Useful multiparticle entanglement and sub shot-noise sensitivity
in experimental phase estimation
We experimentally demonstrate a general criterion to identify entangled states useful for the estimation of an unknown phase shift with a sensitivity higher than the shot-noise limit. We show how to exploit this entanglement on the examples of a maximum likelihood as well as of a Bayesian phase estimation protocol. Using an entangled four-photon state we achieve a phase sensitivity clearly beyond the shot-noise limit. Our detailed comparison of methods and quantum states for entanglement enhanced metrology reveals the connection between multiparticle entanglement and sub shot-noise uncertainty, both in a frequentist and in a Bayesian phase estimation setting.
The field of quantum enhanced metrology is attracting increasing interest GiovannettiNatPhot11 () and impressive experimental progress has been achieved with photons RarityPRL90 (); MitchellNat04 (); KacprowiczNP2010 (); NagataSci07 (); Xiang_2010 (), cold/thermal atoms AppelPNAS2009 (), ions LeibfriedSci04 () and Bose-Einstein condensates GrossNat10 (); RiedelNat10 (). Several experiments have demonstrated phase super resolution MitchellNat04 (); LeibfriedSci04 (), which, if observed with a high visibility of the interference fringes, allows to utilize the state for quantum enhanced metrology ReschPRL07 (); NagataSci07 (). So far, only few experiments have implemented a full phase estimation protocol beating the shot-noise limit with , where is the number of particles LeibfriedSci04 (); AppelPNAS2009 (); GrossNat10 (). Recently, it has been theoretically shown that sub shot-noise (SSN) phase sensitivity requires the presence of (multi-)particle entanglement PezzePRL09 (); HyllusArXiv10b (). In this letter, we experimentally demonstrate this connection. For an entangled state and a separable state with addressable photons, we measure the quantum Fisher information (QFI) Helstrom67 (), which quantifies the amount of entanglement of the state useful for SSN interferometry PezzePRL09 (). We then show how this entanglement can indeed be exploited by implementing a Maximum Likelihood (ML) and a Bayesian phase estimation protocol, both clearly yielding SSN phase uncertainty.
The usefulness of an experimental state can be quantified by the quantum Fisher information (QFI) Helstrom67 (). A probe state of qubits is entangled and allows for SSN phase estimation if the condition
is fulfilled PezzePRL09 (). Here is the linear generator of the phase shift, and is a Pauli matrix rotating the qubit along the arbitrary direction . The maximal further depends on the hierarchical entanglement structure of the probe state and genuine multiparticle entanglement is needed to reach the Heisenberg limit HyllusArXiv10b (); SI (), the ultimate sensitivity allowed by quantum mechanics. With qubits, 2-particle entangled states have , while for 3-particle entangled states HyllusArXiv10b (); nota_k-ent (). The ultimate limit is which is saturated by the so-called Greenberger-Horne-Zeilinger (GHZ) state GHZ (); GiovannettiPRL06 (); PezzePRL09 ().
A state fulfilling Eq. (1) allows for SSN phase uncertainty due to the Cramer-Rao theorem, which limits the standard deviation of unbiased phase estimation as Cramer_book (); Helstrom67 (); note_resources ()
The first inequality defines the Cramer-Rao lower bound (CRLB). Here is the true value of the phase shift, is the number of repeated independent measurements, and
The Fisher information depends on the conditional probabilities to obtain the result in a measurement when the true phase shift is equal to . It is bounded by the QFI PezzePRL09 (); Helstrom67 (), the equality being saturated for an optimal measurement . From Eqs (1) and (2) and from the bounds for multi-particle entanglement, we can infer that, if the experimentally obtained of a -qubit state exceeds the value for -particle entanglement, one can achieve a phase sensitivity better than that achievable with any -particle entangled state of any qubits nota_k-ent ().
For the experimental demonstration, we use the symmetric four-photon entangled Dicke state KieselPRL07 (); notaFock () and the separable state observed from multiphoton parametric down conversion WieczorekPRL09 () [Fig. 1 a)]. Here , () refer to the horizontal (vertical) polarization of a photon in the spatial mode , and . From the measured density matrices ( and SI ()) we deduce a fidelity of for and for (errors deduced with Poissonian count statistics) and also the QFI determining the suitability of the experimentally observed states for phase estimation. For the ideal Dicke state , the QFI reaches its maximum value, , when for all () HyllusPRA10 (). In the experiment, this choice leads to , at the maximal value achievable with 3-particle entanglement. An optimization over the local directions HyllusPRA10 (), leads to the slightly higher value , detecting useful 4-particle entanglement with standard deviations. Sure enough, using a witness operator it is possible to prove 4-particle entanglement in a simpler way KieselPRL07 (); SI (). With only a subset of the tomographic data we obtain a witness expectation value of , proving 4-particle entanglement with a significance of 40 standard deviations SI (). However, witness operators merely recognize entanglement, whereas our criterion directly indicates the state’s applicability for a quantum task. The separable state ideally allows for sensitivity at the shot-noise limit, . The experimental density matrix leads to , a value close to the expected separable limit (the optimized value being ).
In order to demonstrate that the precision close to the one predicted by can indeed be achieved in practise, we experimentally implement a phase estimation analysis with the input states and . Our interferometric protocol transforms the probe state by using the halfwave-plates and phase shifts depicted in Figs 1 a) and b). The unknown value of the phase shift is inferred from the difference in the number of particles, (), in the states and . For the ideal states and the rotation directions , this measurement is optimal, and hence . Experimentally, the optimized direction and measurement can be different because of noise and misalignment. However, for the observed states the expected improvement would be rather small.
The relation between the phase shift and the possible results of a measurement is provided by the conditional probabilities . These are measured experimentally and compared with the theoretical ones for both the separable and the entangled state, as shown in Fig. 2 a)-k).
A fit to the measured conditional probabilities provides , which are used to calculate the Fisher information according to Eq. (3) [see Fig. 2 l)]. As expected, our experimental apparatus can surpass the shot noise limit for a broad range of phase values (where ), and can even exploit useful three particle entanglement (where ).
The phase shift is inferred from the results, , of independent repetitions of the interferometric protocol. We will refer to such a collection of measurements as a single -experiment. In the experiment, we set the phase shift to 9 known values . For each , 12000 results are independently measured and grouped into vectors of length to perform the phase estimation for different values of . Using this data, we implement a ML and a Bayesian phase estimation protocol. While both have been recently used in literature for phase estimation KacprowiczNP2010 (); PezzePRL07 (), here they are compared in detail and applied for the first time to demonstrate SSN phase uncertainty with more than two particles. To display the quantum enhancement and to compare the methods we use the rescaled uncertainty defined below.
In the ML protocol, the estimator of the unknown phase shift is determined as the value maximizing the likelihood function Cramer_book (). For different -experiments it fluctuates with standard deviation , which has to be calculated by repeating a large number of single -experiments. For large , the distribution of approaches a Gaussian centered on and of width saturating the CRLB, Eq. (2) Cramer_book ().
Fig. 3 shows the distributions of the estimator for the phase shift and different values of . As expected, with increasing , the histograms approach a Gaussian shape with standard deviation decreasing as . The width of the histograms is smaller for the Dicke state (red lines) than for the separable state (blue lines). Fig. 4 shows as a function of . For the standard deviation is below the CRLB (Eq. 2) for several values. This is possible because the estimation is biased, i.e., for we have and Cramer_book (); SI (). The bias can be taken into account by replacing the numerator in the CRLB Eq. (2) by . For even smaller , only few different maxima of the likelihood functions can occur, see Fig. 3 a). Then, scatters significantly and hardly allows for an unbiased phase estimate. When , the bias is strongly reduced and the agreement of with the unbiased CRLB is improved significantly. While the bias is still large enough to cause apparent sensitivities below the shot-noise limit for the separable state, for the Dicke state the CRLB is saturated for a large phase interval. This clearly proves that the multiparticle entangled Dicke state created experimentally indeed achieves the SSN phase uncertainty predicted by the CRLB Eq. (2) using the experimentally obtained Fisher information from Fig. 2 l).
A conceptually different phase estimation protocol is given by the Bayesian approach assuming that the phase shift is a random variable. The probability density for the true value of the phase shift being equal to , conditioned on the measured results , is provided by Bayes’ theorem, . To define the a priori probability density we adopt the maximum ignorance principle and take to be constant in the phase interval considered. The Bayesian probability density is then given by . The phase shift can be estimated as the maximum of the probability density as before. However, in contrast to the ML method, the Bayesian analysis allows to assign a meaningful uncertainty to this estimate even for a single -experiment and biased estimators. This can be taken, for instance, as a confidence interval around the estimate, where the area of is equal to (see Fig. 3 d) and SI ()).
Figs 3 d)-f) illustrate how the Bayesian probability density evaluated for a single -experiment becomes Gaussian with a width , already for small values of . In contrast, the ML histograms [Figs 3 a)-c)] approach a Gaussian shape more slowly. We also investigated how the Bayesian analysis performs on average using the same data as in the ML case. The results are shown in Fig. 4 with the rescaled Bayesian uncertainty for various and averaged over several -experiments. For the mean values of the confidences deviate from the CRLB and have a large spread. For , however, the confidences agree well with the CRLB for most values of , for both states.
In conclusion, we have investigated experimentally the relation between SSN phase estimation and the entanglement properties of a probe state. We have identified useful multiparticle entanglement by determining the quantum Fisher information from the tomographical data of a four photon Dicke state. The benefit of such entanglement has been demonstrated by implementing two different phase estimation analyses, both of which saturate the Cramer Rao bound and clearly surpass the shot noise limit. The approach is completely general: it applies for any probe state, is scalable in the number of particles and does not require state selection. Our study thus provides a guideline for the future technological exploitation of multiparticle entanglement to outperform current metrological limits.
We thank N. Kiesel, W. Laskowski, and O. Gühne for stimulating discussions. R.K., C.S., W.W., and H.W. acknowledge the support of the DFG-Cluster of Excellence MAP, the EU projects QAP and Q-Essence, and the DAAD/MNISW exchange program. W.W. and C.S. thank QCCC of the Elite Network of Bavaria and P.H. thanks the ERC Starting Grant GEDENTQOPT.
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