Quantum Optical Metrology of Correlated Phase and Loss
Patrick M. Birchall, Euan J. Allen, Thomas M. Stace, Jeremy L. O'Brien, Jonathan C. F. Matthews, Hugo Cable
QQuantum Optical Metrology of Correlated Phase and Loss
Patrick M. Birchall, Euan J. Allen,
1, 2, ∗ Thomas M. Stace, Jeremy L. O’Brien, Jonathan C. F. Matthews, and Hugo Cable Quantum Engineering Technology Labs, H. H. Wills Physics Laboratory and Departmentof Electrical & Electronic Engineering, University of Bristol, BS8 1FD, United Kingdom. Quantum Engineering Centre for Doctoral Training, H. H. Wills Physics Laboratory and Department of Electrical & Electronic Engineering,University of Bristol, Tyndall Avenue, BS8 1FD, United Kingdom. ARC Centre of Excellence for Engineered Quantum Systems, School of Mathematics and Physics,University of Queensland, Saint Lucia, Queensland 4072, Australia. (Dated: October 3, 2019)Optical absorption measurements characterize a wide variety of systems from atomic gases to in-vivo diag-nostics of living organisms. Here we study the potential of non-classical techniques to reduce statistical noisebelow the shot-noise limit in absorption measurements with concomitant phase shifts imparted by a sample.We consider both cases where there is a known relationship between absorption and a phase shift, and wherethis relationship is unknown. For each case we derive the fundamental limit and provide a practical strategy toreduce statistical noise. Furthermore, we find an intuitive correspondence between measurements of absorptionand of lossy phase shifts, which both show the same scope for precision enhancement. Our results demonstratethat non-classical techniques can aid real-world tasks with present-day laboratory techniques.
The precision of optically measuring an object is limited byfundamental fluctuations in the optical field due to the statis-tical quantum nature of light [1]. When using laser light asan optical probe, the limit of this statistical noise is the shot-noise limit which can be reduced by increasing probe intensityor by enhancing interaction with the sample. However, somesystems are incompatible with increased intensities, for ex-ample if light causes undesired technical effects [2, 3] or thesample to deform [4, 5]. If high-intensity light cannot be usedthen shot-noise will limit the achievable precision [2, 3, 6].Whilst of a fundamental origin, shot-noise is not the ul-timate quantum limit — non-classical probes can be used toexceed the shot-noise limit [7]. Many previous theoretical andexperimental studies have investigated potential benefits ofusing non-classical states for phase estimation in the presenceof loss [8–14], and for loss estimation [5, 10, 15–21]. In ad-dition, a number of studies have investigated quantum boundsfor multiparameter estimation including unitary (phase) [22–24] and non-unitary (phase, loss, de-phasing) [25–27] chan-nels.At a fundamental level, changes in absorption over a narrowspectral range must be accompanied by changes in refractiveindex (and hence phase shifts), as governed by the Kramers-Kronig relations [28]. It is therefore important to considerhow the estimation capabilities of any strategy are affectedby correlation between these two variables. Here we addressthis and seek a unified understanding of quantum strategiesfor measuring absorption and phase of a single mode. Specifi-cally, we consider estimating an unknown parameter χ , whichgoverns both phase θ ( χ ) ∈ [0 , π ) and loss − η ( χ ) ∈ [0 , imparted by a channel Λ χ which we call correlated phase andloss estimation (CPLE). Formally, Λ χ is defined by its actionon a basis of coherent probe states | α (cid:105) Λ (cid:55)→ |√ η e i θ α (cid:105) . Lossy-phase estimation ( ∂ χ η = 0 where ∂ • ≡ ∂∂ • ) and loss estimation( ∂ χ θ = 0 ) [16–18] are special cases of CPLE.We first find the fundamental upper bound on the precisionachievable with CPLE, which is quantified using the quantum Fisher information (QFI) per input photon. We investigatethe saturability of this bound using squeezed coherent states,which can readily be generated experimentally [29]. We alsoconsider direct absorption estimation (DAE), where η ( χ ) isto be estimated but its relationship to θ ( χ ) is not known andtherefore the information contained in the phase cannot be ac-cessed. We conclude by investigating multi-pass strategies forCPLE and DAE, and by investigating the advantage attainablein all cases by current experimental capabilities. Fundamental limit for CPLE —
We use the establishedFisher information (FI) formalism to provide bounds on pre-cision for estimating an unknown parameter χ encoded withina quantum state (cid:37) χ : χ ) ≤ F χ M ( (cid:37) χ ) ≤ F χ ( (cid:37) χ ) . Inequality 1 is the Cr´amer–Rao bound (CRB) [30] and re-lates the variance of unbiased estimates
Var( χ ) to the FI F χ M ( (cid:37) χ ) = (cid:80) i p ( i | χ ) [ ∂ χ log p ( i | χ )] . The FI is a functionof the probabilities p ( i | χ ) = tr( m i (cid:37) χ ) , given by the measure-ment of (cid:37) χ , with a positive-operator valued measure (POVM) M = { m i } and (cid:80) i m i = . Inequality 2 is the quantumCRB [31] which relates F χ M ( (cid:37) χ ) to its maximum value F χ (the QFI) which is found by optimizing over all POVMs [32]. F serves as a measurement basis independent evaluation ofthe information that (cid:37) χ contains on χ . When χ is encodedonto a pure probe state by unitary U χ | ψ (cid:105) = | ψ χ (cid:105) the QFI be-comes (cid:0) (cid:107)| ∂ χ ψ χ (cid:105)(cid:107) −|(cid:104) ψ χ | ∂ χ ψ χ (cid:105)| (cid:1) where | ∂ • ψ (cid:105) ≡ ∂ • | ψ (cid:105) and (cid:107) • (cid:107) is the 2-norm.Loss enacts a non-unitary evolution. Ref. [13] showed thatfor such a non-unitary map Λ χ acting on a pure state | ψ (cid:105) , F [Λ χ ( | ψ (cid:105) )] = min U χ (cid:0) F [ U χ | ψ (cid:105) S | (cid:105) E ] (cid:1) , (1)where U χ is a unitary dilation of the channel, acting on a largerHilbert space containing system mode S and environmentmode E , and satisfying Λ χ ( • ) = tr E (cid:2) U χ ( • S ⊗ | (cid:105)(cid:104) | E ) U † χ (cid:3) . a r X i v : . [ qu a n t - ph ] O c t For lossy-phase estimation U χ can be chosen such that F [ U χ | ψ (cid:105) S | (cid:105) E ] provides informative bounds on the achiev-able precision dependent only on the mean number of probephotons (cid:104) ˆ n (cid:105) in ≡ (cid:104) ψ | ˆ n S | ψ (cid:105) [13].Seeking an upper bound on the precision for CPLE wechoose a unitary dilation of S , with a single free enviromentalparameter ς which dictates the phase imparted onto E . Thisdilation takes the form U χ,ς = U ( θ, ς ) U ( η ) where U ( η ) and U ( θ, ς ) enact system loss (1 − η ) and phase θ of U re-spectively. These unitaries are given by U = exp[ i ˆ H ξ ( η )] , ˆ H = i (ˆ a † S ˆ a E − ˆ a † E ˆ a S ) , ξ ( η ) = arccos(2 η − and U =exp[ i ˆ H ( ς ) θ ] , ˆ H ( ς ) = ˆ n S + ς ˆ n E . We verify that U χ is a dila-tion of Λ χ in Supplementary Material A [33]. In Supplemen-tary Material B [33] we show that for | Ψ χ,ς (cid:105) ≡ U χ,ς | ψ (cid:105) S | (cid:105) E : F χ ( | Ψ χ,ς (cid:105) ) = ( ∂ χ θ ) (cid:16) (cid:107)| ∂ θ Ψ χ,ς (cid:105)(cid:107) − |(cid:104) Ψ χ,ς | ∂ θ Ψ χ,ς (cid:105)| (cid:17) + ( ∂ χ η ) (cid:16) (cid:107) | ∂ η Ψ χ,ς (cid:105)(cid:107) − |(cid:104) Ψ χ,ς | ∂ η Ψ χ,ς (cid:105)| (cid:17) . (2)For any probe state, the second term in Eq. (2) is given by: ( ∂ χ η ) (cid:16) (cid:107)| ∂ η Ψ χ,ς (cid:105)(cid:107) − |(cid:104) Ψ χ,ς | ∂ η Ψ χ,ς (cid:105)| (cid:17) = ( ∂ χ η ) (cid:104) ˆ n (cid:105) in η (1 − η ) , which is independent of ς [33]. Therefore, the optimal ς isgiven by minimization of (cid:107)| ∂ θ Ψ χ,ς (cid:105)(cid:107) −|(cid:104) Ψ χ,ς | ∂ θ Ψ χ,ς (cid:105)| , inaccordance with Eq. (1). This same expression was minimizedin Ref. [13] and hence has the same optimal value: ς opt = 1 − Var in (ˆ n ) (cid:14) [(1 − η ) Var in (ˆ n )+ η (cid:104) ˆ n (cid:105) in ] , (3)with Var in (ˆ n ) = (cid:104) ψ | ˆ n S | ψ (cid:105)− ( (cid:104) ˆ n (cid:105) in ) . Therefore the limit wehave found for CPLE is simply the sum of the limits on QFIfor phase estimation (first term) and loss estimation (secondterm) [21]. Inserting ς opt (Eq. (3)) into Eq. (2) yields: F χ ( (cid:37) χ ) ≤ ( ∂ χ θ ) (cid:20) η (cid:104) ˆ n (cid:105) in Var in (ˆ n )(1 − η ) Var in (ˆ n ) + η (cid:104) ˆ n (cid:105) in (cid:21) + ( ∂ χ η ) (cid:104) ˆ n (cid:105) in η (1 − η ) ≤ (cid:104) ˆ n (cid:105) in η ( ∂ χ θ ) + ( ∂ χ η ) η (1 − η ) =: Q χ . (4)where the last expression depends only on (cid:104) ˆ n (cid:105) in . Q χ denotesthe maximum information available on χ for any quantumprobe and measurement, and therefore the bound we aim tosaturate. We note that phase estimation benefits from super-Poissionian statistics in pure states, Var in (ˆ n ) ≥ (cid:104) ˆ n (cid:105) in [32],while loss estimation benefits from sub-Poissionian statistics, Var in (ˆ n ) ≤ (cid:104) ˆ n (cid:105) in [18] — however, a probe state cannot haveboth properties. This suggests the inequality in Eq. (4) maynot be saturable. However, we show that this bound can besaturated. Probe states for CPLE —
Having found the fundamentallimit for CPLE, we next seek an effective strategy for ex-perimentally achieving this bound using single-mode Gaus-sian states and homodyne measurements. These were recently shown to be optimal for lossy-phase estimation in the largephoton number limit [35].Single-mode Gaussian states are specified by a displace-ment vector d comprised of means, d i = (cid:104) ˆ x i (cid:105) , and a matrix Γ comprised of covariances, Γ ij = (cid:104) ˆ x i ˆ x j +ˆ x j ˆ x i (cid:105)−(cid:104) ˆ x i (cid:105)(cid:104) ˆ x j (cid:105) , ofthe quadrature operators ˆ x = (ˆ a † + ˆ a ) and ˆ x = i (ˆ a † − ˆ a ) [36, 37]. Homodyne measurement of a single-mode state pro-vides a measurement of the ˆ x quadrature [38]. An arbi-trary single-mode pure Gaussian state can be defined by thesqueezing ˆ S ( r, φ ) = exp[ r ( e − i φ ˆ a − e i φ ˆ a † )] , displacement ˆ D ( α ) = exp[ α (ˆ a † − ˆ a )] , and rotation ˆ R ( ϕ ) = exp( i ˆ a † ˆ aϕ ) operators acting on vacuum: | ψ G (cid:105) = ˆ R ( ϕ ) ˆ D ( α ) ˆ S ( r, φ ) | (cid:105) where all arguments are real and the mean number of pho-tons within the state is: (cid:104) ˆ n (cid:105) = α + sinh ( r ) . The ac-tions of squeezing, displacement, rotation (phase shift) andloss modify d and Γ [39]. | ψ G (cid:105) will be transformed by Λ χ to ˜ (cid:37) = Λ χ ( | ψ G (cid:105) ) with ˜ d = R ( ϕ + θ ) (cid:18) α √ η (cid:19) and ˜ Γ = R ( ϕ + φ/ θ ) 14 (cid:18) η e − r +1 − η η e r +1 − η (cid:19) R (cid:62) ( ϕ + φ/ θ ) ,where R ( • ) = (cid:18) cos • − sin • sin • cos • (cid:19) is the rotation matrix [38].Throughout the following, tildes over variables refer to prop-erties of the state after Λ χ has been applied. d and Γ of | ψ G (cid:105) can be observed by setting η = 1 and θ = 0 in ˜ d and ˜ Γ .The QFI of a single-mode Gaussian state ˜ (cid:37) is [40]: F χ (˜ (cid:37) ) = tr[(˜ Γ − ∂ χ ˜ Γ ) ]2(1 + ˜ P ) + 2( ∂ χ ˜ P ) − ˜ P + ( ∂ χ ˜ d ) (cid:62) ˜ Γ − ( ∂ χ ˜ d ) , (5)where ˜ P = tr(˜ (cid:37) ) is the purity. Directly optimising the QFIof a Gaussian state for lossy-phase estimation provides sub-optimal use with homodyne measurement [41]. Because ofthis, we optimize information related to the parameter depen-dence on displacement vector ˜ d , in the third term of Eq. (5).For lossy-phase estimation it was shown that this informationis accessible through homodyne detection and thus we seek tomaximise this term by varying the probe | ψ G (cid:105) .To do this, the squeezing angle φ should be set such that ∂ χ ˜ d is parallel to the direction of minimum uncertainty in theoutput state i.e. aligned with the eigenvector of ˜ Γ with small-est eigenvalue ˜ V min = [ e − r η + (1 − η )] / . A state satisfyingthis condition is plotted in Fig.1. In this case, the informationcontained in displacement vector D is given by D := ( ∂ χ ˜ d ) (cid:62) ˜ Γ − ( ∂ χ ˜ d ) = (cid:107) ∂ χ ˜ d (cid:107) (cid:14) ˜ V min . (6)The output can be measured using homodyne detection to pro-duce a signal which has a FI of D +( ∂ χ ˜ V min ) / (2 ˜ V min ) [30],which shows that D is a quantity which can be accessed witha practical measurement. Using an adaptive feedback strategy(e.g. [42]), the squeezing and homodyne angles can be set ar-bitrarily close to their optimal values. ∂ χ ˜ d = ( ∂ χ θ ) ∂ θ ˜ d +( ∂ χ η ) ∂ η ˜ d where the two terms are always orthogonal, there-fore: Λ FIG. 1:
Phase-space representation of the transformation of ini-tial state (cid:37) to ˜ (cid:37) : after passing through the channel Λ with transmis-sion η and a phase shift of θ . (cid:37) is squeezed in the optimal directionaligned with ∂ χ ˜ d . The red curve is the homodyne signal when thephase of the local-oscillator is optimized for the measurement. (cid:107) ∂ χ ˜ d (cid:107) = (cid:107) ( ∂ χ θ ) ∂ θ ˜ d (cid:107) + (cid:107) ( ∂ χ η ) ∂ η ˜ d (cid:107) = α [4 η ( ∂ χ θ ) + ( ∂ χ η ) ] / η, where α is the coherent amplitude of the input state (Fig. 1)and ( ∂ χ θ ) and ( ∂ χ η ) appear in the same proportions as in Q χ (Eq. (4)). It can be observed from Eq. (5) that the QFIachieved with an unsqueezed coherent state as probe is D (cid:12)(cid:12) r =0 = (cid:104) ˆ n (cid:105) in [4 η ( ∂ χ θ ) + ( ∂ χ η ) ] /η := S χ , (7)which limits the best precision achievable using classicalprobes with a single pass through Λ χ — the standard quan-tum limit (SQL).Combining ˜ V min and Eq. (6) we find D = ( (cid:104) ˆ n (cid:105) in − n sq ) 4 η ( ∂ χ θ ) + ( ∂ χ η ) η [ e − r η + (1 − η )] . (8)where α = (cid:104) ˆ n (cid:105) in − n sq has been used and n sq = sinh ( r ) isthe number of photons contributing to the squeezing of the in-put state. As (cid:104) ˆ n (cid:105) in grows, D / (cid:104) ˆ n (cid:105) in will converge to the quan-tum limit we have found in Eq. (4) i.e. lim (cid:104) ˆ n (cid:105) in →∞ D / (cid:104) ˆ n (cid:105) in = Q χ / (cid:104) ˆ n (cid:105) in if two conditions are satisfied: First, n sq needs to bea vanishing proportion of the total number of probe photons lim (cid:104) ˆ n (cid:105) in →∞ n sq / (cid:104) ˆ n (cid:105) in = 0 . Second, n sq needs to be unboundedwith increasing (cid:104) ˆ n (cid:105) in , which will ensure e − r vanishes. InSupplementary Material C [33] we describe a state with finite,and arbitrary, (cid:104) ˆ n (cid:105) in for which F χ ( (cid:37) ) = Q χ , demonstrating Q χ is a saturable upper bound (though not of genuine practicalutility).Therefore, we have found that there is no trade-off in the in-formation encoded on a state by the phase and loss of a chan-nel. This is in contrast to the task of estimating phase andloss when there is no correlation [43] which displays a neces-sary trade-off in the precision to which each parameter couldbe estimated. Our results also contrast with those reported inRef. [44], which assume total energy of a probe state includingany reference or ancilla (which does not expose the sample)as the resource. With this assumption it was found for the lowphoton-number regime that there is a trade-off in the sensi-tivity of the probe state to either loss or phase. Our choiceof resource (the total optical power incident on the sample) is relevant when the sample is delicate. The total optical powerin a probe often constitutes a small fraction of the total energyneeded for example to generate the quantum probe.For finite (cid:104) ˆ n (cid:105) in , D can be optimised by choosing the bestvalue of n sq . The optimal amount of squeezing is derived inthe Supplementary Material D [33] to be n sq = (cid:16)(cid:112) − η − η (cid:104) ˆ n (cid:105) in − (cid:17) − η ) (cid:16)(cid:112) − η − η (cid:104) ˆ n (cid:105) in − η (cid:17) , (9)which results in D = Q χ η − (cid:104) ˆ n (cid:105) in + (cid:112) − η − η (cid:104) ˆ n (cid:105) in − η − (cid:104) ˆ n (cid:105) in . In Fig. 2a the optimal D for a selection of different values of (cid:104) ˆ n (cid:105) in is plotted over η ∈ (0 , . The range of (cid:104) ˆ n (cid:105) in = 10 i , i ∈{ , , ..., } scale to large numbers but corresponds to low en-ergy e.g. photons at λ = 500 nm equates to × − J.The plot shows that Gaussian states with modest energies canprovide large precision gains for CPLE.
Probe states for DAE —
We now turn to DAEs, which referto measurements of absorption which do not exploit informa-tion about any phase imparted by a sample. Previously, a limiton QFI was found for transmission estimation where no phaseis imparted by the sample i.e. θ = 0 [17], and Fock stateswere identified as optimal for this [18]. This bound appliesequally for DAE since Fock states are invariant under phaseshifts. Since θ is uncorrelated with η and unknown, it can-not increase the QFI associated with η [31], and therefore thelimit on QFI for DAE is Q (cid:12)(cid:12) ∂ χ θ =0 , ∂ χ η =1 := Q η . Similarly S χ (cid:12)(cid:12) ∂ χ θ =0 , ∂ χ η =1 := S η , is the SQL for DAE [18]. Howeverwhen a Gaussian probe is used, DAE is inequivalent to CPLEwith ∂ χ θ = 0 since the probe state will be transformed byany phase shift present. For instance, the strategy for CPLEdescribed above using Gaussian states does not work for DAEas the correct homodyne measurement setting depends on thephase imparted by the sample — we therefore seek an alter-native strategy.Intensity measurements are unaffected by the phase of thedetected light, and therefore provide a way to decouple the ef-fects of sample absorption and any phase shift. To find usefulstrategies for DAE, we consider the statistical information N contained measurement of the mean intensity which will bedetected (cid:104) ˆ n (cid:105) out = η (cid:104) ˆ n (cid:105) in , which can be found most simply us-ing standard error propagation: N := 1 / Var( η ) = ( ∂ η (cid:104) ˆ n (cid:105) out ) (cid:14) Var out (ˆ n )= ( (cid:104) ˆ n (cid:105) in ) (cid:14) (cid:2) η Var in (ˆ n ) + η (1 − η ) (cid:104) ˆ n (cid:105) in (cid:3) , (10)which applies for arbitrary states. Considering only the meanintensity ensures complex measurement and estimation pro-cedures are not needed and N plays a role analogous to FI.Loss reduces the amplitude of a Gaussian state, and soa natural probe state to consider for DAE is an amplitude-squeezed Gaussian state, | ψ G (cid:105) (cid:12)(cid:12) φ =0 . Noting that Var in (ˆ n ) =
56 7 8 a ) c ) n sq = 4 • ⌘ b ) DQ NQ ⌘ DQ = NQ ⌘ • = 1 • = 1 • =-1 h ˆ n i in = 10 • h ˆ n i in = 10 • FIG. 2:
Comparing strategies for CPLE and DAE with their respective quantum limits:
Within each plot the red dashed line shows theSQL. a) Amount of statistical information D encoded onto the displacement vector (mean number of photons) of a squeezed coherent state forCPLE , normalised to the quantum limit Q χ . The inset shows that even for very low absorption these states approach the quantum limit formodest energies. Statistical information plotted for varying input mean photon number operating with the optimal squeezing value presentedin Eq. (9). b) Amount of statistical information N encoded onto the displacement vector (mean number of photons) of a squeezed coherentstate for DAE, normalised to the quantum limit Q η . Statistical information plotted for varying input mean photon number operating with theoptimal squeezing value presented in Eq. (9). c) Amount of statistical information D ( N ) encoded onto the displacement vector (mean numberof photons) of a squeezed coherent state for CPLE (DAE), normalised to the quantum limit Q χ ( Q η ) when α is large. Γ − + ( (cid:104) ˆ n (cid:105) in − n sq ) e − r for an amplitude squeezedGaussian state [45], and tr Γ = O ( n sq ) . Asymptotic optimal-ity lim (cid:104) ˆ n (cid:105) in →∞ N / (cid:104) ˆ n (cid:105) in = Q η / (cid:104) ˆ n (cid:105) in can be achieved if n sq is unbounded (to ensure e − r vanishes) and also a vanishingproportion of (cid:112) (cid:104) ˆ n (cid:105) in . This ensures that the photon numbervariance of the input state contributes negligibly to the de-nominator of expression on the second line of Eq. (10). Alsoshown in Eq. (10) is that in order to maximize N , the pho-ton number variance of the input state should be minimisedfor a given (cid:104) ˆ n (cid:105) in independently of η . In Fig. 2b the optimal N for a selection of different values of (cid:104) ˆ n (cid:105) in is plotted over η ∈ (0 , . (see Supplementary Material E [33] for the opti-mization). This plot shows that Gaussian states with modestenergies can provide large precision gains for DAE. Multi-pass strategies —
Rather than using non-classicalstates, it is sometimes possible to increase precision beyondthe SQL by sending a classical (coherent state) optical probethrough the sample multiple times [35]. Recently it wasshown that, for lossy-phase estimation, multi-pass strategiescould obtain 60 % of the quantum limit on FI for a given num-ber of photons incident upon the sample over all passes andfor any values of the phase shift and loss. In SupplementaryMaterial F [33] we extend this result and show that multi-pass strategies provide exactly the same benefits for CPLEand DAE as they do for lossy-phase estimation. This exactcorrespondence holds even when lossy components are usedto perform the multi-pass strategy. Practical application —
At present the highest amount ofoptical squeezing demonstrated is 15 dB [46] ( n sq = 7 . ). Byexplicitly considering large α we can quantify the quantumadvantage, ∆ , squeezing brings to both CPLE and DAE: ∆ = lim α →∞ N (cid:14) S η = lim α →∞ D (cid:14) S χ = 1 e − r η + (1 − η ) , (11)observing that the enhancement provided for both DAE andCPLE is the same. The precision gains which squeezing brings to probe states with large α is plotted in Fig. 2.c.For CPLE, Eq. (11) encouragingly indicates that a smallamount of squeezing can substantially increase the precisionof a measurement. Generating and detecting Fock states isa non-trivial task and as such only low photon number Fockstates have been generated [47, 48]; these states may proveuseful for the measurement of samples which are damaged byvery few photons. The Gaussian probe state we have stud-ied can be created by the displacement of a squeezed vacuumstate to contain much larger amounts of power [49], benefitingabsorption measurements far beyond the few photon regime.We highlight Ref. [6] which reported absorption measure-ments with 10 µ W of incident laser light ( photons persecond) at 633 nm to detect the presence of single molecules.Using a balanced photodetector the effective intensity fluctua-tions in the laser light were reduced to the shot-noise limit.Using Eq. (11) and taking η to be . , 15 dB of squeez-ing in this experiment would reduce the contribution to themean-squared error (MSE) from fundamental fluctuations bya factor of 12.5. This is 79% of the advantage provided byusing photons per second in Fock states. Alternatively,the same precision could be achieved with a factor of 12.5reduction in input intensity. Conclusion —
Our results further indicate that for estimat-ing parameters of linear optical transformations with non-unittransmissivity, the information which can be encoded in thecoarse-grained properties of a state, such as the mean inten-sity or mean quadrature value, is very close to the fundamen-tal limit on the information which can be encoded on an entirestate [14, 35]. We anticipate the quantum limit on CPLE andour Gaussian state strategy can be generalized to multiparam-eter estimation problems [50] and perhaps even to precisionestimation of general-linear mode transformations.
Acknowledgments —
We thank J.P. Dowling and D.H.Mahler for helpful discussions. This work was supportedby EPSRC, ERC, PICQUE, BBOI, US Army Research Of-fice (ARO) Grant No. W911NF-14-1-0133, U.S. Air ForceOffice of Scientific Research (AFOSR) and the Centre forNanoscience and Quantum Information (NSQI). E.J.A. wassupported by the Quantum Engineering Centre for DoctoralTraining, EPSRC grant EP/L015730/1. J.L.OB. acknowl-edges a Royal Society Wolfson Merit Award and a RoyalAcademy of Engineering Chair in Emerging Technologies.J.C.F.M. and J.L.O’B acknowledge fellowship support fromEPSRC. J.C.F.M. acknowledges support from ERC startinggrant ERC-2018-STG803665. T.M.S. was supported by theBenjamin-Meaker visiting fellowship and the ARC Centre ofExcellence in Engineered Quantum Systems CE17. ∗ Electronic address: [email protected][1] V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhancedmeasurements: beating the standard quantum limit,”
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SUPPLEMENTARY MATERIAL A: CHANNEL DILATION
Letting the system mode be mode one, ˆ a † S = ˆ a † and the environment mode be mode two, ˆ a † E = ˆ a † the unitary dilation statedin the main text is U χ,ς = U ( θ, ς ) U ( η ) (12)where U = exp[ i ˆ H ξ ( η )] , ˆ H = i (ˆ a † ˆ a − ˆ a † ˆ a ) , ξ ( η ) = 2 arccos( √ η ) and U = exp[ i ˆ H ( ς ) θ ] , ˆ H ( ς ) = ˆ n + ς ˆ n . Thetransfer matrix associated with U is: e − i ξ ( η ) Y/ = (cid:18) cos [ ξ ( η ) / − sin [ ξ ( η ) / ξ ( η ) /
2] cos [ ξ ( η ) / (cid:19) = (cid:18) √ η −√ − η √ − η √ η (cid:19) . (13)where Y = (cid:18) − ii (cid:19) is the Pauli Y matrix. Similarly, the transfer matrix associated with U is: (cid:18) e i θ e i ςθ (cid:19) (14)such that the transfer matrix of U χ,ς is (cid:18) e i θ √ η − e i θ √ − η e i ςθ √ − η e i ςθ √ η (cid:19) . (15)The element T is an attenuation by η and a phase shift θ as desired. SUPPLEMENTARY MATERIAL B: BOUNDING THE QUANTUM FISHER INFORMATION FOR CORRELATED PHASE ANDLOSS ESTIMATION
In this section we continue to use the notation ˆ a † S = ˆ a † and ˆ a † E = ˆ a † introduced in Section A. We start with an expression ofthe QFI for pure states: F ( | Ψ χ,ς (cid:105) ) = 4 (cid:0) (cid:104) ∂ χ Ψ χ,ς | ∂ χ Ψ χ,ς (cid:105) − |(cid:104) Ψ χ,ς | ∂ χ Ψ χ,ς (cid:105)| (cid:1) . (16)Focusing on the second term in Eq. (16): (cid:104) Ψ χ,ς | ∂ χ Ψ χ,ς (cid:105) = (cid:104) ψ, |U − ( ∂ χ U ) | ψ, (cid:105) = (cid:104) ψ, |U − (cid:110) i (cid:104) ( ∂ χ θ ) ˆ H + ( ∂ χ η )( ∂ η ξ ) U ˆ H U − (cid:105) U (cid:111) | ψ, (cid:105) = i ( ∂ χ θ ) (cid:104) ψ, | U − ˆ H U | ψ, (cid:105) + i ( ∂ χ η )( ∂ η ξ ) (cid:104) ψ, | ˆ H | ψ, (cid:105) = i ( ∂ χ θ ) (cid:104) ψ, | U − ˆ H U | ψ, (cid:105) (17)where in the first step we used ∂ χ U = ( ∂ χ U ) U + U ( ∂ χ U )= ( ∂ χ θ ) i ˆ H e i ˆ H θ U + U ( ∂ χ η )( ∂ η ξ ) i ˆ H e i ˆ H ξ = i (cid:104) ( ∂ χ θ ) ˆ H + ( ∂ χ η )( ∂ η ξ ) U ˆ H U − (cid:105) U (18)and in the second step we used (cid:104) ψ, | ˆ H | ψ, (cid:105) = 0 which is due to the form of ˆ H = i (ˆ a † ˆ a − ˆ a † ˆ a ) . Expanding the first termin Eq. (16) gives: (cid:104) ∂ χ Ψ χ,ς | ∂ χ Ψ χ,ς (cid:105) = (cid:104) ψ, | (cid:0) ∂ χ U − (cid:1) ( ∂ χ U ) | ψ, (cid:105) = ( ∂ χ θ ) (cid:104) ψ, | U − ˆ H U | ψ, (cid:105) + [( ∂ χ η )( ∂ η ξ )] (cid:104) ψ, | ˆ H | ψ, (cid:105) + ( ∂ χ θ )( ∂ χ η )( ∂ η ξ ) (cid:104) ψ, |{ ˆ H , U − ˆ H U }| ψ, (cid:105) (19)where { A, B } = AB + BA is the commutator. By using U − ˆ a † U = √ η ˆ a † − (cid:112) − η ˆ a † U − ˆ a † U = (cid:112) − η ˆ a † + √ η ˆ a † U − ˆ a U = √ η ˆ a − (cid:112) − η ˆ a U − ˆ a U = (cid:112) − η ˆ a + √ η ˆ a (20)we can calculate: U − ˆ H U = ( η + ς − ης ) ˆ n + η ( ς − (cid:112) − η ˆ n + ( ς − (cid:112) η (1 − η ) (cid:16) ˆ a † ˆ a + ˆ a † ˆ a (cid:17) . (21)As neither ˆ n nor the ˆ n operator will populate mode two with photons we can see that (cid:104) ψ, | ˆ n ˆ H | ψ, (cid:105) = (cid:104) ψ, | ˆ n ˆ H | ψ, (cid:105) = (cid:104) ψ, | ˆ H ˆ n | ψ, (cid:105) = (cid:104) ψ, | ˆ H ˆ n | ψ, (cid:105) = 0 (22)therefore, letting γ = ( ς − (cid:112) η (1 − η ) , we can evaluate the last term in Eq. (19): (cid:104) ψ, |{ ˆ H , U − ˆ H U }| ψ, (cid:105) = γ (cid:104) ψ, |{ ˆ H , (ˆ a † ˆ a + ˆ a † ˆ a ) }| ψ, (cid:105) = γ i (cid:104) ψ, | (cid:20)(cid:16) ˆ a † ˆ a (cid:17) − (cid:16) ˆ a † ˆ a (cid:17) (cid:21) | ψ, (cid:105) = 0 . (23)Putting Eq. (19), Eq. (17) and Eq. (23) together we can observe that the QFI of | Ψ χ,ς (cid:105) is: F ( | Ψ χ,ς (cid:105) ) = 4 (cid:0) (cid:104) ∂ χ Ψ χ,ς | ∂ χ Ψ χ,ς (cid:105) − |(cid:104) Ψ χ,ς | ∂ χ Ψ χ,ς (cid:105)| (cid:1) = 4( ∂ χ θ ) (cid:104) (cid:104) ψ, | U − ˆ H U | ψ, (cid:105) − |(cid:104) ψ, | U − ˆ H U | ψ, (cid:105)| (cid:105) + 4 [( ∂ χ η )( ∂ η ξ )] (cid:104) ψ, | ˆ H | ψ, (cid:105) (24)where the first line of second equality is equal to the QFI for phase estimation multiplied by a factor of ( ∂ χ θ ) associated withchanging variables from the phase θ to χ : (cid:104) ∂ θ Ψ χ,ς | ∂ θ Ψ χ,ς (cid:105) − |(cid:104) Ψ χ,ς | ∂ θ Ψ χ,ς (cid:105)| = (cid:104) ψ, | U − ˆ H U | ψ, (cid:105) − |(cid:104) ψ, | U − ˆ H U | ψ, (cid:105)| = −(cid:104) ˆ n (cid:105) in (1 − η ) η ( ς − + Var in (ˆ n )( η + ς − ης ) (25)where the second line has been calculated using the similarity transformations in Eq. (20). Similarly, the expression on thesecond line of the second equality in Eq. (24) is the QFI for loss estimation multiplied by a factor of ( ∂ χ η ) associated withchanging variables from transmissivity η to χ : (cid:104) ∂ η Ψ χ,ς | ∂ η Ψ χ,ς (cid:105) − |(cid:104) Ψ χ,ς | ∂ η Ψ χ,ς (cid:105)| = ( ∂ η ξ ) (cid:104) ψ, | ˆ H | ψ, (cid:105) = 14 η (1 − η ) (cid:104) ˆ n (cid:105) in . (26)As this expression is independent of ς , the QFI of the purified state F ( | Ψ χ,ς (cid:105) ) is minimised by minimising the term related tothe phase information in Eq. (25). This can be minimised by differentiating with respect to ς and setting the resultant expressionto zero yielding: ς opt = 1 − Var in (ˆ n )(1 − η ) Var in (ˆ n ) + η (cid:104) ˆ n (cid:105) in corresponding to the minimised expression: (cid:104) ˆ n (cid:105) in η (1 − η ) + (cid:104) ˆ n (cid:105) in η/ Var in (ˆ n ) ≤ η (cid:104) ˆ n (cid:105) in − η . (27)Putting together Eqs. (24), (26), (27) we can see the QFI of the state (cid:37) χ = Λ χ ( | ψ (cid:105) ) after the channel Λ χ is bounded by: F ( (cid:37) χ ) ≤ min ς [ F ( | Ψ χ,ς (cid:105) )]= ( ∂ χ θ ) (cid:20) η (cid:104) ˆ n (cid:105) in Var in (ˆ n )(1 − η ) Var in (ˆ n ) + η (cid:104) ˆ n (cid:105) in (cid:21) + ( ∂ χ η ) (cid:104) ˆ n (cid:105) in η (1 − η ) ≤ (cid:104) ˆ n (cid:105) in η ( ∂ χ θ ) + ( ∂ χ η ) η (1 − η ) . (28) SUPPLEMENTARY MATERIAL C: SATURATING THE QUANTUM LIMITS WITH FINITE MEAN PHOTON NUMBERINPUT STATES
In this section we show that probe states can have a finite mean number of photons and saturate the quantum limits on bothCPLE, such that F ( (cid:37) ) = Q χ , and DAE, such that F ( (cid:37) ) = Q η . To demonstrate this mathematical statement, and not in searchor practical strategies, we consider a probe state of the form: (cid:37) p = lim n →∞ (cid:104) pn | ψ n (cid:105)(cid:104) ψ n | ⊗ | (cid:105)(cid:104) | + (cid:16) − pn (cid:17) | (cid:105)(cid:104) | ⊗ | (cid:105)(cid:104) | (cid:105) (29)in which the first system is the state of an optical mode incident upon the sample and the second system is a qubit. The state | ψ n (cid:105) has a mean number photon number of n and | (cid:105) denotes the vacuum. Therefore (cid:37) p contains an average of p photons. Inthe main text of this chapter it was shown that there exist states with a QFI that is asymptotically equivalent to the quantumlimit for both CPLE and DAE. We consider | ψ n (cid:105) to be a state which is asymptotically equivalent to the quantum limit i.e. lim n →∞ [ F ( | ψ n (cid:105) ) / Q • ] = 1 where here Q • is the quantum limit on n mean photon number states. Due to the linearity of QFIover terms which have support on orthogonal subspaces [8], and the fact that the quantum limit is directly proportional to themean number of photons in the input state, the QFI of (cid:37) p is F ( (cid:37) p ) = lim n →∞ [( Q • /p ) n × ( p/n )] = Q • where here Q • is the quantum limit on p mean photon number input states. We note that | ψ n (cid:105) could be a Gaussian state makingthis strategy equivalent to the ones discussed in the main text except sometimes no probe photons are used; such a change canonly yield a superficial advantage and it will require more trials before the regular Cr´amer–Rao is saturated; therefore to make aprecise measurement this type of modification will not reduce the mean number of photons which are incident upon the sample.The purpose of this section is merely to show that the bounds on QFI which we have found are tight. SUPPLEMENTARY MATERIAL D: OPTIMISING D In this appendix we optimize D = (cid:107) ∂ χ ˜ d (cid:107) (cid:14) ˜ V min , for a state which is squeezed in the optimal direction, given in the main textEq. (8) over the amount of squeezing n sq (or equivalently r ) for a given mean number of photons (cid:104) ˆ n (cid:105) . Restating D : D = ( (cid:104) ˆ n (cid:105) in − n sq ) 4 S χ [ e − r η + (1 − η )] . (30)we note that D is will be optimized by the same value of n sq whether the speed of the transfer amplitude is due its phasedependence or loss dependence. In Ref. [35] (cid:107) ∂ θ ˜ d (cid:107) (cid:14) ˜ V min was optimised over n sq for a state squeezed in the optimal direction.As the additional loss dependence of the displacement vector does not change the optimal amount of squeezing, the optimisedvalue of: n sq = (cid:16)(cid:112) − η − η (cid:104) ˆ n (cid:105) in − (cid:17) − η ) (cid:16)(cid:112) − η − η (cid:104) ˆ n (cid:105) in − η (cid:17) = O (cid:16)(cid:112) (cid:104) ˆ n (cid:105) in (cid:17) (31)0for phase estimation reported in [35] will be the optimal value for CPLE. Inserting this value for n sq into the main text Eq. (8)yields D = Q χ η − (cid:104) ˆ n (cid:105) in + (cid:112) − η − η (cid:104) ˆ n (cid:105) in − η − (cid:104) ˆ n (cid:105) in . (32)This expression was used to generate Figure 2 a). SUPPLEMENTARY MATERIAL E: OPTIMISING N In this section we optimise N for an amplitude squeezed Gaussian with a finite mean number of photons. Here we restate N : / ∆ η = ( ∂ η (cid:104) ˆ n (cid:105) out ) (cid:14) Var out (ˆ n ) ≡ N = ( (cid:104) ˆ n (cid:105) in ) (cid:14) (cid:2) η Var in (ˆ n ) + η (1 − η ) (cid:104) ˆ n (cid:105) in (cid:3) . (33)As remarked in the main paper when discussing direct absorption estimation (DAE), the task of minimising N for a given (cid:104) ˆ n (cid:105) in is equivalent to minimising the number variance of the input state. Using the expressions given in [45] we arrive at the numbervariance of an amplitude squeezed pure Gaussian state: Var in (ˆ n ) = 2 (cid:104) ˆ n (cid:105) in n sq − (cid:104) ˆ n (cid:105) in (cid:113) n sq ( n sq + 1) + (cid:104) ˆ n (cid:105) in + 2 (cid:113) n sq ( n sq + 1) + n sq . (34)The above expression can be minimised, for a given (cid:104) ˆ n (cid:105) in , by differentiating with respect to n sq and equating to zero. Theresulting equation cannot be solved analytically for n sq but it can be solved for (cid:104) ˆ n (cid:105) in : (cid:104) ˆ n (cid:105) in = (cid:18) n sq + 2 (cid:113) n sq ( n sq + 1) + 1 (cid:19) (cid:18) n sq (4 n sq + 3) + (cid:113) n sq ( n sq + 1) (cid:19) . (35)This expression is clearly monotonic in n sq for n sq ≥ , therefore for a given (cid:104) ˆ n (cid:105) in the optimal value of n sq will be unique andcan easily be found using numerical methods. This was the approach used to generate the curves in Figure 2 b). SUPPLEMENTARY MATERIAL F: MULTI-PASS STRATEGIES
In this section of the appendix we consider the utility of multi-pass interrogation techniques for CPLE and DAE using tech-niques analogous to those presented in the main paper. First we will consider CPLE. The application of a channel, which impartsa phase θ ( χ ) has a transmissivity of η , k times in series results in an overall transition amplitude of T ( k ) = ( √ η e i θ ) k . The squareof the speed of this transition amplitude is S χ ( k ) = | ∂ χ T ( k ) | = η k − k [ S χ ( k = 1)] (36)where S χ ( k = 1) is the speed of the transition amplitude for a single application of channel. We can insert this modifiedtransition amplitude speed, together with a modified transmissivity of η k , into Eq. (7) of the main text to find the capabilities ofclassical states and into the final line of Eq. (4) of the main text to find the capabilities of optimal quantum states for CPLE inmulti-pass set-ups. Since the capabilities of classical states and the capabilities of optimal quantum states are modified by thesame factor due to the multiple passes, independently of whether it is phase or loss variation contributing to the speed of thetransition amplitude, we can conclude that multi-pass strategies will be just as effective for CPLE as for lossy phase estimationin the preceding chapter. We can also conclude that the optimal number of passes will be the same for CPLE as they were forlossy phase estimation. Specifically, when there is no loss introduced by the apparatus, only by the sample itself, the number ofpasses used to maximise the precision for a given number of lost photons should be adjusted such that η k opt ≈ (the analyticexpression is given in Birchall et al. [35]) in order to maximise the precision per lost photon. The ratio of achievable precisionbetween optimal quantum multi-pass strategies and classical multi-pass strategies for CPLE with a given number of incidentphotons will also be the same as it is for lossy-phase estimation. Therefore the maximum reduction in RMSE one can achievefor CPLE using optimal quantum techniques is ≈ [35].If the elements used to do perform a multi-pass strategy have additional inefficiencies encountered during state preparation η p , state detection η d and in each round trip η r , then the speed of the overall transition amplitude will be reduced by a factor of1 (cid:113) η p η d η k − r . As before, since this factor is independent of whether the initial speed of the transition amplitude is due to phasevariation or loss variation, CPLE is affected by these additional losses in the same way that lossy phase estimation is. Thereforewe can conclude that for CPLE with imperfect components, as long as the round trip loss − η r is less than the combined statepreparation and detection loss η r > η p η d , then non-classical techniques cannot provide more than a reduction in RMSEover classical techniques. This was shown for lossy phase estimation in [35].Multi-pass strategies also enhance DAE in the same way that CPLE is enhanced. The limit on QFI for DAE can be obtainedfrom the limit on QFI for CPLE by setting ( ∂ χ θ = 0) and ( ∂ χ η = 1) such that Q η = Q (cid:12)(cid:12) ∂ χ θ =0 , ∂ χ η =1 . Similarly, the capabilitiesof quantum states in combination with multi-pass setups for DAE are given by the the capabilities of quantum states in combina-tion with multi-pass setups for CPLE when ( ∂ χ θ = 0) and ( ∂ χ η = 1) . The capabilities of classical states and multi-pass setupsis also given by the expressions for CPLE when ( ∂ χ θ = 0) and ( ∂ χ η = 1)= 1)