Distributed quantum phase estimation with entangled photons
Li-Zheng Liu, Yu-Zhe Zhang, Zheng-Da Li, Rui Zhang, Xu-Fei Yin, Yue-Yang Fei, Li Li, Nai-Le Liu, Feihu Xu, Yu-Ao Chen, Jian-Wei Pan
DDistributed quantum phase estimation with entangledphotons
Li-Zheng Liu , , , ∗ , Yu-Zhe Zhang , , , ∗ , Zheng-Da Li , , , Rui Zhang , , , Xu-Fei Yin , , Yue-Yang Fei , , , Li Li , , , Nai-Le Liu , , , Feihu Xu , , , Yu-Ao Chen , , & Jian-Wei Pan , , Hefei National Laboratory for Physical Sciences at the Microscale and Department of ModernPhysics, University of Science and Technology of China, Hefei 230026, China Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics,University of Science and Technology of China, Shanghai 201315, China Shanghai Research Center for Quantum Sciences, Shanghai 201315, China ∗ These authors contribute equally.
Distributed quantum metrology can enhance the sensitivity for sensing spatially distributedparameters beyond the classical limits. Here we demonstrate distributed quantum phaseestimation with discrete variables to achieve Heisenberg limit phase measurements. Based onparallel entanglement in modes and particles, we demonstrate distributed quantum sensingfor both individual phase shifts and an averaged phase shift, with an error reduction up to1.4 dB and 2.7 dB below the shot-noise limit. Furthermore, we demonstrate a combinedstrategy with parallel mode entanglement and multiple passes of the phase shifter in eachmode. In particular, our experiment uses six entangled photons with each photon passingthe phase shifter up to six times, and achieves a total number of photon passes N = 21 atan error reduction up to 4.7 dB below the shot-noise limit. Our research provides a faithful a r X i v : . [ qu a n t - ph ] F e b erification of the benefit of entanglement and coherence for distributed quantum sensing ingeneral quantum networks. Introduction. —
Quantum metrology exploits the quantum mechanical effects to increase thesensitivity of precision sensors beyond the classical limit . By using the entanglement or coher-ence of the quantum resource , it can achieve higher precision of parameter estimation below theshot-noise limit (SNL), and its sensitivity can saturate the Heisenberg limit , which is believedto be the maximum sensitivity achievable over all kinds of probe quantum state.It is well known that many important applications can be regarded as sensor networks withspatially distributed parameters, which is often referred to distributed quantum metrology . Inthe framework of distributed quantum metrology, an important class of estimation problems is con-cerned with the sensing of individual parameters. Typically, the Heisenberg limit can be achievedin both continuous-variable and discrete-variable states . However, recently, there has been in-creasing interest in the study of multiparameter estimation, particularly in the linear combinationof the results of multiple simultaneous measurements at different locations (or modes), for exam-ple, averaged phase shift. For instance, estimating the averaged phase shift among remote modesis the fundamental building block to construct a quantum-enhanced international timescale (worldclock) ; in classical sensing, averaged phase shift is widely used to evaluate the gas concentrationof a hazardous gas in a given area
20, 21 , where the entangled network can substantially enhance thesensing accuracy. In such a case, estimating the parameters separately in modes is not optimal.Even with particle entanglement in each mode, the root-mean-square error (RMSE) for the esti-2ation of the linear combination of multiple parameters is restricted to √ M /N , where N denotesthe total number of entangled particles among all M modes with M ≤ N . However, the ultimateHeisenberg limit is /N . In contrast, the entanglement among modes in an entangled network cansubstantially enhance the sensitivity for multiparameter estimation
13, 22 . Remarkable experimentshave demonstrated distributed quantum metrology with multi-mode entangled continuous-variablestate
23, 24 .Recently, it has been shown that if the distributed sensors are entangled in both modes andparticles, it is possible to achieve the ultimate Heisenberg limit . Here, we refer to entangle-ment in both modes and particles as the parallel strategy. Furthermore, it has been shown that thesequential scheme — a single probe interacting coherently multiple times with the sample — canbe used to reach the same Heisenberg limit
4, 10 , and this has been shown to be an optimal parame-ter estimation strategy in several applications
29, 30 . Also, the combination of sequential scheme andparallel entanglement can even outperform parallel strategy for the estimation of multiple parame-ters in the presence of noise
10, 31–33 . We refer this combination as the combined strategy.In this Article, we perform the experimental demonstration of discrete-variable distributedquantum metrology for both individual phase shifts and averaged phase shift. In the parallel strat-egy for estimating individual phase shifts, by preparing three high-fidelity two-photon entangledsources, we demonstrate three individual phase-shift measurements, where the distributed sen-sors of (mode 1, mode 2, mode 3) achieve a super-resolution effect with RMSE reductions up to(1.44 dB, 1.43 dB, 1.43 dB). In the parallel strategy for estimating averaged phase shifts, by con-3tructing a high-fidelity multiphoton interferometer, we compare the sensitivities for the scenariosof modes entangled/separated and particles entangled/separated, which are referred to as MePe,MePs, MsPe, MsPs, respectively. The results show that compared with the SNL of MsPs, thedistributed sensors of (MePe, MePs, MsPe) can achieve a precision of RMSE reduction up to (2.7dB, 1.56 dB, 1.43 dB) for the estimation of an averaged phase shift across three modes. In thecombined strategy, by interacting the photons with the phase shifter multiple times in each mode,we perform a demonstration for the estimation of an unequal-weight linear function of multiplephase shifts across six entangled modes. The experiment realizes a total number of photon-passesat n = 21 with an error reduction up to 4.7 dB below the SNL. Note that the evaluation of thesuper-resolution effect in these experiments uses post selection that does not include the photonlosses . Protocol. —
Let us consider a general scenario in the framework of distributed quantummetrology with M modes. As shown in Fig. 1, we assume that each mode has an unknown phaseshift θ k . In the estimation of individual parameters, we assume the quantity to be estimated isthree individual phase shifts. The probe states in our experiment have the form | φ individual (cid:105) = √ ( | HH (cid:105) + | V V (cid:105) ) for each mode, where H ( V ) denotes the horizontal (vertical) polarization. Inthe multiparameter strategy, we assume that the quantity to be estimated is a linear global function ˆ θ = α T θ , where θ = ( θ , . . . , θ M ) and α = ( α , . . . , α M ) denote, respectively, the vector of phaseshift and the normalization coefficients for the mode k = 1 , . . . , M . α T is the transpose of α . Theunitary evolution is given by, ˆ U ( θ ) = e − i (cid:80) Mk =1 ˆ H k θ k = e − i ˆ H · θ , (1)4here ˆ H = ( ˆ H , . . . , ˆ H M ) are the Hamiltonians governing the evolution. The task is to estimate ˆ θ with a high precision using classical or quantum sensor networks. Here the uncertainty (or error)for the estimation of ˆ θ can be described by ∆ˆ θ = (cid:0) α T Σ α (cid:1) / with multiparameter covariancematrix Σ whose elements are Σ km = E [( θ k − θ est,k )( θ m − θ est,m )] , where θ est,k and θ est,m denote,respectively, the locally unbiased estimator for θ k and θ m for k, m = 1 , . . . , M . E [ X ] denotes theexpectation value of the random variable X .(1) In the parallel strategy, we set three probe modes where the objective function to beestimated is the averaged phase shift ˆ θ = (cid:80) k =1 θ k / , and the Hamiltonians are set to ˆ H k = σ z / for mode k = 1 , , , where σ z is the Pauli matrix. Thus the evolution can be described as ˆ U k = e − iθ k / e iθ k / . (2)The overall probe states in this scheme can be classified as the modes entangled and particles entan-gled (MePe), modes entangled and particles separated (MePs) and modes separated and particlesentangled (MsPe), which have the form of, | φ MePe (cid:105) = 1 √ | HH (cid:105) M | HH (cid:105) M | HH (cid:105) M + | V V (cid:105) M | V V (cid:105) M | V V (cid:105) M ) , (3) | φ MePs (cid:105) = 12 ( | H (cid:105) M | H (cid:105) M | H (cid:105) M + | V (cid:105) M | V (cid:105) M | V (cid:105) M ) ⊗ , | φ MsPe (cid:105) = 12 √ | HH (cid:105) + | V V (cid:105) ) M ( | HH (cid:105) + | V V (cid:105) ) M ( | HH (cid:105) + | V V (cid:105) ) M . (2) In the combined strategy, beside the parallel entanglement among modes, we utilize thecoherence rather than the particle entanglement in each mode. In this case, the essential feature isthat the phase shift is being interacted coherently over many passes of the unitary evolution, ˆ U k . We5mplement a distributed phase sensing with six modes, and the objective functions to be estimatedare unequal weighted linear functions: ˆ θ = (cid:80) k =1 kθ k / . For each modes k , the evolution withmultiple passes j are set to, ˆ U jk = j (cid:89) i =1 ˆ U k , (4)where ˆ U k is same as Eq. (8). To demonstrate this protocol, we define two types of probe states,namely modes entangled and particles coherent (MePc) and modes separated and particles coherent(MsPc), which can be written as | φ MePc (cid:105) = 1 √ | H (cid:105) M | H (cid:105) M | H (cid:105) M | H (cid:105) M | H (cid:105) M | H (cid:105) M + | V (cid:105) M | V (cid:105) M | V (cid:105) M | V (cid:105) M | V (cid:105) M | V (cid:105) M ) , (5) | φ MsPc (cid:105) = ⊗ M Mi = M (cid:18) √ | H (cid:105) + | V (cid:105) ) M i (cid:19) . The projective measurements on the probe states are performed in the σ x basis, which canachieve the maximum visibility for interference fringe
25, 27 . In this setting, the outcome probabilityin the eigenvectors | ± (cid:105) can be written as P ± = 1 ± V ± cos ˆ θ , (6)where V ± is the interference fringe visibility for n -Greenberger-Horne-Zeilinger states
9, 11 . Thewidely adopted elementary bounds on the RMSE are given by the Cramer-Rao bound ∆ˆ θ ≥ α T α √ µ α T F α , where µ denotes the number of independent measurements and F denotes the classi-cal Fisher matrix with elements ( F ) kl = (cid:80) i = ± P i [( ∂/∂θ k ) P i ] [( ∂/∂θ l ) P i ] . The effective Fisher6nformation (FI) , F (ˆ θ ) , can be used to evaluate the estimation sensitivity , and it is given by, F (ˆ θ ) = α T F α ( α T α ) . (7)By combining Eq. (6) with equation Eq. (7), we can calculate the FI for the linear function ˆ θ .Note that the FI is used to quantify that the accuracy for different strategies (Methods) and thecalculations of FI use only the post selected photons, which does not include photon losses. Experimental set-up. —
The experimental set-up is illustrated in Fig. 2a,b. A pulsed ultravi-olet laser—with a central wavelength of 390 nm, a pulse duration of 150 fs and a repetition rate of80 MHz—is focused on three sandwich-like combinations of BBO crystals (C-BBO) to generatethe independent entangled photon pairs in the channel 1 and 2, 3 and 4, and 5 and 6. With thisconfiguration, the photon pairs are generated in the state | φ + (cid:105) = √ ( | HH (cid:105) + | V V (cid:105) ) , where H ( V ) denotes the horizontal (vertical) polarization.In the parallel strategy (Fig. 2a), the initial probe states | φ MePe (cid:105) , | φ MePs (cid:105) , | φ MsPe (cid:105) can be pre-pared by combining three independent spontaneous parametric down conversion (SPDC) sourcesand a tunable interferometer with the platforms as shown in Fig. 2c-e. In the combined strategy(Fig. 2b), the photon in mode k coherently passes through the phase shift k times, and Hong-Ou-Mandel (HOM) interferences between photons 2 and 3 and photons 4 and 5 are applied. In eachchannel, a lens is used to ensure the collimation of the beam. The narrow-bandpass filters with full-width at half-maximum (FWHM) wavelengths of λ FWHM = 4nm are used to suppress frequency-correlated effect between the signal photon and the idler photon. The probe states evolve andpass through quatre wave plate (QWP) and half wave plate (HWP), and finally detected by single-7hoton detectors.The detailed configurations of the tunable interferometer with four inputs and four outputsare shown in Fig. 2c-e. The tunable interferometer consists of two polarizing beam splitters (
PBS and PBS ) controlled by multi-axis translation stages, whose position at left and right correspondsto non-interference and interference between photon 2 and 3 (4 and 5). The three C-BBOs generatethree Einstein-Podolsky-Rosen entangled photon pairs in the states (cid:12)(cid:12) φ +12 (cid:11) , (cid:12)(cid:12) φ +34 (cid:11) , (cid:12)(cid:12) φ +56 (cid:11) . As shownin Fig. 2c, to prepare the state | φ MePe (cid:105) , the positions of both
PBS and PBS controlled by twomulti-axis translation stages are set to the right; the two PBSs will introduce the Hong-Ou-Mandelinterference between photons 2 and 3 and photons 4 and 5. By replacing one of the C-BBOs witha single piece of BBO crystal as shown in Fig. 2d, the state of | φ MePs (cid:105) is produced. We obtain thedown-conversion probability of the prepared uncorrelated state is about p = 0 . . In Fig. 2e,the positions of both PBS and PBS are set to the left, where there is no interference; this leadsto the prepared quantum state of | φ MsPe (cid:105) and | φ individual (cid:105) . They have a typical down-conversionprobability of p = 0 . per pulse and a fidelity 0.9866 ± Results. —
In the estimation of individual phases, each mode occupies a two-photon entan-gled state. We direct the photon 1 and (cid:48) , (cid:48) and (cid:48) , and (cid:48) and (cid:48) to mode 1, mode 2 and mode 3respectively, and introduce the phase shifts with { θ , θ , θ } continuously from to π . The valuesto be estimated are three individual phase shifts { θ , θ , θ } . The results are shown in Fig. 3. Weobtain the visibility { . , . } (cid:48) , { . , . } (cid:48) (cid:48) and { . , . } (cid:48) for mode k = 1 , and 3, and the optimal FIs are 3.88, 3.85 and 3.86, respectively. This also forms the results for8sPe.In the parallel strategy for estimating averaged phase shift, after the evolutions by QWPs andHWP, the coincidence measurements in the basis σ ⊗ x are performed. Fig. 4a shows the observedaverage outcome probability values with θ ramping continuously from to π where we fix θ = π , θ = π . The experimental data are fitted to the function P ± = ± V ± cos(6ˆ θ )2 , where V + = 0 . and V − = 0 . denote the fringe visibilities in the eigenvectors | ± (cid:105) of measurement basis σ x ⊗ .To quantify the sensitivity, we calculate the FI according to Eq. (7). As shown in Fig. 4b, wedemonstrate an enhancement of sensitivity for a range of phase shifts, and the maximum value ofFI is about 20.825 at ˆ θ = π , which represents a 2.70-dB reduction as compared with the SNL ofMsPs.For the protocol with MePs, the photons 1 and (cid:48) , (cid:48) and (cid:48) , and (cid:48) and 6 are directed to mode1, mode 2 and mode 3, respectively. Similar to MePe, we fix the phase shift θ = π , θ = π andchange the phase shift θ continuously from to π . For photons 1, (cid:48) and (cid:48) ( (cid:48) , (cid:48) and 6), theoutcome probability values can be obtained by observing the counts of photons (cid:48) , (cid:48) and 6 (1, (cid:48) and (cid:48) ). Fig. 4c shows the observed outcome probability values. In this case, the fitting function isset to P (cid:48) (cid:48) ± = P (cid:48) (cid:48) ± = ± V ± cos(3ˆ θ )2 . We obtain the fringe visibility in the eigenvectors | λ = ± (cid:105) of measurement basis σ x ⊗ as { . , . } (cid:48) (cid:48) and { . , . } (cid:48) (cid:48) . From Fig. 4d, we alsodemonstrate the sensitivity enhancement, and the maximum value of FI is 12.313 at the phase shift ˆ θ = π/ , which is a 1.56-dB reduction over the SNL of MsPs.With a little difference, the value to be estimated in the strategy of multiparameter estimation9s the linear function ˆ θ = ( θ + θ + θ ) . The fitting function is set to P (cid:48) ± = P (cid:48) (cid:48) ± = P (cid:48) ± = ± V ± cos(2 θ k )2 for mode k = 1 , and 3. The results of the protocol with MsPe are shown in Fig. 4e,f.We obtain the visibility { . , . } (cid:48) , { . , . } (cid:48) &5 (cid:48) and { . , . } (cid:48) &6 for mode k =1, 2 and 3, respectively. The optimal FI of mode 1 is about 3.887, and the optimal FIs of mode2 and mode 3 are 3.832 and 3.877, respectively (Extended Data Figure 1). The results in Fig. 4clearly demonstrate that MePe is the best choice with the highest sensitivity to estimate the globalfunction ˆ θ .Next, we demonstrate the combined strategy with θ ramping continuously from to π . Theexperimental data are fitted to the function P ± = ± V ± cos(21ˆ θ )2 , where V + = 0 . and V − = 0 . denote the fringe visibilities in the eigenvectors | ± (cid:105) of measurement basis σ x ⊗ . As shown inFig. 5a,b, we fit the observed average outcome probability values and calculate the FI accordingto Eq. (7). the maximum value of FI is about 180 at ˆ θ = π , which represents a 4.7 dB reductioncompared with the SNL of MsPc.Finally, we consider the range where we expect to beat the theoretical limit based on theprobe state MePc. We take around 70 measurements to obtain the probabilities of the measurementoutcomes. The estimator ˆ θ = (cid:80) k =1 kθ k / is obtained using the maximum likelihood estimation,which maximizes the posterior probability based on the obtained data. To experimentally obtainthe statistics of ˆ θ , we repeat the process 100 times to get the distribution of ˆ θ , from which thestandard deviation of the estimator δ ˆ θ is obtained. As shown in Fig. 5c, the experimental precision(black dots) saturates the theoretical optimum value.10 iscussion and conclusion. — Our experiment uses post-selection which does not include theexperimental imperfections of probabilistic generation of photons from SPDC and the photon loss.The post selection is a standard technique in almost all (except for ref. ) previous quantum metrol-ogy experiments . In future, with the improvement of collection and detection efficiency , ourset-up can be directly extended to the demonstration of unconditional violation of the SNL formulti parameters. Also, for the combined strategy, we assume that the samples have no absorp-tions and the samples’ phases are uniformly distributed. However, these assumptions do not haveinfluence on the proof-of-concept verification of the super-resolution effect.Overall, we have demonstrated three types of strategies for distributed quantum metrology,by observing the visibility and FI of phase super-resolution. First, we demonstrate the estimationof individual parameters in three modes. All experimental fringes shown in Fig 3 present highvisibility that is sufficient to beat the SNL. Second, by using a tunable interferometer, we estimatean averaged phase shift across three modes. The visibility of (MePe, MePs, MsPe) shown in Fig. 4clearly demonstrates that MePe is the optimal choice with the highest sensitivity to estimate theaveraged phase shift. The maximum value of FI is about 20.825, which beats all the theoreticalbounds for MePs, MsPe and MsPs (Methods). Third, by interacting the photon through the samplesmultiple times in each mode, we demonstrate the combined strategy with parallel entanglementacross six modes and photon passes up to N = 21 . Our results may open a new window for ex-ploring the advanced features of entanglement and coherence in a quantum network for distributedquantum phase estimation, which may find quantum enhancements for sensing applications.11 ata availability The data that support the plots within this paper and other findings of this study are availablefrom the corresponding authors upon reasonable request.
Code availability
The code that support the plots within this paper and other findings of this study are availablefrom the corresponding authors upon reasonable request.
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This work was supported by the National Key Research and Development (R&D) Plan ofChina (under Grants No. 2018YFB0504300 and No. 2018YFA0306501), the National NaturalScience Foundation of China (under Grants No. 11425417, No. 61771443 and U1738140), theShanghai Municipal Science and Technology Major Project (Grant No. 2019SHZDZX01), theAnhui Initiative in Quantum Information Technologies and the Chinese Academy of Sciences.
Author contributions
Z.-D.L., F.X., Y.-A.C. and J.-W.P. conceived and designed the experiments. Z.-D.L., F.X.and Y.-A.C. designed and characterized the multiphoton optical circuits. Z.-D.L., R.Z., X.-F.Y.,L.-Z.L., Y.H., Y.-Q.F and Y.-Y.F carried out the experiments. Z.-D.L, R.Z., F.X. and Y.-A.C.analysed the data. Z.-D.L., X.F., Y.-A.C. and J.-W.P. wrote the manuscript, with input from allauthors. F.X., Y.-A.C. and J.-W.P. supervised the project.16 ompeting financial interests
The authors declare no competing financial or non-financial interests.
Additional information
Correspondence and requests for materials should be addressed to F.X., Y.-A.C or J.-W.P.17 𝜃 k Figure 1:
A sketch for estimating distributed multiparameter.
Each sensor node (white) isequipped with a measurement device where the red dashed line represents a single pass or mul-tiple passes and θ k represents the unknown phase shift in mode k , while the central node (red) isequipped with a source to produce multi-party entangled states. The white dashed lines representthe quantum channel that can be used to distribute photons.18igure 2: Experimental set-up. a,b,
The set-up for the parallel strategy ( a ) and the combinedstrategy ( b ). An ultrafast ultraviolet pump laser passes BBO(s) or C-BBO(s) to produce threedown-converted photon pairs. In each channel, a lens is used to ensure the collimation of thebeam. The narrow-bandpass filters with full-width at half-maximum (FWHM) wavelengths of λ FWHM = 4 nm are used to suppress frequency-correlated effect between the signal photon and theidler photon. The probes prepared by different combinations of the interferometer, are distributedto three modes and then undergo the evolution. Finally, the probes are detected by measurementsystems, consisting of PBS, HWP, QWP and two single-photon detectors. A complete set of six-photon coincidence events are simultaneously registered. c-e The interferometer configura-tions to produce the quantum states of MePe( c ), MePs( d ), MsPe( e ). SC-YVO and TC-YVO represent spatial compensation (SC) and temporal compensation (TC) yttrium orthovanadate crys-tals (YVO ) 19igure 3: Experimental results for estimating single parameters for mode 1, mode 2 andmode 3. a,c,e,
The average outcome probability for mode 1 ( a ), mode 2 ( c ) and mode 3 ( e ) inthe measurement basis σ ⊗ x for two-photon entangled states. The blue (orange) lines represent theaverage outcome probability P (cid:48) + , P (cid:48) (cid:48) + and P (cid:48) ( P (cid:48) − , P (cid:48) (cid:48) − and P (cid:48) − ) for mode 1, mode 2 andmode 3. b,d,f, The FI per trial, fitted from { P (cid:48) , P (cid:48) (cid:48) , P (cid:48) } for mode 1 ( b ), mode 2 ( d ) and mode3 ( f ). The shaded areas correspond to the confidence region, which are obtained from theuncertainty in the fitting parameters. Error bars are calculated from measurement statistics and toosmall to be visible. The blue dashed line is the theoretical limit of FI for the Heisenberg limit. Theblack dotted line is the theoretical limit of FI for the SNL.20igure 4: Experimental results for the parallel strategies of MePe, MePs and MsPe. a,c,e,
The average outcome probabilities in the measurement basis σ ⊗ x , σ ⊗ x , σ ⊗ x for six-photon ( a ),three-photon ( c ) and two-photon ( e ) entangled states. In a , the blue (orange) line represents theaverage outcome probability P (cid:48) (cid:48) (cid:48) (cid:48) ( P (cid:48) (cid:48) (cid:48) (cid:48) − ). In c , the blue (orange) line represents theaverage outcome probability P (cid:48) (cid:48) + ( P (cid:48) (cid:48) − ) and the green (red) line represents the average outcomeprobability P (cid:48) (cid:48) ( P (cid:48) (cid:48) − ). In e , the blue (orange) line represents the average outcome probability P (cid:48) + ( P (cid:48) − ). b,d,f, The FI per trial, fitted from P (cid:48) (cid:48) (cid:48) (cid:48) , { P (cid:48) (cid:48) , P (cid:48) (cid:48) } and { P (cid:48) , P (cid:48) (cid:48) , P (cid:48) } for MePe ( b ), MePs ( d ) and MsPe ( f ). The shaded areas correspond to the 90 % confidence region,which are obtained from the uncertainty in the fitting parameters. Error bars are calculated frommeasurement statistics. The red dot-dashed line is the theoretical limit of FI for MePs. The bluedashed line is the theoretical limit of FI for MsPe. The black dotted line is the theoretical value ofFI for MsPs. 21igure 5: Experimental results for the combined strategy. a , The average outcome probabilityin the measurement basis σ ⊗ x for six-photon to the probe state | φ MePc (cid:105) . The blue (orange) linerepresents the average outcome probability P ( P − ). b , The FI per trial, fitted from P for MePc. The red dot-dashed line is the theoretical limit of FI for MsPc. The purple dotted lineis the theoretical limit of FI for the SNL. c , Observed phase estimation and its uncertainty. Theblack solid line is the theoretical limit of RMSE for MePc. The purple dashed line is the theoreticallimit of RMSE for the SNL. The shaded areas correspond to the 90 % confidence region, which areobtained from the uncertainty in the fitting parameters. The square data points are calculated from7000 detection events. The error bars are discussed in the Methods.22 ethod Sensitivity evaluation
In our experiment, we assume that the form of objective function is ˆ θ = α T θ , where θ = ( θ , . . . , θ M ) and α = ( α , . . . , α M ) denote, respectively, the vector of phase shift and thenormalization coefficients with (cid:80) k α k = 1 . The Hamiltonians are set to H k = σ z / for mode k = 1 , . . . , M , where σ z is the pauli matrix. The unitary operator of mode k can be expressed as U k = e − iθ k / e iθ k / . (8)According to the given evolution, we can determine the sensitivity for different estimation strate-gies. (1) Let us start with the analysis of the parallel strategy. To obtain the optimal sensitivity formodes entangled and particles entangled (MePe), we will first consider the probe state that containentanglement among all of the N photons, that is, the Greenberger-Horne-Zeilinger (GHZ) state: | φ MePe (cid:105) ini = 1 √ (cid:0) Σ Mk =1 | H (cid:105) ⊗ N k + Σ Mk =1 | V (cid:105) ⊗ N k (cid:1) , (9)where N k = α k N denotes the number of photons in mode k = 1 , . . . , M and the total number ofphotons is N = (cid:80) k N k . The probe state after the evolution as described in Eq. (8) is of the form | φ MePe (cid:105) evo = 1 √ (cid:32) M (cid:79) k =1 | H (cid:105) ⊗ N k + e i (cid:80) Mk =1 N k θ k M (cid:79) k =1 | V (cid:105) ⊗ N k (cid:33) . (10)The projective measurements on the probe state are performed in the σ x basis, which can achievethe maximum visibility for interference fringe
9, 11 . In this setting, the outcome probability in the23igenvectors | ± (cid:105) are P MePe ± = 1 ± V ± cos(Σ Mk =1 n k θ k )2 = 1 ± V ± cos( N ˆ θ )2 , (11)where V ± denotes the fringe visibilities in the eigenvectors | ± (cid:105) of measurement basis σ x ⊗ .Following the above expression, the FI of | φ MePe (cid:105) can be calculated as F MePe (ˆ θ ) = V N sin ( N ˆ θ )1 − V cos ( N ˆ θ ) . (12)It is easy to see that the Heisenberg limit δ ˆ θ = 1 / √ F MePe = 1 /N can be achieved when the noiseis free ( V = 1) .We then consider the sensitivity for modes entangled and particles separated (MePs). For thepurpose of our study, the equal weight linear function is considered and the probe state reach theoptimal sensitivity can be written as | φ MePs (cid:105) ini = 12 (cid:0) Σ Mk =1 | H (cid:105) k + Σ Mk =1 | V (cid:105) k (cid:1) ⊗ NM , (13)where we assume N/M is an integer. After the evolution, this probe state becomes to | φ MePs (cid:105) evo = 12 (cid:16) ⊗ Mk =1 | H (cid:105) k + e i (cid:80) Mk =1 θ k ⊗ Mk =1 | V (cid:105) k (cid:17) ⊗ NM . (14)Since the objective function is ˆ θ = α T θ , and | φ MePs (cid:105) evo is a product state of
N/M identical M -mode entanglement states, the theoretical limit for MePs is the sum of the FI of these N/M states,that is, δ ˆ θ = 1 / √ F MePs = 1 / √ M N .Indeed, the protocol for modes separated and particles entangled (MsPe) can be viewedas estimating the parameters separately in each mode . The sensitivity is converged to δ ˆ θ =1 / √ F MsPe = 1 / (cid:16)(cid:80) Mk =1 N k (cid:17) , which is equal to the sum of of single-parameter sensitivities.24n our experiment, we set the number of photons N = 6 and the number of modes M = 3 .Therefore, according to above conclusions, the FI are F MePe = 36 , F MePs = 18 and F MsPe = 12 when the noise is free ( V = 1) .(2) In the combined strategy, we utilize the coherence rather than the particle entanglementin each mode. In this case, the essential feature is that the phase shift is being interacted coherentlyover many passes of the unitary evolution. This process can be described as follows | φ MePc (cid:105) = 1 √ (cid:0) Σ Mk =1 | H (cid:105) k + Σ Mk =1 | V (cid:105) k (cid:1) θ −→ √ (cid:16) Σ Mk =1 | H (cid:105) k + e i (cid:80) Mk =1 θ k Σ Mk =1 | V (cid:105) k (cid:17) , (15)where n k = α k n denotes the number of interactions in mode k = 1 , . . . , M and the totalnumber of interactions is n = (cid:80) k n k . When the noise is free ( V = 1) , the Heisenberg limit canbe achieved for MePc, that is, δ ˆ θ = 1 / √ F = 1 /n . The sensitivity for mode separated and particleconherent (MsPc) is converged to δ ˆ θ = 1 / √ F MsPe = 1 / (cid:0) Σ Mk =1 n k (cid:1) . In our experiment, we setthe number of interactions n = Σ k =1 k = 21 and the number of modes M = 6 , and thus the FI are F MePc = 441 , F MsPc = 91 . Error analysis
To obtain the standard deviation of the value of phase shifter, we take k measurement sets,and each set contains around s coincidence events. In our experiment, around 7000 coincidenceevents are measured and divided into 100 groups for each phase shifter. By using maximumlikelihood method, the standard deviation δ ˆ θ is then obtained from the outcome probability, which25re calculated from these coincidence events. The error for this experimentally obtained δ ˆ θ is wellapproximated by δ ˆ θ = δ ˆ θ/ (cid:112) s − . Extended Data Figure 1
Figure 6:
Experimental results for the parallel strategies of MsPe for mode 2 and mode 3. a,c,
The average outcome probability in the measurement basis σ ⊗ x for two-photon entangled states( (cid:48) (cid:48) and (cid:48) respectively). Blue (orange) lines represent the average outcome probability P (cid:48) (cid:48) + and P (cid:48) ( P (cid:48) (cid:48) − and P (cid:48) − ) for mode 2 and mode 3. b,d, The fisher information per trial, fitted from P (cid:48) (cid:48) and P (cid:48) for MsPs, respectively. The shaded areas correspond to the90%