Distributed Phase Estimation at the Heisenberg Limit with Classical Light

Quantum metrology, such as quantum phase estimation, can surpass classical sensing limits, reaching the Heisenberg‐scaling precision. So far, this kind of metrology has been thought to be only implementable with the quantum systems, which, however, are fragile to environmental noise and hardly contribute to the practical detections. Herein, it is demonstrated both theoretically and experimentally that the parameter encoded by the optical phase can also be estimated at the Heisenberg scaling in classical optics. Inspired by the quantum‐entanglement‐enhanced sensing scheme, the estimation is performed by using classically correlated beams as probes, and obtaining the probes readout after their interaction with the target system. Because the correlated beams considered are spatially separable, a distributed phase estimation scheme is given, which can sense the linear combinations of the phase shifts induced by distinct systems. The results of our experiments show an error reduction up to 3.89 dB below the classical limit when the correlated beam number for probing is 6, approaching the Heisenberg limit. Compared with quantum strategies, the proposal shows a better robustness against the environmental disturbance and keeps their performances even when the correlated beam number is relatively large. Hence, it indicates promising practical applications in the future.


Introduction
Parameter estimation is one essential step for exploring the physical world.[3][4][5][6] Nowadays, numerous detections dive into the realm of extreme cases, such as the super-resolution microscopy for the medical diagnostics at molecular level, [7][8][9] hyperfine spectroscopy of atomic states, [10][11][12] high-precision interferometry for the gravitational wave, [13][14][15][16] or stable sensor networks across thousands of kilometers for global positioning, [17][18][19] leading to an unparalleled need for estimating the relevant parameters at higher levels of precision.Traditionally, the precision in terms of the statistical errors can be improved by simply repeating the measurement and averaging the outputs.For instance, we use a beam of classical light as a probe to estimate an optical phase and we repeatedly read out the intensity.According to the central limit theorem, the error of the average finally converges to Δσ= ffiffiffiffi N p when the number of the repeats N is sufficiently large.Here, Δσ is the error of a single measurement of one probe with the same light source in a constant integral time.The statistical scaling of errors with 1= ffiffiffiffi N p is also known as the shot noise limit (SNL), which serves as the fundamental bound for improving the precision of the estimation through independent measurements.
By far, one of the most promising candidates surpassing the SNL is the quantum metrology.[25][26][27][28][29][30][31][32][33][34] For example, a recent work about the distributed quantum phase estimation with entangled photons has shown a better precision beyond the classical limit. [32]In this work, Greenberger-Horne-Zeilinger (GHZ)-like multiphoton-entangled states are prepared as the input probes of the measurement system.The input photons interact with the samples to be estimated and carry the phase information.Using projective measurement, the authors extract the target information from entangled photons and then estimate the distributed phase parameters.In this experiment, the authors use photon counting devices to do the measurement and thus each photon in the entangled states is regarded as a single probe.Results have shown that the sensitivity of this quantum scheme surpasses the classical limit and even scales at K À1 , which is usually called the Heisenberg limit (HL). [2,3]Here, K is the number of probes (photons) used in the experiment.A more detailed summary of this quantum scheme can be found in Section S1 (Supporting Information).
Albeit the quantum estimation schemes have their attractive advantages, there are still some unavoidable difficulties when building a practical quantum measurement system.On the one hand, if we want the entanglement scheme to realize a considerable improvement compared to the classical one, we always need to prepare certain multiparticle-entangled state as the input state.However, the preparation of large entangled state with high fidelity is extremely challenging. [26,35,36]On the other hand, the quantum system is so fragile that the results can be easily affected by the environment noise.[39][40] Therefore, the practical application of quantum-enhanced measurement scheme is quite limited.
In this work, inspired by the quantum-enhanced measurement scheme, we design a distributed measurement scheme based on classical light.Using the classical correlations induced by different degrees of freedom (DOFs) of the classical light, e.g., spatial distribution and polarization, we reduce the standard estimation error of the parameters encoded in the optical phase to the scale of N À1 .Here, N represents the number of correlated beams, which is also considered as the probes of the measurement process.The sensitivity of our scheme shows a better result compared with the classical N-time-repeats experiment, which uses the same optical power laser beams as the probes.The precision improvement between our scheme and the N-time repeats experiment is similar to the improvement between the quantum-enhanced scheme and the N individual photon measurement.This means our scheme achieves a Heisenbergscaling phase estimation.Furthermore, we propose and analyze a global phase estimation protocol.Unlike the quantum estimation schemes, our scheme shows a better robustness against the environment noise, which would benefit the practical applications in the future.

Theoretical Scheme
Let us first specify the distributed phase estimation problem.Suppose that there are M propagation modes, which are spatially separable and will introduce unknown phase shifts θ 1 , : : : , θ M to the probes after the interaction with them.The problem of estimation we concern about is how to estimate these phase shifts at the Heisenberg scaling with classical optics.As shown in Figure 1a, our basic scheme also consists of three parts: the source part, the encoding part, and the correlated detection part.In the source part, we introduce the classically correlated states of the beams as our probes for sensing the parameters.After interacting with the modes, the unknown values of the phase shifts are carried out by the probe states in the encoding part.Finally, in the correlated detection part, the probe readouts are obtained by The orange lines represent the standard error obtained from the curves of P AE .The blue-dashed line is the theoretical limit of the standard error for the Heisenberg limit (HL).The black-dashed line is the theoretical limit of the standard error for the shot noise limit (SNL).
local detectors and then post-processed according to our requirements.Such a distributed phase estimation scheme with classical light has a correspondence with the quantum entanglement scheme.We illustrate details of the processes as follows.

Source State Preparation
43][44][45][46][47][48][49][50][51] Furthermore, with some special designs, we can construct an analogy of quantum entanglement, which is usually considered as a precious resource to many problems.For instance, the polarization of a single classical beam can be denoted by jEÞ ¼ c H jhÞ þ c V jvÞ , which is analog to the quantum state of a two-level system.Here, jhÞ and jvÞ represent the horizontal and vertical polarization basis, respectively.Complex numbers c H and c V are the projective component of state jEÞ on jhÞ and jvÞ respectively, denoted by c H ¼ ðhjEÞ and c V ¼ ðvjEÞ like the Dirac inner product of the quantum states.hj ð ( vj ð ) are the Hermitian conjugate of jhÞ (jvÞ), and jc [44][45][46][47][48][49] As described in refs.[42-49], we can build a classical optical field corresponding to the Bell state in quantum system by using two laser beams E 1 and E 2 with different wavelengths.The forms of the beams system are taken as Following the expression of the single classical beam jEÞ ¼ c H jhÞ þ c V jvÞ, here these two classical beams can be denoted by j2EÞ ¼ 2 À1=2 ½jhÞ a jvÞ b þ jvÞ a jhÞ b .In ref. [47], the authors have performed the Clauser-Horne-Shimony-Holt test in this system and the experiment yields the strongest violation of Bell's inequalities.It is verified that this classical beams system have a good correspondence with the two-qubit state, including the entangled states.By establishing the correlation among the polarization states of the multiple beams, one can also obtain the state analog to the N-qubit quantum state as shown in ref. [51].Now, based on the understanding of classical correspondence with an N-qubit state, we construct a classical analogy of the product state of M quantum N-GHZ states jψi Here, ⊗ denotes the tensor product operation, jhÞ ⊗N denotes the product of N state vector jhÞs, and the same goes for other similar terms.M represents the number of modes in the distributed estimation scheme.The product operation is easy to achieve since we only need to make sure every single mode evolves individually and do the multiplication at last.So we focus on constructing the classical analogy of the N-GHZ state ðjhi ⊗N þ jvi ⊗N Þ= ffiffi ffi 2 p in each mode as shown in Figure 1b.In each mode, we use N beams to encode the N-GHZ state, which is expressed by where f H ðr, tÞ and f V ðr, tÞ are orthonormal basis of an optical DOF.They could be the frequency components, optical paths, etc., satisfying where Ω represent the domain of the orthonormal relation.We define the correlation function as c i 1 i 2 : : : i N :¼∫ Q N n¼1 ðe n ⋅ E n ÞdΩ, which correspond to the complex coefficient α i 1 i 2 : : : i N ¼ ½ i 1 j i 2 j : : : i N jjψ h h h in the N-qubit state.Here, e n is the projection direction of the E n 's polarization state.Using this mapping relation, we can calculate and find that the N beams system in Equation (1) corresponds to the N-GHZ state.Therefore, we denote this N beams system in one mode as the state ½jhÞ ⊗N þ jvÞ ⊗N = ffiffi ffi 2 p , which is regarded as the analogy of the N-GHZ state ðjhi ⊗N þ jvi ⊗N Þ= ffiffi ffi 2 p .As Figure 1b shows, the N beams correspond to the N qubits in quantum scheme, and they are the N probes for sensing a single parameter θ m : After the construction of classical analogy in each mode, we simply do the multiplication and can easily get the overall classical state: which is the mimic of the product state of M quantum N-GHZ states.
Inspired by the quantum-enhanced measurement scheme, the precision of phase estimation can be improved obviously and is close to the HL when the initial probe state is maximally entangled state like N-GHZ state (see Equation (S1) in Section S1, Supporting Information).Similarly in our theory scheme, we prepare N-correlated beams as the probes to estimate each phase parameter.In the practical experiment, we can apply the optical path DOF to implement f H ðr, tÞ and f V ðr, tÞ.Using beam splitter (BS), polarizing beam splitter (PBS), halfwave plate (HWP), and other devices, the preparation of the initial state described by Equation ( 2) would be relatively easier.A specific design is shown in Experimental Verification.

Parameter Encoding
As shown in Figure 1b, one of parameter to be estimated θ m is detected by the state . Particularly, the parameter is represented by the phase shift of a sample, and finally encoded in the probe state.Inspired by the quantum-enhanced measurement scheme, we also set that the interactions between the sample and the N beams satisfy the relation f H ðr, tÞ ! e Àiθ m =2 f H ðr, tÞ, and f V ðr, tÞ ! e iθ m =2 f V ðr, tÞ More generally, for sensing M parameters θ 1 ; : : : ;θ M , the interactions between the corresponding samples and the copies of probe state ½jhÞ ⊗N þ jvÞ ⊗N = ffiffi ffi 2 p are described by the similar relation with Equation (3).The only differences are the specific subscript m of θ m , which ranges from 1 to M. Hence, by using the total correlation function of the M Â N beams mentioned in the state preparation part, the process for sensing M parameters can be given by the following evolution of the state We can easily find that the aforementioned evolution has a correspondence with that of the photonic state in quantum scheme (see Equation (S3) in Section S1, Supporting Information), which indicates that we realize an equivalent parameter-encoding process comparing with the controlled unitary evolution in quantum scheme.

Measurement
Following the aforementioned manners, one of the parameters, such as θ m , can be estimated by using the normalized intensity P m AE , given by where as marked in Figure 1b.According to the source state preparation part, P m AE can be measured by the correlation we mentioned.Suppose that the beams encoding the output state given by Equation ( 4) are denoted by E 0 n with n ranging from 1 to M Â N.Then, P m AE can be obtained in the following steps.First, pick out the N beams E 0 n m (n m ¼ ðm À 1ÞN þ 1; …, mN) as the probes interacting with the mth mode and encoding θ m .Second, record the local projections h ⋅ E 0 n m and v ⋅ E 0 n m .Third, multiply the N projections on the h direction as well as the N projections on the v direction, and integrate the product, as shown by the "Post Process" in the right side of Figure 1a.Hence, the expression j∫ ) is the vital one, and our solution is to use Mach-Zehnder interferometer (MZI) for homodyne detection, [51] specifically described as follows.For the E 0 n m with a specific a m , an ancillary beam . Finally, h ⋅ E 0 n m (or v ⋅ E 0 n m ) can be obtained by measuring the intensity difference of the two light signals.The procedure of the aforementioned actions would become quite simple when the DOF of our beams are properly set.For example, one can consider using the DOF of optical paths.Then, f H and f V correspond to the different propagation paths for light.Hence, according to our design, the light in each path of the beam is purely horizontal or vertical polarized.Therefore, we can directly use piezoelectric ceramics for global phase modulation without considering the influence of polarization state.In the next section, we conduct such an experiment to verify our proposal using the DOF of optical paths.Furthermore, it can be noticed that the process for obtaining P m AE has a correspondence with the coincidence counting measurement in the quantum scheme.And also, it is easy to find that the form and result of the correlated function in Equation ( 5) are consistent with the projective measurement of the quantum scheme shown by Equation (S4) in Section S1, Supporting Information.
Finally, we briefly discuss the phase parameter estimation process.Because P m AE is the function of phase parameter θ m , the value of θ m can be estimated by the measurement value of P m AE .Based on Equation ( 5), P m AE reflects the magnitude of correlated components j AE 1Þ that can be observed in the output state given by Equation ( 4).Therefore, by using the picture of statistical process, P m AE can be interpreted as quantity proportional to the chance or the probability of finding the component.Then, the process we estimate θ m according to P m AE turns to be a classical parameter estimation problem in statistics.Using maximum likelihood estimation method, we can always get to the precision limit with sufficient times of measurement. [52]One more thing to add is that the estimation process described here is similar to that one in the quantum scheme (see the measurement part in Section S1, Supporting Information).
To describe the precision of this measurement scheme and compare it with the HL, we obtain the standard error of the measurement result through the calculation of Fisher information (FI).In classical statistics, suppose that there is a probability distribution fP i ðθÞg, where θ is the parameter to be estimated, the definition of FI can be written as The standard error has an approximate relation with the FI that σðθÞ ¼ 1= ffiffiffiffiffiffiffiffiffiffiffi FIðθÞ p (more details about FI can be found in ref. [52]).In fact, the estimation method for our classical scheme is analog to that applied in the quantum scheme.We illustrate the data processing method specifically later.
In our proposal, it needs to be emphasized that our detectors could only record the intensity in a constant time but not the photon numbers, so that the basic detecting element is a beam of light.The normalized intensity P m AE is regarded as a measure of N-correlated probes.Therefore, the FI calculated here is a little different from what is usually considered in a quantum scheme.We only compare the precision of our scheme with the N-time repeats measurement because both of them regard a laser beam with certain optical power as one basic probe.Considering that the result of our correlated measurement is affected by noise in practical experiment, here we give a hypothetical form of the correlated function in a noisy environment where parameter V M m ∈ ð0, 1Þ is the interference fringe visibility, which describes the influence of environmental noise.When the environment is completely free of noise, V M m ¼ 1.The experimental data Pm AE are fitted to Equation (7).Then, we can easily figure out the standard error through the calculation of FI As Equation ( 8) shows, our classical scheme could approach the precision of the HL theoretically. Figure 1c-h gives a more concrete explanation about this.Suppose that our experiment is relatively noiseless, here we take fringe visibility parameter V = 0.9995.Figure 1c-e shows the theoretical curves of correlation function P AE ðθÞ for the single-parameter measurement when the number of the correlated beams interacting with the corresponding mode, N, is 2, 6, and 12.The corresponding standard error results are shown in Figure 1f-h

Experimental Verification
To experimentally validate our theoretical scheme, we investigate the phase estimation problem in two scenarios.One is the single-phase parameter estimation by using jφ 2,1 Þ, jφ 4,1 Þ, and jφ 6,1 Þ, and other is distributed phase parameter estimation by using jφ 2,3 Þ.
Figure 2a shows the experimental setup for single-phase parameter estimation by using jφ 2,1 Þ, jφ 4,1 Þ, and jφ 6,1 Þ.Here, we mainly instruct the setup for the scheme of jφ 2,1 Þ.The other two can be easily performed by extending the setup of the jφ 2,1 Þ scheme.Corresponding to the schematic illustration in Figure 1a, our experiments implement the three parts we present in the section of theory scheme.For implementing the state preparation part, a laser with a central wavelength of 632.8 nm is introduced, polarized by a beam displacer (BD).Then, the polarization state of the light is rotated to 45°by an HWP, and divided into two sub-beams by a BS thereafter.The two sub-beams are employed to further give the light E 1 and E 2 of the state jφ 2,1 Þ, according to the setup given by Equation ( 1) and (2).Next, each sub-beam is divided by a PBS, generating the two basis of the optical path DOF denoted by f H and f V .After that, we obtain the probe beams 1), and complete the state preparation part.
For implementing the parameter-encoding part, the unknown phase shift is effectively induced by the piezoelectric ceramics.It is added to the f V v component of E 1 and E 2 , marked by the yellow square in Figure 2a and corresponding to the process given by Equation (3).For implementing the measurement part, the local oscillators for obtaining the polarization projection are generated by the non-polarizing beam splitter (NPBS) after the PBS.The employment of the NPBS can ensure a 50:50 division of the polarized light at a higher precision than the BS.Then, the interference of the local oscillators and the output beams are realized by the four NPBSs, and results are given by the intensity difference between the light signals output by each NPBS.Generally the results of the local projections h ⋅ E 0 n and v ⋅ E 0 n ðn ¼ 1; 2Þ are complex numbers.One way to detect the real and imaginary part of the local projections is to introduce a phase delay of 0 and π=2, respectively, to the local oscillators, and then measure the intensity difference.Lastly, multiply the two results obtained by the f H path of corresponding output light E 0 1 and E 0 2 , and the same goes for the two results obtained by the f V path of E 0 1 and E 0 2 .The value of P AE can be obtained by adding up the two products or subtracting one from the other, which is theoretically illustrated by Equation ( 5).Such a procedure is the simplified version of the strategy we propose in the measurement part of the theoretical scheme section.
To estimate parameter θ when θ belongs to ½0; 2π, we finally introduce a phase difference of ½0; 2π and ½π=2; 5π=2 in the local oscillator path, respectively.This is realized by using a piezoelectric ceramic modulated reflector to change the optical length.To perform the schemes of jφ 4,1 Þ and jφ 6,1 Þ, one can simply repeat the setup scheme of jφ 2,1 Þ, as shown by the black box in Figure 2a.The only difference is that multiplication of the interference results has to involve those from E 0 1 to E 0 4 (for jφ 4,1 Þ) or from E 0 1 to E 0 6 (for jφ 6,1 Þ).Through the parametric fitting methods described near Equation ( 7), we calculate the corresponding standard error of the data obtained by all the schemes.
The results of our experiments are shown in Figure 2b-g.Figure 2b-d shows the measurement results of the correlation function P AE ðθÞ.The red and black solid lines correspond to the theoretical values of P þ ðθÞ and P À ðθÞ, respectively, and the red and black data points are the corresponding experimental results.We find that the experimental and theoretical values are in good agreement.According to the experimental results, the interference fringe visibility obtained by our fitting are V N¼2 ¼ 0.9859, V N¼4 ¼ 0.9816, and V N¼6 ¼ 0.9996, which indicates that the environmental noise has little disturbance to our experiment.The standard error curves we calculated using the parameter V are shown in Figure 2e-g.We focus on the improvement between our scheme and N-time repeats measurement, which use the same optical power laser beams as the probes.The minimum standard errors described by SNL should be 1= ffiffi ffi 2 p , 1= ffiffi ffi 4 p , and 1= ffiffi ffi 6 p , respectively, when N ¼ 2; 4, and 6 as the blue-dotted lines show.And the HL gives the results that the minimum standard errors should be 1=2, 1=4, and 1=6, correspondingly, as the black-dotted lines show.In our classical scheme, the experimental values of the minimum standard errors obtained for the three cases are σ N¼2 ¼ 1= ffiffiffiffiffiffiffiffi ffi .Compared with the classical noise limit SNL, we give a reduction in measurement standard error of 1.44, 2.93, and 3.89 dB, respectively.The results we obtained are consistent with the trend of the 1=N relation given by the HL.Therefore, we verified that the accuracy of our constructed classical optical field measurement scheme is able to exceed the classical noise limit SNL and is consistent with the accuracy results of the HL.
Figure 3a shows the experimental setup for distributed multiparameter estimation by using jφ 2,3 Þ. Modes 1, 2, and 3 mark the three different phase parameters to be estimated, each of which is measured with only two probes or two correlated beams.The experimental process is similar to the single-parameter estimation as shown in Figure 2a.Because the beams we set are spatially separable, the estimation can be performed across a long distance.
The correlation function curves are shown in Figure 3b-d.The experimental results are consistent with our theoretical prediction.It is easy to find that our results at different modes are very similar, which indicates that we have the same precision across the distribution area in our classical scheme.In these three modes, the visibilities of interference fringe obtained by our fitting are V M 1 ¼ 0.9994, V M 2 ¼ 0.9996, and V M 3 ¼ 0.9997.Using these parameters, we figure out their standard error curves as shown in Figure 3e-g.Still we only focus on the improvement between our scheme and N-time repeats measurement.The minimum standard error described by SNL should be 1= ffiffi ffi 2 p when N ¼ 2, while the HL gives the result that it should be 1=2.In our classical scheme, the experimental values of the minimum standard errors obtained for the three modes are  -d).The blackand blue-dashed lines are the limit of accuracy described by the HL and the SNL, respectively.
, and σ M 3 ¼ 1= ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3.9976 p .Compared with the classical noise limit SNL, we give a reduction in measurement standard error of 1.50 dB approximately.Such results indicate that our measurement scheme can achieve distributed phase estimation with the same accuracy as the HL predicts.
In addition, we can also use other input state for phase estimation and receive a precision better than the SNL, such as jφ MN,1 Þ and jφ M, N Þ.In our proposal, we still take N beams as the probe array for sensing one single mode.Using jφ MN,1 Þ as the input indicates that the probe beams for sensing all the modes are correlated like a large GHZ state.In addition, using jφ M,N Þ as the input indicates that the probe beams of the same order in the arrays for sensing different modes are correlated as the GHZ state, while the ones in the array for sensing one mode are not.The specific theoretical details of these two input state schemes can be found in Section S2, Supporting Information.In fact, the measurement process we give for these two input states is also similar to that in the quantum scheme (see Section S1, Supporting Information, for more details). [32]ere, we use these two input states to estimate the average of the phase parameters when M ¼ 3 and N ¼ 2, as shown in , respectively.In our classical scheme, the experimental , which represents 3.899 and 2.383 dB reduction compared with the SNL, respectively.The results show that the measurement scheme we designed can also implement a multiparameter distributed measurement process for various input states, and the precision is the same as the best quantum measurement scheme.

Global Phase Estimation
In the previous discussions, we have demonstrated that the precision of our scheme approaches the Heisenberg scaling described by the FI.][55][56][57][58][59] In other words, it only indicates the ability to distinguish a small difference around a known point.For a general phase estimation problem, it is more important to measure a phase parameter without any prior knowledge.In ref. [39], the author proposed a practical quantum method to realize the global Heisenberg scaling measurement.Here, we show that this kind of global phase estimation can also be implemented by using the aforementioned scheme.
In the first step, we prepare the input state jφ 2 N , 1 Þ ¼ 2 À1 ½jhÞ ⊗2 N þ jvÞ ⊗2 N with N-correlated beams, and encode the phase parameter θ in each one.Then, we measure the normalized intensity where AE1j ¼ ½ðhj . The experimental realization of state preparation, parameter encoding and measurement is similar with what we did in Section 3. If P þ,1 > P À,1 , we set Inductively, in the kth step, we prepare the input state )correlated beams, and encode the phase parameter θ in each one.Then, we encode an extra phase parameter θ 0 k ¼ ÀA 1 2 Àkþ1 πÀ A 2 2 Àkþ2 π À : : : À A kÀ1 2 À1 π.After that, we measure the normal- . The previous process is repeated until we get A Nþ1 in the (N þ 1)th step.The final estimation of phase θ should be This protocol has a good correspondence with the quantum method described in ref. [39].We use (N À k þ 1)-correlated beams to construct the input state and encode the phase parameters θ and θ 0 k to obtain the final state 1 ffiffi 2 p ½e Àið2 NÀkþ1 θþθ 0 k Þ=2 jhÞ ⊗2 NÀkþ1 þ e ið2 NÀkþ1 θþθ 0 k Þ=2 jvÞ ⊗2 NÀkþ1 , which corresponds to the quantum final state U ÀA 1 2 Àkþ1 πÀ : : : j1Þ in the kth step.And also, the normalized  7) and the correlation function curves in (a,b).The black-and blue-dashed lines are the limit of accuracy described by the quantum scheme and the SNL, respectively.intensity measurement P AE,k corresponds to the projective measurement in the basis fjþi, jÀig in the quantum protocol.Therefore, precision of the previous estimation should be consistent with that achieved by the quantum protocol.In ref. [39], the author defined two notions of Heisenberg scaling in global estimation respectively based on the average estimation error R:¼E θ,θ ½sin 2 ð θ À θÞ and the limiting distribution of estimation error P θ fj θ À θj > cg, where c is a certain threshold.The Heisenberg scaling is achieved if R scales as n À1 and c scales as Oðn À1 Þ, where n is the number of resources consumed (while in this protocol n ¼ 2 Nþ1 À 1).To experimentally validate the global phase estimation protocol, we set N ¼ 1 and estimate the phase parameter θ in ½0, 2πÞ.The output expected values and the frequency distribution of the estimation error are shown in Figure 5.
Figure 5a shows the output expected values, and Figure 5b corresponds to the frequency distribution of the estimation error.Here, the division value is taken as π=2, because we set N ¼ 1.In Figure 5a, the black solid line represents the result when the expected value is exactly the true value.The red plus sign points are the experimental outputs, and each point is the average value of three times readout.It is seen clearly that the experimental output values are basically close to the true values.The measuring range is obviously larger than that of the local phase estimation protocol with three entangled probes.It should be noted that we do not show the result when the true value close to 2π.The reason is that the result of phase estimation is a multivalued function and there is no essential difference between 0 and 2π.However, their values of number are quite different.This problem is expected to be solved if we use more probes to increase the precision.From Figure 5b, it is found that the average estimation error in the experiment is R exp ¼ E θ,θ ½sin 2 ð θ À θÞ ¼ 0.1949 based on the error distribution.In ref. [39], the theoretical result of the quantum protocol in the noiseless case should be R theo ¼ 2 ÀðNþ2Þ ¼ 0.125.Comparing our experimental results with the theoretical case, some deviation appears.There are at least two reasons for explaining it.One is that the authors use some approximation in the calculating process and the theoretical result works better when N is large.However, we only set N ¼ 1 in our experiment.The other reason is that we only choose several data points with equal interval in ½0, 2πÞ.The statistical property is a little different from the case where the unknown phase parameter is subject to the uniform distribution on the continuous set.Comparing our experimental results with the previous phase estimation schemes, the precision of our experiment is better than the SNL.Let us consider the precision bounded by the SNL in a traditional scenario.If the expected value θ is a Gaussian random variable, the average value is θ and the variance σ 2 ¼ 1=n ¼ 1=3.Therefore, the estimation error R SNL % 0.2433 and we have R theo < R exp < R SNL .This means that we can achieve the global phase estimation by using the special designed classical optical system and improve the sensing limit compared with the previous classical scheme.

Discussion and Conclusion
In our experiment, we use a single parameter V to describe the influence of practical ambient noise.Even though our description of the noisy model is relatively simple, we could obviously find that the practical noise in our experiment has little influence to our measurement system because of the robustness of classical light.At this point, quantum scheme shows a different result since the quantum-entangled state is easy to be affected due to the decoherence.In the result of ref. [32], the fringe visibility V decreases significantly with the increase of entangled qubit number N, which makes the actual precision lower than the theoretical limit.In the result of our experiment, on the one hand, the value of parameter V is always close to 1 as theory scheme described.On the other hand, parameter V does not change obviously with the increase of the correlation number N. It shows that, compared with the quantum entanglement scheme, our classical light scheme is more robust and remains good sensitivity especially when the number of the correlated beam N is sufficiently large.In summary, we propose a new way of distributed phase estimation with classical light.Our theoretical and experimental results show that the measurement precision of phase estimation can be further improved with the help of classical correlation, which is analog to the schemes enhanced by the quantum entanglement.Particularly, the error of the estimation by our schemes has been decreased to the HL scale, surpassing the traditional SNL showing the advance of the quantum schemes.In addition, we have proposed a global phase estimation protocol based on our classical optical platform, and the results have shown that our scheme could also implement the global phase estimation and the precision is better than the SNL cases.Compared with the quantum schemes, our classical light schemes are more robust to the ambient noise and have better sampling efficiency without photon loss.It is expected to be directly applied to the existing measuring instruments and benefit the practical detection process in the future.

Experimental Section
Instrument Specification: In our experiment, the laser (Thorlabs, HNL210LB) with a central wavelength of 632.8 nm was utilized to prepare source states.We used BD (Thorlabs, BD40) to polarize the initial beam with a transmission rate at 90%.The HWP (Daheng Optics, GCL-0604) rotated the linear polarized state to 45°.Then the beam was divided into two sub-beams by a 50:50 BS (Daheng, GCC-4010).PBSs (Daheng Optics, Gcc-4020) were used to generate the two basis of optical path DOF denoted by f H and f V .NPBSs (Daheng Optics, GCC-4031) were used to generate the local oscillator denoted by E LO , with a transmission rate at 92%.We used power meters (Thorlabs, PM160) as the detectors to obtain the results.
Data Processing: A specific process for obtaining the correlation function P AE was described in the measurement part in our theoretical scheme.Here, we only briefly introduced the calculation process of FI.After obtaining the correlation function curves, we used the nonlinear curve fitting function supported by Origin, fitting our experimental data into the noisy model as expressed in Equation (7).We figured out the parameter of the interference fringe visibility V.And then, we could calculate the FI and the standard error.

Figure 1 .
Figure 1.a) Schematic illustration of the distributed phase estimation with classical light.b) An illustration of the single-parameter sensing involved in the distributed scheme.c-h) Theoretical results of the correlation function P AE and the standard error of the single-phase parameter estimation scheme for c,f ) N ¼ 2, d,g) N ¼ 6, and e,h) N ¼ 12. c-e) The black (red) lines represent the correlation functionP þ ðP À Þ. f-h)The orange lines represent the standard error obtained from the curves of P AE .The blue-dashed line is the theoretical limit of the standard error for the Heisenberg limit (HL).The black-dashed line is the theoretical limit of the standard error for the shot noise limit (SNL).
. The SNL and the HL are shown as black and blue dotted lines on the figures respectively.For the P AE ðθÞ curves, since the precision is proportional to the slope, it indicates that the precision increases when the correlated beam number N is getting larger.The minimum standard errors indicated by the SNL are 1= ffiffi tively, when N ¼ 2; 6; 12.The HL gives the result that the minimum standard errors could be 1=2, 1=6, and 1=12, correspondingly.In our classical scheme, the theoretical minimum standard errors are 1= ffiffiffiffiffiffiffiffiffiffiffi 3are close to the HL.Compared with SNL, our scheme offers an error reduction of 1.50, 3.89, and 5.39 dB.Therefore, the precision of our estimation scheme is better than the traditional N-time repeats measurement and is close to the Heisenberg scaling, which is similar to the improvement of quantum-enhanced schemes compared with N individual photon measurement.

Figure 2 .
Figure 2. a) Experimental setup for single-phase parameter estimation when M ¼ 1 and N ¼ 2; 4; 6.In our actual experimental process, the phase shift θ introduced by sample S1 is realized via the piezoelectric ceramic-modulated reflectors in Mach-Zehnder interferometers (MZIs).The optical power of all beams after beam splitter (BS) is about 110 μW.b-d) The results of the correlation function P AE of the previous experiment.The red and black solid lines in the figure correspond to the theoretical curves P þ ðθÞ and P À ðθÞ, respectively, and the red and black scattered points are the experimental results.e-g) The results of standard error curves.The black solid lines are the theoretical result when considering the interference fringe visibility V = 0.9999, and the red scattered dots are the experimental results obtained by fitting calculations based on Equation (7) and the correlation function curves in (b-d).The blackand blue-dashed lines are the limit of accuracy described by the HL and the SNL, respectively.

Figure 4 .
schemes can be found in Section S2, Supporting Information.In fact, the measurement process we give for these two input states is also similar to that in the quantum scheme (see Section S1, Supporting Information, for more details).[32]Here, we use these two input states to estimate the average of the phase parameters when M ¼ 3 and N ¼ 2, as shown in Figure 4.The correlation function curves are shown in Figure 4a,b.Similarly, the experimental values and the theoretical values are in good agreement.According to the experimental results, the interference fringe visibilities obtained by our fitting are V MN,1 ¼ 0.9996, V n¼1 M,N ¼ 0.9994, and V n¼2 M,N ¼ 0.9995.The standard error curves we calculated using the parameter V are shown in Figure 4c,d.The SNL gives the minimum standard error as 1= ffiffi ffi 6 p .While, using these input state, the quantum scheme gives the minimum standard error as 1= ffiffiffiffiffi 36 p and 1= ffiffiffiffiffi 18 p, respectively.In our classical scheme, the experimental

Figure 3 .
Figure 3. a) Experimental setup for distributed multiparameter estimation when M ¼ 3 and N ¼ 2. The optical power of all beams after BS is about 110 μW.b-d) The results of the correlation function P AE of the previous experiment.The red and black solid lines in the figure correspond to the theoretical curves of P þ ðθÞ and P À ðθÞ, respectively, while the red and black scattered points are the experimental results.e-g) The results of standard error curves.The black solid lines are the theoretical result when considering the interferometric fringe visibility V = 0.9999, and the red scattered dots are the experimental results obtained by fitting calculations based on Equation (7) and the correlation function curves in (b-d).The black-and blue-dashed lines are the limit of accuracy described by the HL and the SNL, respectively.

Figure 4 .
Figure 4. Results of the average phase parameters estimation for jφ MN,1 Þ and jφ M, N Þ states when M ¼ 3 and N ¼ 2. a,b) The correlation function curves P AE .The red and black solid lines in the figure correspond to the theoretical curves P þ ðθÞ and P À ðθÞ, respectively, while the red and black scattered points are the experimental results.c,d) The standard error curves.The black solid line represents the theoretical result when considering the interferometric fringe visibility V = 0.9999, and the red scattered dots are the experimental results obtained by fitting calculations based on Equation (7) and the correlation function curves in (a,b).The black-and blue-dashed lines are the limit of accuracy described by the quantum scheme and the SNL, respectively.

Figure 5 .
Figure 5. Experimental results of the global phase estimation protocol when N ¼ 1 and n ¼ 2 Nþ1 À 1 ¼ 3. a) The output expected value.The black solid line in the figure shows the result when the expected value is exactly the true value.The red plus sign points are the experimental outputs.Each point is the average value of three times readout.The yellow background area represents the measuring range of the local phase estimation protocol when three entangled probes are employed.b) The frequency distribution of the estimation error θ À θ.
where the two beams E a and E b correspond to the two qubits in the Bell state.The first-order correlation of electric fields c ij ∶ ¼ E Ã ai ðr, tÞE bj ðr, tÞ ði, j ¼ h, vÞ corresponds to the coefficient α ij ¼ ½ a ij b hjjjψi ab h in the two qubits state jψi ab ¼ P i,j¼h,v α ij jiij ji.Using this mapping relation, we find that such a classical optical field consisted of E a and E b corresponds to the Bell state jψ