Squeezing-induced quantum-enhanced multiphase estimation

We investigate how squeezing techniques can improve the measurement precision in multiphase quantum metrology. While these methods are well-studied and effectively used in single-phase estimations, their usage in multiphase situations has yet to be examined. We fill this gap by investigating the mechanism of quantum enhancement in the multiphase scenarios. Our analysis provides theoretical and numerical insights into the optimal condition for achieving the quantum Cramer-Rao bound, helping us understand the potential and mechanism for quantum-enhanced multiphase estimations with squeezing. This research opens up new possibilities for advancements in quantum metrology and sensing technologies.

Although the mechanism of squeezing for quantumenhanced metrology in single-phase estimations is well understood, its application to multiphase estimations remains unexplored.Recent attempts to incorporate squeezing into sensor networks for multiphase estimations [26] and using variational squeezing optimization [27,28] have been reported.However, these approaches do not fully elucidate the underlying mechanism of quantum enhancement.
In this work, we explore how nonlinear spin squeezing enhances multiphase estimations.We focus on a scenario where a three-dimensional magnetic field interacts with an ensemble of N identical two-level systems.The precision is bounded by the quantum Cramér-Rao bound (QCRB), which is attainable for single-phase estimations but remains unsaturated for multiphase esti-mations.Previous studies have claimed the saturation of the QCRB for multiphase estimation, however they did not provide the mechanism behind it [29,30].Here, we clearly explain the mechanism behind this saturation by examining the sysmetry in ensemble systems.We begin by discussing the saturation of the QCRB for an ideal quantum state characterized by a multi-GHZ entanglement state, which includes GHZ states in all three spatial directions.We then investigate a realistic case where the quantum state is a single GHZ entanglement in a certain direction.We elucidate the mechanism behind quantumenhanced precision in this context and assess the impact of noise.Our findings contribute to a better understanding of quantum-enhanced mechanisms in metrology and facilitate the development of quantum sensors and imaging technologies.
Quantum-enhanced with a multi-GHZ probe state.-Weexamine a 3D vector field ϕ = (ϕ x , ϕ y , ϕ z ) ⊺ that requires estimation.This field interacts with a probe of N spin-1/2 particles through a Hamiltonian where J = (J x , J y , J z ) is an angular momentum, and µ , µ = {x, y, z}.Inspired by single-phase estimations, where the probe state is prepared in a superposition of the maximum and minimum eigenstates of the Hamiltonian H [31], we consider a multi-GHZ probe state |Ψ⟩ as where N is the normalization constant, Here, |λ max µ ⟩ and |λ min µ ⟩ are two eigenstates of J µ corresponding to the maximum and minimum eigenvalues λ max and λ min , respectively.This approach has demonstrated Heisenberg scaling in the noiseless case [30].
The performance of an unbiased estimator is determined by the covariance matrix C(ϕ).This matrix has an ultimate lower bound known as the quantum Cramér-Rao bound (QCRB) [36], i.e., M C(ϕ) ≥ I −1 , where M denotes the repeated measurements [37].The QCRB is attainable in single-phase estimations [31].However, to achieve this bound in multiphase estimations, it is necessary (but not sufficient) that Im[⟨Ψ|A ⊺ A|Ψ⟩] = 0 (see App. C).To quantify this necessary condition in our case, we define a matrix D as and derive the Frobenius norm ∥D∥ F = µν |D µν | 2 .When ∥D∥ F = 0, it implies that Im[⟨Ψ|A ⊺ A|Ψ⟩] = 0 or in other words, the QCRB in multiphase estimations is attainable.
In Fig. 1a, we show ∥D∥ F as a function of N , with ϕ x = ϕ y = ϕ z = ϕ.The numerical results indicate that ∥D∥ F ≈ 0 for large N , such as, ∥D∥ F ∝ 10 −4 to 10 −14 for N = 35 to 50, as shown in the inset Fig. 1.This result can be explained by the symmetry of the wavefunction |Ψ⟩ as follows.
Without loss of generality, let us analytically examine this result in the limit of ϕ → 0. See detailed calculation in App.D. Note that the ensemble of spins exhibits permutation symmetry, represented by the Dicke basis, and can be implemented using collective (global) operators.At ϕ → 0, we have A ≈ J .Then, each element of the matrix D is given by For µ = ν, we have D µν = 0.For µ ̸ = ν, we expand Eq. ( 5) into the Dicke basis {|m⟩} as where J = N/2.For odd N , the nonzero terms are D yz and D zy = −D yz , which gives This term is nonzero when |Ψ⟩ is asymmetric under |±J⟩ and |±(J −1)⟩, leading to ∥D∥ F ̸ = 0.If |Ψ⟩ is symmetric, or the asymmetry of |Ψ⟩ becomes negligible, e.g., large N , then D yz = 0, which results in ∥D∥ F = 0.The analysis is the same for even N , yielding the same result for D.
In Fig. 1b, we plot the probability 2 , which represents the projection of the probe state onto the z axis.In this example, the asymmetry is present for N = 15 as shown by the purple circles, while it is symmetric for N = 16.Additional examples with N = 17, 18 can be found in App.D. Correspondingly, Fig. 1a indicates that ∥D∥ F > 0 at N = 15 and ∥D∥ F ≈ 0 at N = 16.As N becomes large, the asymmetry becomes negligible, as seen with N = 40 and N = 41 in this example, resulting in ∥D∥ F ≈ 0.
To understand the symmetry-asymmetry behavior, we next analyze the spin fluctuation, quantified by the variance ∆ 2 .It is given by ∆ of the angles θ and φ in the spherical coordinate system.The fluctuation decreases as the number of spins N increases as indicated in Fig. 1c.In Fig. 1c, we compare spin fluctuations for small and large N .Apparently, for small N , the large fluctuations in all |ψ µ ⟩ components lead to the overlap and interference, causing deformation and easier symmetry breaking in |Ψ⟩.Conversely, for larger N , the asymmetry is small and becomes negligible.The overlap and interference are visible in the Husimi distribution shown at the bottom of Fig. 1c, highlighted by the red arrows and circles.Notably, the symmetry-asymmetry behavior is not strictly tied to an odd or even number of spins.
Finally, given that the condition is met, we analyze the total variance |∆ϕ| 2 = Tr[C(ϕ)], which is now expressed as Tr[I −1 ]/M .Figure 1d illustrates the total variance as a function of N , showing a Heisenberg scaling similar to that in Ref. [30].
Particularly, we use squeezing techniques to compress the probe state |ψ z ⟩, following by encoding the phases and reverting (echo), as shown in Fig. 2a.The nonlinear squeezing methods are represented by the operators x ; U TAT = e −itχ(J 2 x −J 2 y ) ; where χ represents the magnitude of the spin-spin interaction, Ω stands for the rate of rotation about the y axis, and we also introduce Λ = N χ/Ω.When χ ≫ Ω or Λ ≫ N , then TNT simplifies to OAT.Hereafter, we set Λ/N = 0.02 to investigate the effect of TNT.These transformations are experimentally confirmed [21][22][23][24][25].
The estimation scheme is as follows.We first induce squeezing on the probe state |ψ z ⟩ using U k for k = {OAT, TAT, TNT}, followed by the phasing unitary U (ϕ) = e −iH(ϕ) .We also set ϕ x = ϕ y = ϕ z = ϕ for numerical calculation.Then, we apply an inverted dynamic U −r k , where r ∈ R is an arbitrary constant [16], resulting in the final state In general, U −r k does not affect the QFIM as following In Fig. 2c, we plot |∆ϕ| 2 as a function of the squeezing angle χt for N = 100.We observe that each type of squeezing achieves a minimum value at a certain angle.The OAT result saturates at its optimal angle, the TAT result attains the highest precision at a small optimal angle χt, and the TNT result exhibits nonlinear oscillations as χt increases.
In Fig. 2d, we plot the squeezing parameter ξ 2 as a function of χt, following the definition by Kitagawa and Ueda [40].More details can be found in App.F. For single-phase estimations, ξ 2 is proportional to the variance, i.e., ξ 2 ∝ |∆ϕ| 2 [39,41].In our case, we observe a similar behavior for ξ 2 and |∆ϕ| 2 , where ξ 2 min aligns with |∆ϕ| 2 min .As χt changes continuously, the squeezing parameter tends to increase after reaching its minimum value and does not return to the minimum.
To understand the precision-enhanced mechanism, in Fig. 2b we examine the Husimi distribution function of the squeezed state at various χt points.In the OAT case, the probe state evolves from coherent to squeezing in the x-y plane at an angle determined by χt [40].When χt increases, these components stretch and rotate toward the y-axis (see also App.G).When they overlap, interference occurs, resulting in bright and dark points in the Husimi distribution function.Consequently, the squeezed state spreads in both ±z and ±y directions, reducing the total variance until it stabilizes as no additional information is added.
Conversely, in the TAT case, the squeezing occurs in the x-y plane at ±45°for |±z⟩, respectively (see App. G).As χt increases, these components extend in those directions.Due to spherical symmetry, we rotate the state 45°a long the z axis, causing it to distribute in both the ±x and ±y directions.At the optimal point, the state evenly spreads across all ±x, ±y, ±z directions, resembling the multi-GHZ state |Ψ⟩.This causes both |∆ϕ| 2 and ξ 2 to reach their minimum values, as indicated by the orange arrows.More visualization about this case can be found in App.G and the animation (animation.pm4).
The explanation for the TNT case is similar to that of OAT.In this case, the squeezed state distributes along the ±z, −x, and +y axes and also interfere with each other.
In Fig. 2e, we analyze the TAT case and calculate the Frobenius norm ∥D∥ F at the optimal χt for different N , confirming the condition ∥D∥ F = 0.The increasing value with N is merely a random fluctuation on this scale.Finally, we examine the variance in Fig. 2f.The variance with TAT reverts to or improves upon that of the |Ψ⟩ scenario, while the variance with OAT only achieves the SQL, and the variance with TNT transitions from SQL to HL.
Quantum-enhanced under noise.-Wenext examine the case with dephasing noise during the encoding process.Under this noise, a quantum state ρ evolves to where Here, 0 ≤ ϵ ≤ 1 is the noise probability.See detailed calculations in App.H.
In Fig. 3a, we first optimize χt at ϵ = 0 to obtain the minimum |∆ϕ| 2 .After that, we keep (χt) opt constance and examine how |∆ϕ| 2 changes with ϵ.The result is shown by the red curve.As trivial, increasing ϵ results in an increase in |∆ϕ| 2 , which decreases the precision.Since (χt) opt is fixed, we denote this case as "without correction."Next, we attempt to improve the result by optimizing χt for each ϵ.Concretely, at each ϵ, we apply the TAT transformation and optimize χt to minimize |∆ϕ| 2 .As depicted in Fig. 3a with blue triangles, this correction has a minimal impact for small ϵ, but it slightly enhances the precision for larger ϵ.
To quality the enhancement, we define a ratio R as where "w/wo" represents "with/without" correction.For ϵ ≤ 0.4, R remains at zero.However, for larger ϵ, R ranges from 0 to slightly over 3%.Although squeezing can enhance precision in noisy conditions, the effect is minimal.Figure 3b illustrates the enhancement mechanism.Initially, the red curve depicts the case at ϵ = 0, where we adjust χt to find the optimal (χt) opt , resulting in the best |∆ϕ| 2 , indicated by the red star.In the presence of noise, the red curve shifts upward, indicated by the blue line.Maintaining (χt) opt leads to an increase in |∆ϕ| 2 (blue star).However, further adjustment of χt yields another optimal point, which reduces |∆ϕ| 2 (magenta star).Although the correction effect is not significant, it can lead to further improvements in squeezing-based corrections.
Conclusion.-We studied the mechanism behind the improving precision in multiphase estimations using nonlinear squeezing techniques.We started with a spin ensemble in a GHZ state along a specific axis.We utilized squeezing techniques such as one-axis twisting and twist-and-turn to transform the quantum state to the other axes, thereby increasing precision along those axes.On the other hand, two-axis twisting extended the ensemble to all other axes, resulting in quantum enhancement across all investigated directions.Understanding this mechanism can aid in the design of more effective quantum sensors based on squeezing techniques.This work is supported by the JSPS KAKENHI Grant Number 23K13025.All numerical computations in this study were done using the tqix code [42,43].
Then, we have The right hand side of (C.1) yields where In our model, we first note that for individual GHZ states, i.e., |Ψ⟩ = |ψ µ ⟩, µ = x, y, z, condition (C.5) is always satisfied regardless of the number of spins.This result is due to the symmetry presented in the bases |λ max µ ⟩ and |λ min µ ⟩.When |Ψ⟩ is a summation of all components, as shown in Eq. ( 2), and with a large value of N , the asymmetry in |Ψ⟩ is negligible.Thus, the condition (C.6) still holds, as demonstrated in the main text.
Appendix D: QCRB saturating under the multi-GHZ case Now, we analytically demonstrate the saturating of condition (C.6) in the limit of ϕ → 0. We emphasize that for a small phase ϕ, numerical results indicate that D ≈ 0. However, we do not prove this case here.)Starting from A, we have For ϕ = 0, we have A ≈ J .Then, each element of matrix D is given by We identify all nonzero terms D µν ∀µ, ν, and examine the conditions under which they become zero.Utilizing the permutation symmetry of the probe system [35], and in the absence of noise, the probe can be represented in the Dicke basis |J, m⟩, where J = N/2, and −J ≤ m ≤ J, or denoted as |m⟩ for short.The angular momentum operators are defined by spin-J operators, where We expand Eq. (D.2) using the Dicke basis as For µ = ν, it is evident that D µν = 0.When µ ̸ = ν, we distinguish between two cases: odd N and even N .In the case of odd N , the nonzero terms are D yz and D zy = −D yz , where The blue and magenta terms will cancel out for m = ±k, ∀k ∈ {1/2, • • • , J − 1}.The only nonzero terms occur when m = ±J, which is For small N , it is evident that D yz ̸ = 0 when the probe state |Ψ⟩ is asymmetry under | ± J⟩ and | ± (J − 1)⟩.Inversely, if |Ψ⟩ is symmetric, the term [⋆] in (D.7) vanishes, leading to D yz = 0.As N grows large, the asymmetry of |Ψ⟩ becomes negligible, resulting in D yz = 0.
In the case of even N , the nonzero terms are D xz and D zx = −D xz .These can be calculated using the same method and exhibit the same behavior.
In Fig. 4, we plot the probability P (m) = |⟨m|Ψ⟩| 2 for N ranging from 15 to 18 and their corresponding ∥D∥ F .Within this range, only the wavefunction with N = 16 is symmetric, resulting in ∥D∥ F ∝ 10 −15 .The other cases exhibit asymmetry, leading to nonzero ∥D∥ F .
FIG. 2. (a)A metrology approach starts by preparing a GHZ state in the z direction, |ψz⟩, followed by a squeezing transformation, phasing, and reverting transformations.This sequence yields a quantum state containing all necessary information for phase estimation.(b) Visualization of the Husimi distribution for OAT, TAT, and TNT.(c) Plot of |∆ϕ| 2 as a function of χt for OAT, TAT, and TNT cases.(d) Plot of ξ 2 (dB) as a function of χt for OAT, TAT, and TNT cases.(e) Log-log plot of ∥D∥F as a function of N for the optimal TAT case.The increasing value with N is merely a random fluctuation on this scale.(f) Log-log plot of optimal |∆ϕ| 2 as a function of N for OAT, TAT, and TNT cases, compared with the multi-GHZ case |Ψ⟩, SQL, and HL.N is fixed at 100 for (b-d).

FIG. 3 .
FIG. 3. (a) Plot of |∆ϕ| 2 as a function of the noise probability ϵ for two cases of the correction and without correction.The corresponding enhancement ratio R is shown on the right column.For high noise levels, an increase of up to 3% is observed.Data represented for N = 40.(b) Illustration demonstrating the squeezing correction.