In this paper, we are interested in detecting the presence of a nearby phase-sensitive object, where traveling light works out under a low-photon loss rate. Here we investigate the optimal quantum phase estimation with generalized multi-component Schrödinger cat states. In addition, we show the optimal conditions of the generalized multi-component cat states for the phase estimation in a lossless scenario. We then demonstrate that the generalized multi-component cat states can beat the performances of the NOON and two-mode squeezed vacuum states in the presence of small loss, while maintaining the quantum advantage over the standard quantum limit, attainable by coherent states. Finally, we propose a generation scheme of the entangled multi-component cat states with current or near-term optical technologies.
In this paper, we are interested in detecting the presence of a nearby phase-sensitive object, where traveling light works out under a low-photon loss rate. Here we investigate the optimal quantum phase estimation with generalized multi-component Schrödinger cat states. In addition, we show the optimal conditions of the generalized multi-component cat states for the phase estimation in a lossless scenario. We then demonstrate that the generalized multi-component cat states can beat the performances of the NOON and two-mode squeezed vacuum states in the presence of small loss, while maintaining the quantum advantage over the standard quantum limit, attainable by coherent states. Finally, we propose a generation scheme of the entangled multi-component cat states with current or near-term optical technologies.
It was recently shown that an entangled coherent state, which is a superposition of two different coherent states, can surpass the performance of noon state in estimating an unknown phase shift. This may hint at further enhancement in phase estimation by incorporating more component states in the superposition of resource state. We here introduce a four-headed cat state (4HCS), a superposition of four different coherent states, and propose its application to quantum phase estimation. We demonstrate the enhanced performance in phase estimation by employing an entangled state via the 4HCS, which can surpass that of the two-headed cat state (2HCS), particularly in the regime of small average photon numbers. Moreover, we show that an entangled state modified from the 4HCS can further enhance the phase estimation, even in the regime of large average photon number under a photon-loss channel. Our investigation further extends to incorporate an increasingly large number of component states in the resource superposition state and clearly show its merit in phase estimation.
We extend our previous study of squeezed two-component Schrödinger cat states to a phase-referenced single-mode family with an arbitrary number N of coherent components and general relative phases. Within the established circular-state construction, we derive a closed complex overlap kernel for the ordered action of displacement, phase rotation, and squeezing, retaining exact finite-amplitude normalization. We distinguish finite-displacement orthogonality from local phase-estimation sensitivity quantified by quantum Fisher information (QFI). For equal-phase states, we compare N=2,…,9 at a common mean photon number and include coherent-state and squeezed-vacuum benchmarks. A bound on the higher photon-number sectors establishes the fixed-energy asymptote FQ,N=4nˉ(N−nˉ)+o(1). Its leading term agrees with the known QFI of a vacuum--N-photon superposition. The resulting decrease of the ideal local quantum Cramér--Rao bound does not establish unlimited attainable precision: prior phase information, sample size, phase-reference resources, optical loss, state preparation, and measurement choice remain relevant.
Multiple-phase estimation exploiting quantum states has broad applications in novel sensing and imaging technologies. However, the unavoidable presence of lossy environments in practical settings often diminishes the precision of phase estimations. To address this challenge, we propose an optimal multiple-phase estimation scheme that is inherently robust against photon loss, ensuring a persistent quantum advantage across all levels of photon loss. The scheme employs a multi-mode definite photon-number (DPN) state with weights optimized for given levels of photon loss. We theoretically demonstrate that the DPN state can sustain quantum enhancement in estimation precision under all levels of photon loss, compared to the classical benchmark that employs a coherent state input. The proposed scheme using DPN states generalizes earlier studies employing NOON states, which are only optimal when photon loss is small. We believe that our study, demonstrating persistent robustness to photon loss, paves the way for significant advancements in quantum-enhanced sensing technologies, enabling practical applications and quantum advantages in real-world scenarios.
The simultaneous multiparameter estimation problem using a class of multimode entangled states is investigated in this paper. Specifically, the problem of optical phase imaging is considered, and the quantum probe is taken to be a balanced coherent superposition of components with an arbitrary quantum state in one mode and vacuum states in all the other modes, which is a generalization of the multimode NOON state. The analytical form for the quantum Cram\'er-Rao bound (QCRB) is presented, which shows the performance by providing a lower bound of the estimation uncertainty. It is shown that the NOON state has the worst performance among those in the class of the entangled states considered. We also analyze in detail four different scenarios, which are the NOON state, the entangled coherent state, the entangled squeezed coherent state, and the entangled squeezed vacuum state. From the comparison among these four states, we find that when the mean photon number is fixed, the squeezed vacuum state has the smallest QCRB, followed by the squeezed coherent state, entangled coherent state, and NOON state. Finally, we illustrate that the balanced entangled state can perform better than a more generalized unbalanced form studied in previous works for certain scenarios.
Abstract Multi-mode NOON states can quantum-enhance multiple-phase estimation in the absence of photon loss. However, a multi-mode NOON state is known to be vulnerable to photon loss, and its quantum-enhancement can be dissipated by lossy environment. In this work, we demonstrate that a quantum advantage in estimate precision can still be achieved in the presence of photon loss. This is accomplished by optimizing the weights of the multi-mode NOON states according to photon loss rates in the multiple modes, including the reference mode which defines the other phases. For practical relevance, we also show that photon-number counting via a multi-mode beam-splitter achieves the useful, albeit sub-optimal, quantum advantage. We expect this work to provide valuable guidance for developing quantum-enhanced multiple-phase estimation techniques in lossy environments.
In 1981, Caves pointed out that the phase sensitivity of a Mach-Zehnder interferometer with single-mode inputs is bounded by the shot-noise limit. The quantum Fisher information analysis shows that this statement holds true for the scenario where two antisymmetric phase shifts occur in two arms, but it is invalid for the scenario where an unknown phase is embedded in one of two arms. In this paper, we focus on the phase sensitivity directed against the latter scenario. The optimal single-mode input is discussed by analyzing common states, including displaced squeezed states, displaced number states, squeezed number states, Schrödinger cat states and completely mixed states. We find that the best choice is a squeezed vacuum state and show the specific measurement scheme which is capable of saturating the corresponding phase sensitivity limit. In addition, we study the effects of several realistic factors-anti-squeezing noise, photon loss and dark counts-on the phase sensitivity. Our results suggest that sub-shot-noise-limited phase sensitivity is attainable with low noise or loss, which paves the way for practical metrology.