometer with a squeezed vacuum and a laser input, we analyze how coarse graining in homodyne detection affects the precision of phase estimation. We evaluate the Fisher information under various coarse-graining conditions and determine, in each case, an optimal estimation strategy that saturates the Cramér-Rao bound. Remarkably, even extremely coarse-grained measurement -- with only two bins -- enables phase estimation beyond the standard quantum limit and even achieves a precision that follows the Heisenberg scaling. We experimentally demonstrate quantum-enhanced phase estimation under coarse-grained homodyne detection. To determine an optimal estimation strategy, we employ the method of moments and present calibration procedures that enable its application to general experimental settings. Using only two bins, we observe a quantum enhancement of 1.2 dB compared to the classical method using the ideal measurement, improving towards 3.8 dB as the bin number increases. These results highlight a practical pathway to achieving quantum enhancement under the presence of severe experimental imperfections.
Abstract We study the metrological performance of the superposition of Greenberger-Horne-Zeilinger state and Twin Fock state (SGT) in quantum phase estimation. The analytical quantum Fisher information (QFI) of an Nqubit SGT state in noninteracting environment has been derived and with the increasing ratio β in SGT state it emerges a turning point at βt = q 1 − 1 N · q2 3 , where the smallest QFI FQ = (N 2 + 2N)/3 is achieved. Interestingly, we find the 4-qubit SGT state performs better than TF state in all ratio region and further achieves the Heisenberg limit at β = √ 3/2. Working in the Mach-Zehnder interferometry, we have provided the optimal metrology schemes saturated by quantum CRLB for all kinds of SGT states and found the same βt also existed in the selection of phase generator and measured parity operator. Moreover, we have considered the decoherence effects and with the increasing decay the QFI in amplitude damping and phase damping channel decreases to the number of qubits involved, while it decreases to zero in depolarizing channel. Specially, the 4-qubit SGT state of β = √ 3/2 is found to perform better than GHZ state in phase damping channel. Our work enriches the metrological useful quantum resources.
Phase sensing with entangled multi-qubit states in the presence of noise is a central theme of modern quantum metrology. The present work investigates Dicke state superposition probes for quantum phase sensing under parameter encoding generated by one- and two-body interaction Hamiltonians. Under linear collective-spin encoding, near-optimal Dicke superposition states are shown to exhibit significantly enhanced robustness against phase damping noise compared with Greenberger--Horne--Zeilinger (GHZ), W-superposition, and balanced Dicke states, while maintaining favorable metrological performance under realistic decoherence channels. For two-body interactions, optimal probe states maximizing the quantum Fisher information are identified. Their noise resilience and metrological scaling behaviour under phase damping, amplitude damping, and global depolarizing channels are analyzed. The associated near-optimal Dicke superposition states are found to exhibit improved resilience to phase damping, for the system sizes considered. These results establish tailored near-optimal Dicke state superposition probes as versatile and noise-resilient resources for Heisenberg and super-Heisenberg quantum phase sensing governed by one- and two-body interactions.