Fisher information of a squeezed-state interferometer with a finite photon-number resolution

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1 PHYSICAL REVIEW A 95, ) Fisher information of a squeezed-state interferometer with a finite photon-number resolution P. Liu, 1 P. Wang, W. Yang,,* G. R. Jin, 1, and C. P. Sun, 1 Department of Physics, Beijing Jiaotong University, Beijing , China Beijing Computational Science Research Center, Beijing , China Received 4 December 016; published 13 February 017) Squeezed-state interferometry plays an important role in quantum-enhanced optical phase estimation, as it allows the estimation precision to be improved up to the Heisenberg limit by using ideal photon-number-resolving detectors at the output ports. Here we show that for each individual -photon component of the phase-matched coherent squeezed vacuum input state, the classical Fisher information always saturates the quantum Fisher information. Moreover, the total Fisher information is the sum of the contributions from each individual -photon component, where the largest is limited by the finite number resolution of available photon counters. Based on this observation, we provide an approximate analytical formula that quantifies the amount of lost information due to the finite photon number resolution; e.g., given the mean photon number n in the input state, over 96% of the Heisenberg limit can be achieved with the number resolution larger than 5 n. DOI: /PhysRevA I. ITRODUCTIO uantum-enhanced optical phase estimation through a Mach-Zehnder interferometer MZI) is important for multiple areas of scientific research [1 7], such as imaging, sensing, and high-precision gravitational waves detection. The MZIbased optical phase estimation consists of three steps [see, e.g., Fig. 1a)]. First, a two-mode input state of the light is prepared. Second, the light passes successively through a beam splitter, the unknown relative phase shift ϕ between the two arms of the MZI, and another beam splitter, and it evolves to the output state. Third, the output state is measured for many times, and the outcomes x ={x 1,x,...,x v } are processed to construct an unbiased estimator ˆϕx) to the unknown parameter ϕ [8,9]. The estimation precision is quantified by the standard deviation ϕ ˆϕx) ϕ). By using optimal data-processing techniques to extract all the information contained in the data, the estimation precision from v 1 repeated measurements is given by the Cramér- Rao lower bound [8,9]: ϕ CRB 1/ vfϕ), where F ϕ) is the classical Fisher information CFI) for the measurement scheme used. Given the input state, maximizing F ϕ) over all possible measurement schemes gives the quantum Fisher information FI) F and hence the quantum Cramér-Rao bound ϕ CRB 1/ vf [10 14], which sets an ultimate precision for estimating the unknown phase shift ϕ. Usually the precision ϕ CRB improves with increasing number of photons n contained in the input state. Using a coherent-state of light as the input, the achievable phase sensitivity per measurement is limited by the classical or shot noise) limit δϕ v ϕ 1/ n, asthefif O n). To improve the precision beyond the classical limit 1/ n), it is necessary to employ quantum resources, such as entanglement and squeezing in the input state [1 7]. Inthis context, the squeezed states of light play an important role and have been widely studied in the past few decades ever since * wenyang@csrc.ac.cn grjin@bjtu.edu.cn cpsun@csrc.ac.cn the pioneer work of Caves [1], who showed that by feeding a coherent state α into one port of the MZI and a squeezed vacuum ξ into the other port, the unknown phase shift can be estimated with a precision beyond the classical limit. Pezzé and Smerzi [15] further suggested that the previously used phase estimator based on the averaged relative photon number is not optimal. When the injected fields are phase matched, i.e., the phases of two light fields θ a and θ b obeying cosθ b θ a ) =+1, the FI can reach the Heisenberg scaling O n ) for a given mean photon number n = α + sinh ξ. More importantly, this FI can be saturated by the CFI for ideal photon counting measurements. Consequently, by using the optimal data-processing technique such as the maximumlikelihood estimation or Bayesian estimation) to process these measurement outcomes, the phase estimation precision can attain the Heisenberg limit δϕ CRB = δϕ CRB 1/ n. Recently, Lang and Caves [16] proved that given the total average photon number n of the input state, if a coherent light is fed into one input port of the MZI, then the squeezed vacuum is the optimal state to inject into the second input port. Liu et al. [17] have analyzed the phase-matching condition PMC) that maximizes the FI in the squeezed-state interferometer, where a superposition of an even or odd number of photons is injected from one port of the interferometer and any input state from another. An important requirement of these theoretical works [15,16] is to take into account all the photon-counting events, which in turn requires photon-number-resolving detectors with perfect number resolution [18]. However, on the experimental side, the best detector to date can resolve the number of photons only up to four [19,0]. This makes it unclear whether or not the Heisenberg limit of the estimation precision can still be achieved by using realistic photon detectors with an upper threshold on the number resolution. To bridge this gap between the theory and experiments, it is of interest to investigate the experimentally achievable estimation precision when the total number of photons being detected is limited, i.e., = 1 + res, where res / determines the number resolution by a single photon-counting detector. Since the existence of an upper threshold res essentially amounts to discarding the information contained in photon-counting /017/95)/038410) American Physical Society

2 P. LIU, P. WAG, W. YAG, G. R. JI, AD C. P. SU PHYSICAL REVIEW A 95, ) FIG. 1. a) Photon-counting measurement at output ports of the MZI fed with a coherent state α and a squeezed vacuum ξ, and the -photon state ψ BS, postselected by the number of photons being detected = 1 + = J. b) For a given J = / = 10, the probability distribution p μ = J,μ ψ BS against μ = 1 )/ [ J, + J ]andx = α / tanh ξ. At a certain value of the ratio x opt) blue line), the distribution shows two symmetric peaks at μ =±J, indicating the appearance of a path-entangled OO state. c) The fidelity between the -photon state and an ideal OO state and d) the FI of the -photon state, with their values calculated at x = x opt) blue solid line) and x FI) red line with crosses); see text. The inset in c) indicates xfi) xopt) /. events with the number of photons larger than res,itis therefore important to investigate the distribution of the FI and CFI in the -photon components of the coherent squeezed vacuum input state and calculate how much the FI is kept with a finite number resolution. In addition, studying the distribution of the FI and CFI in the -photon components also helps to understand the phase estimation precision in recent postselection experiments. When the MZI is fed by the coherent squeezed vacuum, the state after the first beam splitter of the MZI contains a small fraction of the path-entangled OO state [1,], which is a well-known -photon nonclassical state that allows the phase estimation precision to achieve the Heisenberg limit [3 9]. In the limit α, ξ 1, Afek et al. [] have demonstrated -fold oscillations of the coincidence rates for up to 5, manifesting the appearance of -photon OO states. However, the generation probability of a -photon OO state decreases dramatically with increasing, e.g., the fivephoton count rate 3 per 100 s []. Therefore, it is desirable to study the overall estimation precision when such small generation probabilities are included, since there are general conclusions that the generated state under postselection cannot improve the precision for estimating a single parameter when the total number of input photons are included see, e.g., Refs. [30 3]). In this paper, we investigate the distribution of the FI and CFI in the different -photon components of the coherent squeezed vacuum input state and provide the achievable estimation precision by using imperfect photon counters with an upper threshold res for the photon number resolution. Under the PMC cosθ b θ a ) =+1, we show that the CFI always saturates the FI for each individual-photon component. Consequently, when the detectable number of photons is upper bounded by res, the phase estimation precision δϕ CRB is always equal to δϕ CRB and both of them are determined by the sum of the CFI or equivalently the FI for each -photon component with up to res.forthe commonly used optimal input state with α sinh ξ n/[15 17], photon-counting measurement with ideal photon detectors res ) gives the CFI or the FI F id),opt n, leading to the Heisenberg limit of the estimation precision [15 17]. For finite photon number resolution, we provide an approximate analytical expression that quantifies the amount of lost information, which predicts that over 96% of the ideal FI can be achieved as long as res 5 n. Compared with the ideal case i.e., α sinh ξ ), we find that the optimal input state contains more coherent light photons than that of the squeezed light. II. FIITE -PHOTO STATE UDER POSTSELECTIO As illustrated schematically by Fig. 1a), a postselection scheme for creating path-entangled OO states has been proposed by injecting a coherent state of light and a squeezed vacuum into a Mach-Zehnder interferometer [1,]. This scheme has been demonstrated by Afek et al. [] inthe limit α, ξ 1. However, the generated -photon state in postselection cannot improve the precision for estimating 0384-

3 FISHER IFORMATIO OF A SUEEZED-STATE... PHYSICAL REVIEW A 95, ) an unknown phase shift, since the CFI is weighted by the generation probability [30]. It is therefore important to investigate whether or not a sum of each -component for up to a finite number can beat the shot-noise scaling O n). To answer this question, in this section we first derive explicit form of the -photon state generated by postselection. ext, we calculate the quantum) Fisher information for the -photon state, which determines the ultimate precision on the phase estimation. A. The fidelity of the -photon state and the OO state Without any loss and additional reference beams, the input state can be expressed as a superposition of -photon states [14], i.e., α a ξ b = G ψ, where G denotes the generation probability of a finite -photon state, and = 1 + is the number of photons postselected by the photoncounting events { 1, }. In Fock basis, the -photon state is given by ψ = 1 G [/] c k θ a )s k θ b ) k,k a,b, 1) where m,n a,b m a n b, and the sum over k is up to [/] = 1)/ for odd ), or /foreven), because of an even number of photons that injected from the port b. ote that the probability amplitudes of the coherent state and the squeezed vacuum c m θ a ) = m α and s n θ b ) = n ξ depend explicitly on the phases of two input light fields θ a and θ b see Appendix A). Furthermore, the generation probability G is also the normalization factor and is given by G = [/] c k s k = e α cosh ξ ) tanh ξ R x), ) where we have introduced a ratio x α / tanh ξ and a polynomial R x) = [/] k)! k)!k!) x) k, 3) which obeys R 0) =!/[/)!] for even, and R 0) = 0 for odd, similar to the Hermite polynomials at x = 0. In the limit α, ξ 1, the ratio can be approximated as x α / ξ, and its square is indeed the two-photon probability of the coherent state divided by that of the squeezed vacuum []. The explicit form of the -photon state crucially depends on the relative phase difference between the squeezing parameter ξ and the coherent-state amplitude α. Following Refs. [15,17], we consider the PMC, i.e., cosθ b θ a ) = +1, for which Eq. 1) can be reexpressed as ψ = expiθ a ) ψ, where ψ denotes the -photon states with real amplitudes for details, see Appendix A). After the first beam splitter, the -photon state becomes ψ BS = e iπj x / ψ = +J μ= J J,μ ψ BS J,μ, 4) where, for brevity, we have introduced the eigenstates of J z, i.e., J,μ J + μ,j μ a,b, with J = / and μ [ J, + J ]. Under the PMC, the probability amplitudes of ψ BS can be written as J,μ ψ BS = e iθ a e iπμ J )/ pμ, 5) which depends solely on the phase of the coherent-state light θ a and the probability distribution see Appendix A) p μ J,μ ψ BS [ [/] = 1 R x) d J μ,j k ) ] π k)! k! k)! x)/ k, 6) where dμ,v J ϕ) are the elements of Wigner s d-matrix [33,34]. It is interesting to note that for a given, the probability distribution depends only on the introduced ratio x = α / tanh ξ, hereinafter denoted by p μ = p μ x). Figure 1b) shows the probability distribution as a function of μ for a large enough. Atx = 0, i.e., a pure squeezed vacuum being injected, the probability distribution is almost a Gaussian, due to p μ 0) = [dμ, J J π/)] exp μ /J ). As x increases, the -photon state always shows a symmetric probability distribution i.e., p μ = p +μ ). One can see this directly from Eq. 6), where d μ,v J ϕ) = 1)J v d+μ,v J π ϕ); see, e.g., Refs. [33,34]. Physically, the symmetric probability distribution arises from the fact that the -photon state ψ contains only an even number of photons in mode b, i.e., ψ J y ψ =Im ψ a b ψ =0, which in turn leads to ψ J y ψ = ψ BS J z ψ BS = p +μ p μ )μ = 0 μ 0 7) and hence p μ = p +μ. This symmetry enables us to write an explicit expression of the -photon state, ψ BS = e iθ a e i π μ J ) p μ x) μ 0 J,μ +e iπμ J, μ ), 8) which is indeed a superposition of the path-entangled states J,μ +e iπμ J, μ ), where the relative phase e iπμ comes from Eq. 5). For a certain value of x, the probability distribution p μ x) reaches its maximum at μ = ±J =±/, indicating ψ BS ψ OO = J, + J + e iπj J, J )/, with the fidelity given by F OO ψ OO ψ BS = pj x). 9) Clearly, the fidelity depends on the ratio x and the number of photons being detected =J ). For a given, maximizing the fidelity with respect to x, one can obtain the optimal value of the ratio, denoted hereinafter as x opt). For small, ithas been obtained x opt) = 1 for =, 3), 3 = 4), and.016 = 5); see Ref. []. When 1, the optimal value of x is about /, for which F OO 8/ see Ref. [1] and Table I). In Fig. 1c) we show the optimal value of the fidelity F OO x opt) ) as a function of the blue solid line), which coincides with Ref. []. From Eqs. 4) and 9),

4 P. LIU, P. WAG, W. YAG, G. R. JI, AD C. P. SU PHYSICAL REVIEW A 95, ) TABLE I. For a given, the fidelity F OO and the FI F, depend solely on the ratio x α / tanh ξ and reach a maximum and x FI), respectively. For =, 3, F OO = F, / = 1atx opt) = x FI) = 1; while for = 4, F OO = F, / = at = x FI) = 3. at x opt) x opt) , xfi).016, , , , , , , F OO, F, / 0.941, , , , , , , x opt) one can also see that before the first beam splitter, ψ itself at x = x opt) approaches the OO state expiπj x /) ψ OO, which shows the polarization along ±J y. B. The Fisher information of the postselected -photon state We now investigate the CFI of the -photon state in the photon-counting measurements and show that it always equals to the FI under the PMC. To this end, we first calculate the FI of the phase-encoded state exp iϕj y ) ψ, where the unitary operator represents sequent actions of the first beam splitter, the phase-shift accumulation in the path, and the second 50:50 beam splitter at the output ports, as illustrated in Fig. 1a).Dueto ψ J y ψ =0, it is easy to obtain the FI [10 14]: F, = 4 ψ Jy ψ =4 ψ BS +J J ψ BS z = 4 μ p μ, μ= J 10) where ψ BS denotes the -photon state after the first beam splitter and its probability distribution p μ x) has been given by Eq. 6). Similar to the fidelity, one can see that the FI depends on the ratio x and the number of photons. Forthe cases =, 3, and 4, both of them reach maximum at x opt) because of the relation F, = F OO x) + ) p J 1 x) +, 11) where p 0 x), p 1/ x), and p 1 x) are vanishing at x = x opt). When 5, however, {p μ x)} with μ <J provide nonvanishing contributions to the FI. umerically, we find that F, reaches its maximum at x FI), which is slightly smaller than x opt) see Table I). In Fig. 1d) we plot the maximum of the FI as a function of and find F, ext, we consider the photon-counting measurements over the phase-encoded state exp iϕj y ) ψ and calculate the CFI. Again, we consider the PMC and rewrite the -photon state as ψ =expiθ a ) ψ, where ψ is given by Eq. 1) with θ a = θ b = 0. ote that the probability amplitudes of ψ and hence that of exp iϕj y ) ψ are real, which result in the conditional probabilities see Appendix A): P μ ϕ) = J,μ e iϕj y ψ = [ J,μ e iϕj y ψ ], 1) where μ = 1 )/ [ J, + J ] and J = 1 + )/ = /. Obviously, for a given, there are + 1 outcomes with their probabilities satisfying the normalization condition μ P μ ϕ) = ψ ψ =1. Due to the real probability amplitudes, J,μ exp iϕj y ) ψ R, we further obtain P μ ϕ) ϕ = P μ ϕ) J,μ ij y )exp iϕj y ) ψ R, indicating that J,μ J y exp iϕj y ) ψ is purely imaginary for each μ. This is the key point to obtain the CFI: F ϕ) = +J μ= J = 4 [ P μ ϕ)/ ϕ] +J μ= J P μ ϕ) [ J,μ J y e iϕj y ψ ] = 4 ψ J y ψ =F,, 13) where F, is the FI of the phase-encoded state exp iϕj y ) ψ under the PMC, given by Eq. 10). As one of main results of this work, Eq. 13) indicates that as the input state, ψ at x = x FI) could provide a global phase estimation at the Heisenberg scaling [35], as F ϕ) = F, However, this scaling is defined with respect to the number of photons being detected. Furthermore, ψ is postselected by the -photon detection events with the generation probability G, which is usually very small as 1[seeFig.a)]. Indeed, purely with the -photon detection events i.e., totally + 1 outcomes with a definite ), one cannot improve the accuracy for estimating an unknown phase shift, since the CFI is weighted by the generation probability [30], i.e., G F ϕ). For the input α a ξ b with a given mean photon number n = α + sinh ξ, one can see that the weighted CFI for different values of can reach only the classical limit O n), as depicted by Fig. c), where we considered the special case α,ξ R, for which the PMC is naturally fulfilled and therefore F ϕ) = F,. III. THE TOTAL FISHER IFORMATIO In order to improve the estimation precision, all the detection evens { 1, } have to be taken into account in the photon-counting measurements, which gives ideal result of the CFI [10 14]: F id) ϕ) = +J J =0 μ= J [ PJ,μ ϕ)/ ϕ] = P J,μ ϕ) G F ϕ), =0 14)

5 FISHER IFORMATIO OF A SUEEZED-STATE... PHYSICAL REVIEW A 95, ) FIG.. a) The generation probability G, b) the FI of each -photon state F,, c) the weighted FI G F,, and d) the total FI F res, n,α ), for n = 5fixedand res = 3 n solid circles), 5 n squares), 10 n open circles), and red solid lines). The last case is given by Eq. 16), which predicts αopt id) n/ andf,opt n n + 3/). For each a given n [1,10], using the same values of res and maximizing the total FI with respect to α to obtain e) αopt / n, and f) the associated FI F,opt. In c) the peak height of the weighted FI is about n/. The vertical lines in d) are the optimal value of α for different values of res. The dashed line in f) is the classical limit F = n. where we have reexpressed the input state as ψ in = G ψ,sowehave P J,μ ϕ) J,μ e iϕj y ψ in = G P μ ϕ), and P μ ϕ) J,μ exp iϕj y ) ψ, given by Eq. 1). ote that the total CFI is indeed a sum of each -component contribution F ϕ) weighted by G. With only the -photon detection events, the Fisher information is simply given by G F ϕ), as mentioned above. Similar to Eq. 10), we further calculate the total FI of the output state exp iϕj y ) ψ in, which is independent from any specific measurement scheme and is given by F = 4 J y in J y in )[10 14]. For the input state α a ξ b,we obtain J y in = 0 and hence the ideal result of the FI F id) = 4 G ψ Jy ψ = =0 G F,, 15) =0 where F, is the FI of the -photon component. Under the PMC, we have show that for each component F ϕ) = F,, which naturally results in a global phase estimation F id) ϕ) = F id) [35]. According to Refs. [15 17], one can obtain an explicit form of the FI by directly calculating 4 Jy in see also Appendix B): F id) ϕ) = F id) = α e ξ + sinh ξ. 16) Given a constraint on the mean photon number n, the maximum of the FI was found to achieve the Heisenberg scaling F id),opt n n + 3/) O n )[16], provided α sinh ξ n/ 1 [15]; see also the red solid lines in Figs. d) f). However, such a scaling is possible only with exactly perfect photon-number-resolving detectors [18], which enable us to record an infinite number of the photon-counting events; see also Eq. 14). Usually a single number-resolving detector can register only the number of photons up to four [19,0]. It is therefore important to investigate the CFI of each component for up to a finite number of photons being resolvable res. For brevity, we consider the input fields with the real amplitudes and large enough mean photon number i.e., n = α + sinh ξ>1). Since the PMC is naturally fulfilled, the CFI is still a sum of each component with the weight G and equals the FI: res F = 4 G ψ Jy ψ res = G F, =0 res = [/] =0 =0 [ + 4k k)+ 4kα tanh ξ ] [c k 0)s k 0)], 17) where ψ is the -photon state and G = G α,ξ) denotes its generation probability, given by Eqs. 1) and ). Obviously the FI considered here depends on three variables { res,α,ξ}, or equivalently, { res,α, n} for a given n. When res, the ideal result of the FI is recovered see Appendix B). The Heisenberg scaling of the FI can be maintained for large enough res, provided that all the nonvanishing {G F, } are included. To obtain the minimum value of res,weshowg, F,, and G F, against and α under a constraint on n.fromfig.b), one can see that F, increases quadratically with. This is because the FI reaches its maximum F, O ) when α / tanh ξ = x FI) see Table I), which corresponds to α / n 1, i.e., the classical light being dominant for a given n = α + sinh ξ.onthe other hand, the generation probability shows a little complex behavior on ; see Fig. a). Atα = 0, G is nonvanishing at even number of and decreases monotonically with the increase of. When α 1 i.e., G 1 G 0 ), it reaches a

6 P. LIU, P. WAG, W. YAG, G. R. JI, AD C. P. SU PHYSICAL REVIEW A 95, ) maximum at a certain value of and then decreases. Similar to G, the weighted FI G F, reaches a maximum at n and then decreases with the increase of. As depicted in Fig. c), one can also see that the values of G F, tend to vanishing as 5 n, implying res 5 n. To confirm the above result, we maximize Eq. 17) with respect to α for given n and res. Figure d) shows F as a function of α for a fixed n = 5, where res = 3 n the solid circles), 5 n the squares), and 10 n the open circles). When res = the red solid line), the ideal result of the FI is recovered and is given by Eq. 16), which reaches the Heisenberg scaling αopt n/ [15 17]. One can see that the FI with res = 10 n almost follows the ideal result. In Figs. e) and f), we show optimal value of the ratio α / n and the associated FI F,opt = F res, n,αopt ) for each given value of n [1,10], where we take the number resolution res as the same as Fig. d).fromfig.e), one can see that when res > n, the optimal input state contains more coherent light photons than that of the squeezed vacuum. The Heisenberg scaling of the FI is attainable with res 5 n, as depicted by Fig. f). Figure 3 shows F /F id),opt as a function of res/ n for the increase of n from to 0. For each given n, we first maximize the ideal FI with respect to α to obtain αopt id) and F,opt,as depicted by the red lines of Figs. d) f), and then calculate the FI of Eq. 17) using the same input state. Our numerical results show that F /F id),opt increases with res and approaches 1as res n. To quantify how much phase information is kept by a finite cutoff res, we try to find an analytical result of F /F id),opt in the limit n. To this end, we first separate the FI into two terms F = F id) F lost) lost), where F = = res G F, denotes the FI being lost. This expression is the same to Eq. 17), except the sum over res, ). ext, we note that the photon number distribution of the coherent state is much narrow than that of the squeezed vacuum, which enables us to obtain an approximate result of F lost) see Appendix B). Furthermore, the ideal result of the FI can reach its maximum at the optimal condition α = sinh ξ = n/ 1[15 17]. Using the same input, we obtain F F id),opt 1 lim F lost) n x, n) n n + 3/) erf x 1/) e x+1/ π x 1/, 18) where x res / n and erf ) denotes the error function. Our analytical result shows a good agreement with the numerical results; see the solid lines in Fig. 3. When res 5 n, it predicts that over 96% of the ideal FI can be kept, while for res < n/, most of the phase information is lost. Finally, it should be mentioned that coherent-state interferometry has been demonstrated using two visible light photon counters with res = 8[19]. This number resolution is large enough to realize the global phase estimation for the coherentstate input n 1. Based upon a Bayesian protocol [19], the achievable phase sensitivity was found almost saturating a quantum Cramér-Rao bound over a wide phase interval, in agreement with the theoretical prediction F ϕ) = F = n.to realize higher-precision optical metrology, it requires a bright nonclassical light source with larger mean photon number [15], low photon loss [14,36 38], and low noise [39 50], as well as the photon counters with high detection efficiency [51] and large enough number resolution. IV. COCLUSIO In summary, we have investigated optical phase estimation with coherent squeezed vacuum light by using imperfect photon counters with an upper threshold res for the photon number resolution. We show that both the CFI and the FI are the sum of the contributions from individual -photon components, and the CFI always saturates the FI for each individual -photon component. For ideal photon-counting detectors with res, the CFI or the FI attains its maximum F id),opt n when α sinh ξ, leading to the Heisenberg limit of the estimation precision. For the detectors with large enough number resolution res > n, we find that the optimal input state contains more coherent light photons than that of the squeezed vacuum. We present an analytical result that quantifies the amount of lost information and show that over 96% of an ideal FI can be attained as long as res 5 n, while for res < n/, most of the phase information is lost. Our results highlight the important influence of the finite number resolution of photon-counting detectors for optical phase estimation. It is also interesting to explore the performance of other continuous-variable input states, e.g., a product of two squeezed vacuum ξ ξ [5], when realistic photon counters are used. FIG. 3. umerical results of F /F id),opt as a function of res / n for given values of n, using the optimal condition that maximizes Eq. 16). The solid line is given by our asymptotic result [Eq. 18)], and the dashed line is a fitting result for the case n = 0. Both of them indicate that 96% of the ideal FI can be obtained as long as x = res / n 5 the vertical dashed lines). ACKOWLEDGMETS We would like to thank S. Rosen and Y. Silberberg for kind response to our questions. This work has been supported by the SFC Grant os , , , and 11354) and the ational 973 program Grant os. 01CB9104, 014CB91403, and 014CB848700)

7 FISHER IFORMATIO OF A SUEEZED-STATE... PHYSICAL REVIEW A 95, ) G.R.J. also acknowledges support from the Major Research Plan of the SFC Grant o ). P.L. and P.W. contributed equally to this work. APPEDIX A: THE -PHOTO STATE UDER THE PHASE-MATCHIG CODITIO Formally, a single-mode squeezed vacuum of light is defined by ξ =Sξ) 0, with the squeeze operator [53 55]: [ ] 1 Sξ) = exp ξ b ξb ) ) ) = exp e iθ b b 1 b b+ 1 exp e iθ b tanh ξ cosh ξ tanh ξ b ), A1) where ξ = ξ expiθ b ) denotes the complex amplitude of the squeezed vacuum. In the Fock basis, using b 0 =0, the squeezed vacuum can be expressed as ξ = 1 cosh ξ exp e iθ b tanh ξ b ) 0 = + s k k, A) where s k k ξ denote the probability amplitudes of the squeezed vacuum, given by ) k)! s k θ b ) = k! e iθ tanh ξ k b, cosh ξ ) A3) H k 0) or s k θ b ) = e iθ tanh ξ k/ b, k! cosh ξ with the Hermite polynomials H n 0) = 1) n n)!/n! and H n+1 0) = 0. ote that one can obtain an explicit form of the squeezed vacuum using the disentangling formula [53 55], as done in Eq. A1), or, alternatively, directly solving the eigenvalue equation Sξ)b 0 =Sξ)bS ξ) ξ =0 [56]. The single-mode squeezed vacuum contains only an even number of photons and has been generated in experiments [57 63]. We now consider the interferometer fed with the squeezed vacuum from one input port and a coherent-state light from another port. The coherent state is given by α = n c nθ a ) n, with the probability amplitudes c n θ a ) n α =e α / α n e inθ a n!, A4) where α = α expiθ a ) denotes the complex amplitude of the coherent light. In Eqs. A3) and A4), we have written explicitly the phase dependence of the probability amplitudes, purely for later use. Under the phase-matching condition PMC): cosθ b θ a ) =+1, we now calculate the probability amplitudes of the -photon states ψ as c k θ a )s k θ b ) G = 1) k eiθ a e ikθ b θ a ) R x) k)! k! k)! x) k)/ PMC 1) k e iθ a k)! R x) k! k)! x) k)/ e iθ a c k0)s k 0) G, A5) where we have used an explicit form of G, given by Eq. ), and the condition exp[ikθ b θ a )] =+1 for integers k.ote that Eq. 1) can be rewritten as ψ =expiθ a ) ψ, where ψ denotes the -photon states with real amplitudes i.e., θ a = θ b = 0). Finally, we consider a unitary operation exp iϕj η )onthe -photon states ψ, with J η = J x cos η + J y sin η, to obtain Eqs. 5) and 6). Under the PMC, we obtain e iϕj η ψ =e iθ a e iϕj η ψ =e iθ a e iηj z e iϕj x e iηj z ψ = eiθ a G e iηj z e iϕj x [/] e iηj k) c k 0)s k 0) J,J k, A6) where, in the second step, we have used the relation exp iηj z )f J x )expiηj z ) = f J η ), and Eq. 1) with θ a = θ b = 0for ψ, which is expressed in terms of the states J,J k = k a k b. In the eigenbasis of J z,we obtain the probability amplitudes J,μ e iϕj η ψ = eiθ [/] a e iημ e iηj k) c k 0)s k 0) G J,μ e iϕj x J,J k [/] R x) ei π η)μ J ) = eiθ a e ikη k)! k! k)! x)/ k d J μ,j k ϕ), A7) where, in the last step, we have introduced Wigner s d-matrix dμ,v J ϕ). Obviously, for the special case η = 0 and ϕ = π/, we obtain the -photon state after the first 50:50 beam splitter exp iπj x /) ψ and its probability distributions; see Eqs. 5) and 6). For η = π/ and arbitrary ϕ, we can obtain the output state exp iϕj y ) ψ and its probabilities P μ ϕ). APPEDIX B: AALYTICAL RESULTS OF THE UATUM FISHER IFORMATIO In a lossless and noiseless interferometer, the FI of a pure phase-encoded state ψ out =exp iϕg) ψ in is simply given by F = 4 G in G in )[10 14], where G is a Hermitian operator. For the squeezed-state interferometer, as illustrated in Fig. 1a), the input state is the product of a coherent state and a squeezed vacuum, i.e., ψ in = α a ξ b, and the phase shifter is given by G = J x, or J y, where, for brevity, we have introduced the angular-momentum operators J + = J ) = a b and J z = a a b b)/, with the bosonic operators of two light fields a and b

8 P. LIU, P. WAG, W. YAG, G. R. JI, AD C. P. SU PHYSICAL REVIEW A 95, ) According to Ref. [15], the FI of the output state exp iϕj y ) ψ in is optimal when the two injected light fields are phase matched, i.e., the PMC cosθ b θ a ) =+1. Recently Liu et al. [17] have derived a more general form of the PMC for the interferometer U MZI ϕ) = exp iϕj y ), where a superposition of even or odd number of photons is injected from one port and an arbitrary state from another port. To show it clearly, we focus on the PMC cosθ b θ a ) = +1 and calculate the FI of exp iϕj y ) ψ in, F id) = 4 Jy in = a a + b b + a ab b) a b + H.c.) in, B1) where H.c. denotes the Hermitian conjugate. There are two contributions to the FI. First, it is easy to obtain a a + b b + a ab b) in = n a + n b + n a n b, B) with n a = α and n b = sinh ξ being the mean photon number of light fields from two input ports. Second, using the relation S ξ)bsξ) = b cosh ξ b e iθ b sinh ξ, we obtain a b in = α ξ b ξ = n a nb 1 + n b )e iθ b θ a ). B3) Therefore, the ideal result of the FI is given by F id) = n a[1 + n b + n b 1 + n b ) cos θ b θ a )] + n b n a [1 + n b + n b 1 + n b )] + n b, B4) where the equality holds when the PMC is fulfilled: cosθ b θ a ) =+1. Similarly, one can note that the PMC cosθ b θ a ) = 1 is a good choice for the output state exp iϕj x ) α a ξ b, e.g., the phases of the two light fields θ a,θ b ) = 0,π)[1] and π/,0) []. Furthermore, one can simplify the ideal result of the FI as Eq. 16), using the relation 1 + n b + n b 1 + n b ) = e ξ. With a finite number resolution res, we have shown that the CFI and the FI are the same and given by Eqs. 17), which can be rewritten as F = res a =0 res a b =0 res n a n a b =0 [ ) ] a a + α [ca b 0)s b 0) ] tanh ξ [ sb 0) ] n a + n a tanh ξ ) res n a b =0 b [ sb 0) ], where, for brevity, we consider the two light fields with real amplitudes, i.e., θ b = θ a = 0, and the probability amplitudes c n 0) and s k 0) are given by Eqs. A3) and A4). In the above result, we made an approximation B5) res a =0 res a b =0 f a )g b ) [ c a 0)s b 0) ] a =0 f a ) [ c a 0) ] res n a b =0 g b ) [ s b 0) ], B6) where n a = α and the sum over the mode b is still kept, since the photon number distribution of the squeezed vacuum is usually wider than that of the coherent state even for n b < n a )[56,61]. For a finite n a and res, it is easy to obtain the ideal result of the FI as F n a n a + n ) a n b = F id) tanh ξ, B7) where tanh ξ = n b / n b + 1) and F id) is given by Eq. B4). Finally, we consider a finite number resolution with large enough res > n a ), and derive analytical result of the FI. To this end, we first rewrite Eq. B5) asf = F id) F lost) lost), where F quantifies the lost phase information caused by the finite number resolution, given by F lost) = 4 G ψ Jy ψ n a [s k 0)] n a + n ) a k[s k 0)] tanh ξ = res k= res n a k= res n a n a + n ) a tanh ξ) k kdk, B8) tanh ξ res n a πkcosh ξ where, in the last step, we keep only the terms O n ). In addition, we replace the sum over k by an integral and use Stirling s formula k! kπk/e) k. When res n a, it is easy to find F lost) F id) and hence the achievable FI F O n 0 )oro n 1 ), corresponding to an almost complete loss of the phase information or the ultimate estimation precision in the classical limit. To enlarge the FI, we take res > n a and obtain F lost) n [ ) ] a n b B n res n a 1 + e b ) erfc B + res na n B res n a e b B, B9) B3/ n b π n b

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