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Quantum enhanced multiple-phase estimation with multi-mode N00N states

Quantum metrology can achieve enhanced sensitivity for estimating unknown parameters beyond the standard quantum limit. Recently, multiple-phase estimation exploiting quantum resources has attracted intensive interest for its applications in quantum imaging and sensor networks. For multiple-phase estimation, the amount of enhanced sensitivity is dependent on quantum probe states, and multi-mode N00N states are known to be a key resource for this. However, its experimental demonstration has been missing so far since generating such states is highly challenging. Here, we report generation of multi-mode N00N states and experimental demonstration of quantum enhanced multiple-phase estimation using the multi-mode N00N states. In particular, we show that the quantum Cramer-Rao bound can be saturated using our two-photon four-mode N00N state and measurement scheme using a 4 × 4 multi-mode beam splitter. Our multiple-phase estimation strategy provides a faithful platform to investigate multiple parameter estimation scenarios.

Q uantum metrology has attracted intensive interest in recent years, as it allows us to estimate an unknown parameter with enhanced sensitivity over classical approaches. Developments are now directed towards various applications such as microscopy [1][2][3][4][5] , biological imaging 6-10 and sensor network [11][12][13] . In such practical applications, quantum metrology is naturally extended to multiple parameter scenario, since the number of parameters that affect a physical process is generally more than one. For estimating multiple parameters, simultaneous estimation is desirable as it can achieve higher precision than an individual estimation of each parameter using the same amount of resources 14,15 . Furthermore, if a physical system of interest has time dynamics, estimation of multiple parameters has to be done simultaneously. However, unlike single parameter estimation, it is non-trivial to optimize multiple parameter estimation to achieve the maximum sensitivity. Various strategies have been thus proposed to enhance the sensitivity by optimizing either the probe states or the measurement scheme [14][15][16][17][18][19][20][21][22] . For example, quantum strategies exploiting various quantum probe states such as Greenberger-Horne-Zeilinger states 23 , single-photon Fock states 24 , squeezed states 12,25-28 , Holland-Bernett states [29][30][31] , and N00N states 14,32,33 have been extensively studied.
In particular, N00N states have been outstanding to investigate the fundamental quantum limit of quantum metrology given by the Heisenberg uncertainty principle with a fixed number of particles 34 . In a single-phase estimation scheme, N00N states can saturate the Heisenberg limit thanks to its largest number variance between the two modes 34 . Enhanced sensitivity beyond the standard quantum limit has been experimentally demonstrated with N00N states 35,36 . Recently, the concept of N00N states has been extended to generalized multi-mode N00N states to investigate the quantum enhancement in multiple-phase estimation 14 . Moreover, for multiple-phase estimation with limited resources, it has been known that multi-mode N00N states allow achieving the enhanced sensitivity outperforming the other quantum probe states or classical strategies for a multi-mode interferometer 14,15,32 . However, since generation of such multimode quantum probe states is challenging, experimental demonstrations of multiple parameter quantum metrology have been limited to utilizing quantum states other than multi-mode N00N states 12,15,23,24,27,29 .
In this work, we experimentally demonstrate quantum enhanced multiple-phase estimation using a multi-mode N00N state with photon number N = 2 and mode number m = 4. To this end, we propose a scheme for generating multi-mode N00N states. At first, we observe that the measured interference fringes exhibit phase super-resolution as a function of phase differences φ i (i = 1, 2, 3) of the generated 4-mode 2002 state with a measurement scheme using a 4 × 4 multi-mode beam splitter, socalled a quarter 37,38 . Then, we demonstrate quantum enhanced phase sensitivity of the 4-mode 2002 state by analyzing the Cramer-Rao bound (CRB) and the quantum Cramer-Rao bound (QCRB) 14,16,17,39 . Moreover, we show that the measured sensitivity is better than a coherent state jαi, which is a classical probe state, as well as another quantum probe state prepared by singlephoton Fock states 24 . Our results can motivate an investigation into the quantum strategies using multi-mode and multi-particle entanglement to develop the quantum enhanced multiple parameter metrology.

Results
Multiple-phase estimation scenario. Let us begin by introducing the system model for multiple-phase estimation schemes with multi-mode N00N states defined as where N is the number of photons, and they are distributed along m modes 14,32,33 . The multi-mode N00N state is a coherent superposition of all possibilities where N photons in one mode and none in any of the other m − 1 modes 34 . Then, we theoretically analyze the sensing scheme of d = m − 1 unknown multiple phases and the fixed photon number N = 2 with different probe states. A quantum probe state undergoes individual phase shifts φ = {φ 1 , φ 2 , ⋯ , φ d } and the phase shifted state is combined using an m × m multi-mode beam splitter and then detected by photon number-resolving detectors (PNRDs) at each mode. Here, the goal is to minimize the total uncertainty of the phase estimation governed by the CRB and the QCRB. The lower bound of the sum of the variance of each phase estimation given by the CRB and the QCRB is 14,16,17,39 where F C (φ) is the classical Fisher information matrix (CFIM), F Q (φ) is the quantum Fisher information matrix (QFIM), and μ is the number of measurements. Note that F −1 refers to the inverse of the Fisher information matrix F. The first and second inequalities are direct consequences of the CRB and the QCRB, respectively. Note that in our experiments, the CRB can always be saturated, asymptotically in μ, by using maximum likelihood estimator, and the mean of phases are expected to converge to the true values 17,39 . We theoretically analyze the CRB and the QCRB of d multiplephase estimation scheme with two different probe states. The first probe state we consider is a classical state prepared by injecting a coherent state α j i with an average photon number N ¼ 2 into one of the input ports of an m × m multi-mode beam splitter (Fig. 1a), and the other probe state is the two-photon m-mode N00N state (Fig. 1b). The measurement scheme is identical for two probe states. Each mode of the probe states is combined at another m × m multi-mode beam splitter after undergoing phase shifts, and then measured by PNRDs. We provide the CRB and the QCRB values of the total variance ∑ Δφ 2 for two probe states of coherent states and multi-mode N00N states with various number of phases d in Fig. 1c. Note that multi-mode N00N states always have lower phase uncertainty than coherent states, i.e., classical probe states.
Generation of the 4-mode 2002 state. The conceptual diagram of our proposed scheme for preparing the multi-mode N00N state with N = 2 and m = 4 is shown in Fig. 2a. The generation process of 4-mode 2002 state jψ 2 4 i is the following: Here, jΦ þ i ¼ 1 ffiffi iÞ is the triplet Bell state where, for example, j1 H α 0 i denotes the horizontally polarized single-photon state in the mode α 0 . Our scheme can generate ψ 2 4 with a unity conversion probability from the pre-selected initial Φ þ state into ψ 2 4 , thus two-photons always move together, see Methods for the detailed information. Then, prepared ψ 2 4 undergoes phase encoding φ i (i = 1, 2, 3) and it is combined by a quarter as shown in Fig. 2b. The output states are measured by single-photon detectors located at each mode. Note that one output port of the quarter is split into two (b 0 0 and b 0″ ) using a 50/50 fiber beam splitter to implement a pseudo-PNRD. Post-selected two-photon coincidence counts of both output port b 0 0 and b 0″ correspond to the two-photon counting at b 0 with success probability of 1/2. The schematic of experimental setup is shown in Fig. 2c.
As a first step to verify the generation of ψ 2 4 , we investigate the coherence among all four modes of ψ 2 4 by observing the interference fringes between the reference mode (a 0 ) and one of the other modes (a i ). We measure the two-photon coincidence counts on b 0 0 and b 0″ , C b0 0 b0 00 , of two input modes while we block the other two input modes. The phase shift φ i is realized by adjusting the optic axis angle θ i of the half waveplate (HWP) located between two quarter waveplates (QWPs). Note that φ i = 2θ i . As shown in Fig. 3a-c, results of C b0 0 b0 00 clearly reveal two times faster sinusoidal modulations compared to the singlephoton input case due to the λ/2 photonic de Broglie wavelength of two-photon N00N states [40][41][42] .
Then, in order to directly confirm the generation of ψ 2 4 , we measure two-photon probability distributions for the output states when we use all of the input modes a 0 , a 1 , a 2 , and a 3 of ψ 2 4 while we vary one of the phase encoding φ i . The experimental results of the interference fringes from all of the modes are shown in Fig. 3d-f, and they are compared with the theoretical calculations. We obtain the theoretical predictions of and the experimentally reconstructed quarter transition matrix. Then we experimentally measured post-selected coincidence counts of C i See Supplementary Note 1 for the detailed information on a quarter transition matrix and theoretical calculations. The results of Fig. 3d-f shows that the experimentally obtained interference fringes without normalization are very well-matched to the theoretical calculations, thus we can confirm that the prepared input state is ψ 2 4 .
Experimental multiple-phase estimation. Then, we investigate the sensitivity bound of multiple-phase estimation using our prepared ψ 2 4 and an ideal quarter when the CRB saturates the QCRB. We obtain the theoretical two-photon detection probability set of {P l (φ)} (l = 0, 1, . . . , 9) with a set of projectors jg satisfying the normalizing condition ∑ l P l (φ) = 1. Note that P 0 = P 1 = P 2 = P 3 , P 4 = P 5 , P 6 = P 7 , and P 8 = P 9 for all φ (See Methods). Then CFIM is given by where j and k can be 1, 2, and 3. The minimum value of the CRB is obtained to be Tr ½F À1 C ðφÞ = 1.5 where φ 1 ≃ π/2, φ 2 = 0, and φ 3 = π/2, and it saturates the QCRB = 1.5. In order to experimentally estimate the CRB value, we obtained interference fringes by scanning φ 1 near the point where we expected both the CRB and the QCRB to be saturated (φ 2 = −0.07π and φ 3 = 0.52π) for the prepared ψ 2 4 exp probe state (See Methods). Two-photon detection , which were appropriately normalized assuming the following relations Note that the assumed relations are always satisfied for P l (φ) with an ideal quarter and the experimentally reconstructed quarter transition matrix is close to an ideal quarter. Then obtained P exp l ðφÞ are functions of unknown phases φ and used to calculate the derivatives in Eq. (4). The detailed relation between P exp l ðφÞ and P m b i b j are provided in Supplementary Note 3.
Experimentally obtained two-photon detection probabilities are shown in Fig. 4a where our experimental data are well-matched to our fitting function P exp l ðφÞ, which are obtained from ψ 2 4 exp and U q;exp (see "Methods"). The CFIM can be obtained by using P exp l ðφÞ, and the diagonal terms of the CFIM at various φ 1 are plotted in Fig. 4b (see "Methods" for the detailed information). Note that the maximum values of all diagonal terms of the CFIM are 3 at φ 1 ≃ 0.5π with an ideal ψ 2 4 state and an ideal quarter. Then we numerically find the minimum CRB of Tr ½ðF exp C Þ À1 ðφÞ ¼  state and an ideal quarter at φ 1 ≃ 0.5π, φ 2 = 0, and φ 3 = 0.5π.
In our experiment, experimental errors are mainly from nonunity visibility of the observed interference, a phase fluctuation in each arm of an interferometer, a normalization assumption due to lack of superconducting nanowire single-photon detector (SNSPD) channels, and the fact that the quarter (U q;exp ) used in our experiment is slightly different from an ideal quarter U q , see Methods for comparison between U q;exp and U q . In particular, in undergoes the phase shift φ i (i = 1, 2, 3) in each mode. After the phase encoding, a quarter is used for combining four modes of the probe state, and then the two-photon output states are detected using single-photon detectors. In particular, one of the output modes of quarter (b 0 ) is divided into two modes using a FBS for two-photon detection in that mode.  Eq. (4), with an ideal quarter, ∂P l (φ)/∂φ j = 0 should be at the same φ j value for all P l (φ), however, a non-ideal quarter transition matrix U q;exp makes ∂P exp l ðφÞ=∂φ j ¼ 0 happens at slightly different φ j values. One can notice it from Fig. 4a that P do not have their minimum (maximum) value at φ 1 = 0.5π. This is the reason why the minimum value of the CRB is not obtained at φ 1 ≃ 0.5π but φ 1 ≃ 0.47π.
Furthermore, we emphasize that the enhanced sensitivity can be obtained by the prepared ψ 2 4 probe state compared to other quantum probe states using single-photon Fock states ψ Fock proposed in ref. 24 , where 1100 j i state is used as an input state instead of α j i in Fig. 1a (detailed calculations are provided in Supplementary Note 2). Moreover, we theoretically compare the sensitivity bounds between our multi-mode N00N state and an amplitude-unbalanced multi-mode N00N state ψ u , which is proposed in refs. 14,15,39 . The amplitude-unbalanced multi-mode N00N state has the form of . ψ u is known to have the minimum QCRB among the multi-mode N00N states 14,15,39 . In a measurement scheme using a quarter and PNRDs, the QCRB and the CRB values of ψ u with N = 2 and d = 3 are theoretically calculated to be 1.4 and 1.62, respectively. Here, we find that even though ψ u has the lower QCRB of 1.4 than 1.5 of our ψ 2 4 , ψ 2 4 can provide a better sensitivity (smaller CRB) 1.5 than 1.62 of ψ u with a realistic measurement scheme using a quarter. Note that an optimal measurement saturating the QCRB may not be experimentally feasible 14,17 .
In Table 1, we summarize the ideal QCRB and CRB values for various probe states as well as the CRB values obtained from our experimental results. We emphasize that the experimentally obtained CRB value of 1.85 ± 0.01 provides a better sensitivity than the ideal CRB values of 3 for α j i and 2.44 for ψ Fock . Our experiments use the post-selection technique and does not consider the photonic losses due to experimental imperfection 43 and lack of high-efficiency PNRDs 36 . By considering the postselection probability and photon losses, the CRB of our ψ 2 4 state cannot outperform the classical strategy. However, a genuine quantum enhancement can be achieved if one uses the state-ofart technologies such as high-efficiency PNRDs and optimized low-loss optical components with our ψ 2 4 state. Note that the post-selection technique does not affect the proof-of-concept verification of quantum enhancement and post-selection is standard technique used in almost previous quantum metrology experiment 15,23,24,29 . The corresponding quantitative analysis as well as the detailed discussions on photonic losses in our experimental setup are provided in Supplementary Note 4.

Discussion
In conclusion, we proposed a scheme for generating a multimode N00N state and experimentally demonstrate that the prepared quantum probe state is the 4-mode 2002 state by observing various interference fringes shown in Fig. 3 using a quarter and photon number resolving detection using post-selective pseudo-PNRDs. Moreover, we exploit the prepared 4-mode 2002 state as a quantum probe state for simultaneously estimating three phases of a 4-mode interferometer with quantum enhanced sensitivity. Then, we confirm that the CRB obtained by our 4-mode 2002 state and measurement scheme can saturate the QCRB. Our results provide a practical platform to investigate intriguing issues in the field of quantum multiple parameter metrology. At first, we emphasize that our scheme can be extended to generation of higher-mode N00N states. For instance, one can exploit a multiple-path Sagnac interferometer 44 to increase the number of modes from 4 to 4n, i.e., ψ 2 4n , where n is the number of Sagnac interferometers, and then one can estimate up to 4n − 1 phases simultaneously. Note that another scheme for generating multimode N00N states with N ≥ 2 has been theoretically proposed 33 . However, experimental demonstration seems to be challenging within current technology since it requires extremely strong nonlinearity or deterministic generation of multi-photon states. Another interesting future direction would be finding a realistic  measurement scheme for minimizing the CRB. In general, the CRB obtained by measurement scheme using a balanced multimode beam splitter cannot saturate the QCRB, see Fig. 1c with d = 2, 4, 5. Since the optimal measurement saturating the QCRB involves complex multi-photon states, it may not be experimentally feasible 14,17 . Hence, finding an experimentally realistic measurement scheme minimizing the CRB is essential for practical applications. Our results have direct applications for the quantum enhanced phase object imaging requiring a low photon flux 7,45,46 . In addition, our results can pave the way for demonstrating distributed quantum enhanced multiple-phase estimation by increasing the number of phases in local interferometers consisting of distributed quantum sensors 12,23,27 .

4-mode 2002 state preparation.
We used a CW single frequency laser operating at a center wavelength of 780 nm. The polarization of pump laser is set to polarization. The 10 mm-thick type-II periodically poled KTiOPO 4 (PPKTP) crystal with 46.15 μm poling period is located at the center of the Sagnac interferometer, which consists of a dual wavelength polarizing beam splitter (PBS), a dual wavelength HWP whose optic axis is oriented at 45 ∘ , and two dual wavelength mirrors as shown in Fig. 2c 47 . Here, dual wavelength optical components are designed for working at both 780 and 1560 nm photons. The horizontal (vertical) polarization component of the pump laser is transmitted (reflected) at dual wavelength PBS. The vertically polarized pump laser is changed to the horizontal polarization after transmitting a dual wavelength HWP. Then, the both of clockwise and counter-clockwise propagating pump laser have horizontal polarization in front of the PPKTP crystal and they probabilistically create photon pairs having orthogonal polarizations of H j i and V j i, respectively, via degenerate spontaneous parametric down conversion process. Then, after passing through the dual wavelength PBS, the counter propagating photon pair states are emerging to ð HV j iþ e iϕ VH j iÞ= ffiffi ffi 2 p , where ϕ is relative phase between two states. Then, we can prepare Φ þ state using a set of WPs. The measured heralding efficiency of Φ þ state is 37% with our SNSPD whose detection efficiency is 80%. Two photons prepared in Φ þ state simultaneously entered at both input ports of a lateral beam splitter (LBS), then we can observe the Hong-Ou-Mandel (HOM) interference 48 . The path length difference between two photons are matched to minimize the coincidence count rate detected at both output ports. See Supplementary Note 1 for the HOM interference results with Φ þ input state. After the LBS, we can prepare the two-photon N00N state with different polarization (See the second line of Eq. (3)). The horizontally (vertically) polarized photons are transmitted (reflected) at the PBS so that they split into 4 path modes depending on the polarization states. Note that only two-photon states can exist in each mode. By rotating the polarization state of the vertically polarized photons into the horizontal polarization, we can prepare ψ 2 4 .
Theoretical analysis of ψ whereâ y 0 is a creation operator, which creates a single-photon in the input mode a 0 of a quarter. A quarter has four input modes (a 0 , a 1 , a 2 , a 3 ) and four output modes , respectively, as shown in Fig. 2c. The unitary matrices for the phase encoding U φ and the ideal quarter transition U q are given by 37,38 , and U q ¼ 1 2 respectively. After the initial state ψ 2 4 undergoes the phase encoding and the quarter transformation, then the output state ψ out with c 0 = c 1 = c 2 = c 3 , c 4 = c 5 , c 6 = c 7 , and c 8 = c 9 . Then, one can obtain the twophoton detection probability of P l (φ) = |c l | 2 (l = 0, 1, ..., 9), and it satisfies the normalizing condition ∑ l P l (φ) = 1. P l (φ) is a function of φ, and used for theoretical calculations in Fig. 3. See Supplementary Note 3 for the detailed information on P l (φ) = |c l | 2 .
Then experimental two-photon detection probabilities are obtained as P exp l ðφÞ, it is used for fitting curves in Fig. 4a. See Supplementary Note 3 for the detailed calculations on our error analysis.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability
The code used to generate the figures within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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