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arXiv:2610.23398

Quantum Sensitivity of NNN-Component Cat States

Ranjit Singh, Alexander E. Teretenkov

quant-phSubmitted Oct 16, 2026• Announced Oct 19, 2026• 0 citations
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Abstract

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 $N=2,\ldots,9$ 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 $F_{Q,N}=4\bar n(N-\bar n)+o(1)$. 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.

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We extend our previous study of squeezed two-component Schrödinger cat states to a phase-referenced single-mode family with an arbitrary number NNN 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,…,9N=2,\ldots,9N=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)F_{Q,N}=4\bar n(N-\bar n)+o(1)FQ,N​=4nˉ(N−nˉ)+o(1). Its leading term agrees with the known QFI of a vacuum--NNN-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.
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.
We propose two-stage optimized strategies for full-period optical phase estimation with single-mode Gaussian states and Gaussian measurements under a fixed energy constraint. In the first stage (Stage I), displaced squeezed probes and heterodyne measurements provide coarse localization of the phase to a window on the circle. In the second stage (Stage II), squeezed-vacuum probes with adaptive homodyne measurements perform efficient phase estimation inside the selected window. We derive a generalized Cramér-Rao bound for this family of two-stage Gaussian strategies, which contains the contribution from local parameter estimation in Stage II plus an overshoot penalty from coarse localization errors in Stage I. For E <= 25 photons and squeezing limited to 12 dB, protocols using displaced squeezed states in Stage I reduce the optimized two-stage bound relative to protocols using coherent states in Stage I, and remain within a factor of 3 to 30 of the idealized local squeezed-vacuum quantum Cramér-Rao bound.
We theoretically study the phase estimation based on a Mach-Zehnder interferometer (MZI) with a two-mode squeezed coherent state. By maximizing the quantum Fisher information, we find that the quantum Cramér-Rao bounds (QCRB) can reach sub-Heisenberg limit under the phase-matched condition. The optimal phase sensitivity can reach the sub-shot noise limit (SNL) and approach the QCRB by employing the intensity difference detection. Meanwhile, compared with the MZI fed with a coherent plus a single-mode squeezed vacuum state, this scheme can have better performance by adjusting the squeezing parameter and the mean photon number. With the same parameter, our scheme shows more sensitive phase measurement than the SU(1,1) interferometer with a coherent plus a vacuum state. We also show that the phase sensitivity of our proposal can still reach the SNL when the loss of the photon is 36%. This scheme can provide potential applications in optical sensors.
Janus states, defined as coherent superpositions of two single-mode squeezed vacua, provide a simple but genuinely non-Gaussian setting for studying how interference reshapes quantum Fisher information (QFI) beyond the Gaussian squeezed-vacuum picture. Using an exact analytic treatment, we determine the QFI of Janus states and identify the benchmarks under which they can or cannot offer a metrological advantage over the single squeezed vacuum. We find that, under a fair comparison at fixed mean photon number, the single squeezed vacuum remains optimal for principal second-moment squeezing, so no genuine Janus advantage exists at that level. By contrast, within a fixed two-state span, a Janus superposition can simultaneously outperform its constituents in a laboratory quadrature variance and in number-generated phase QFI. We also introduce an operational benchmark based on fixed measured squeezing and show that, at the same observed squeezing level, Janus interference can substantially enhance the QFI for quadratic-generator sensing beyond the pure-Gaussian squeezed-vacuum reference. These results show that the metrological performance of Janus states is controlled not only by quadrature squeezing, but also by higher-order fluctuations and by the benchmark used for comparison.
We consider an optical interferometer with coherent light in one input and a squeezed vacuum in another. Such an interferometer is known to beat the standard quantum limit of sensitivity to the difference of phase shifts in its arms. We find the ultimate limit of precision for such an interferometer by calculating the quantum Fisher information of the mixed quantum state in one of the interferometer's outputs about the difference phase. We show that, in the vicinity of the black fringe, this information is asymptotically close to the quantum Fisher information about this phase for the two-mode readout. We conclude that the single-mode readout is optimal for phase estimation in squeezing-enhanced interferometry and allows for the Heisenberg scaling of precision. We also show that the optimal local measurement in the vicinity of the black fringe consists of amplifying the output field in a phase-sensitive way and measuring its photon number.
A paradigmatic task in quantum metrology is that of phase estimation. For this task it is commonly believed, under reasonable assumptions about prior information, that a quadratic advantage in sensitivity is the maximum possible advantage offered by the use of quantum resources. In this work, we consider a channel that implements a phase shift by an angle 2πk/M2πk/M2πk/M, where kkk is a uniformly random unknown integer, in addition to a small unknown phase shift of interest. We show that, subject to a constraint on the mean photon number per channel use, there is a super-exponential separation between the quantum Fisher information (QFI) attainable using a squeezed vacuum state and that attainable using coherent states. This advantage holds over any state obtained from a coherent state via phase covariant channels. Specifically, when the mean photon number per channel use is 1, the ratio of their QFI grows as Ω[(M/2e)M]Ω[(M/\sqrt{2}e)^M]Ω[(M/2​e)M]. For a mean photon number per channel use of M/log⁡(M)M/\log(M)M/log(M), the squeezed vacuum QFI is Ω(M3/2/log⁡(M))Ω(M^{3/2}/\sqrt{\log(M)})Ω(M3/2/log(M)​), whereas for a coherent state it is at most exp⁡[−Mlog⁡log⁡(M)+O(M)]\exp[-M\log\log(M)+O(M)]exp[−Mloglog(M)+O(M)]. We complement this QFI analysis with non-asymptotic estimation results. For a mean photon number per channel use of M/log⁡MM/\log MM/logM, an explicit squeezed-vacuum protocol estimates the phase to accuracy O(1/M)O(1/M)O(1/M) using O(Mlog⁡M)O(\sqrt{M\log M})O(MlogM​) channel uses, whereas every coherent state protocol requires at least exp⁡[Mlog⁡log⁡M−O(M)]\exp[M\log\log M-O(M)]exp[MloglogM−O(M)] channel uses. We consider the effects of noise and show that the ratio of their QFIs can still grow super-exponentially in MMM. Finally, under fixed optical loss and thermal noise, choosing nˉ=M2/(log⁡M)3/2\bar n=M^2/(\log M)^{3/2}nˉ=M2/(logM)3/2 allows squeezed vacuum with heterodyne detection to achieve accuracy O(1/M)O(1/M)O(1/M) using sub-polynomially many channel uses, whereas every protocol in the specified classical family requires super-polynomially many.