Optimal Quantum Metrology of Distant Black Bodies
OOptimal Quantum Metrologyof Distant Black Bodies
Mark E. Pearce, Earl T. Campbell, and Pieter Kok
Department of Physics & Astronomy, University of Sheffield, Sheffield S3 7RH, United KingdomOctober 16, 2018
Measurements of an object’s temperature areimportant in many disciplines, from astronomyto engineering, as are estimates of an object’sspatial configuration. We present the quantumoptimal estimator for the temperature of a dis-tant body based on the black body radiationreceived in the far-field. We also show how toperform separable quantum optimal estimates ofthe spatial configuration of a distant object, i.e.imaging. In doing so we necessarily deal withmulti-parameter quantum estimation of incom-patible observables, a problem that is poorly un-derstood. We compare our optimal observablesto the two mode analogue of lensed imaging andfind that the latter is far from optimal, evenwhen compared to measurements which are sep-arable. To prove the optimality of the estimatorswe show that they minimise the cost functionweighted by the quantum Fisher information—this is equivalent to maximising the average fi-delity between the actual state and the esti-mated one.
The quantum Fisher information (QFI) is a prevalentfigure of merit in the field of quantum parameter esti-mation [5, 16, 18, 19, 21]. The quantum analogue ofthe Cram´er-Rao bound provides a lower bound to thecovariance matrix of the parameters to be estimated, al-though the attainability of this bound is not guaranteedas for classical statistics [12]. As in the classical case[17], the quantum Fisher information for states thatfollow a Gaussian distribution takes on a closed formthat depends only on the first and second moments andtheir derivatives [7, 25, 34]. The fact that thermal statesare so commonly found in nature and exhibit Gaussianstatistics makes them the ideal testbed for Gaussian
Mark E. Pearce: [email protected] T. Campbell: [email protected] Kok: p.kok@sheffield.ac.uk quantum estimation problems [23, 26].The Cram´er-Rao bound gives a lower bound on thecovariance matrix of unbiased parameter estimates bythe inverse of the Fisher information matrix [17]. Phys-ically, this describes the relationship between the infor-mation obtained about a parameter via a measurementand the uncertainty in an estimate of the parameterfrom the measurement data. Similarly, the quantumCram´er-Rao bound gives a lower bound on estimationuncertainty via the inverse of the QFI [13]. Since theQFI is a property of the state alone and does not de-pend on a particular measurement scheme, the precisionin parameter estimates is determined by the uncertaintyin the state only [4], and is therefore fundamental in na-ture and cannot be reduced by improving measurementapparatus.In classical metrology, the extension to multiple pa-rameters does not entail a significant alteration to thebasic theory. However, in the quantum theory, the pos-sibility of observables being incompatible leads to ad-ditional complications [31]. The task of finding opti-mal observables under such circumstances is far fromtrivial and may involve collective measurements overmany independent copies of the system, which is exper-imentally challenging. In this paper we consider onlyseparable measurements (each system measured inde-pendently) and attempt to find the optimal observablesamong this class of observables.Thermometry is an important method for interrogat-ing the physical world. The temperature of astronomi-cal objects reveals important clues about their nature,e.g. the cosmic microwave background [6] and estimatesof effective stellar temperatures [1]; in both engineeringand living systems, the temperature of components pro-vides an essential diagnostic tool. The temperature ofan object can be measured in multiple ways [24], forexample via estimating the heat flow in direct-contactmeasurements, or by remotely measuring the radiationfield emitted by the object. Typically, the thermal ra-diation field emitted is a black body spectrum. In thispaper we consider the latter method, and present op-timal quantum estimators for the temperature of black
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Quantum2017-07-12, click title to verify a r X i v : . [ qu a n t - ph ] J u l ody emitters.Imaging provides us with important informationabout the spatial configuration of an object. Typically,imaging techniques are not studied for their optimal-ity. Even in high resolution imaging, the focus tends tobe towards increasing the resolution beyond the diffrac-tion limit and rarely are optimal estimators considered.The main obstacle in applying the powerful techniquesof metrology (both classical and quantum) to imagingis in identifying the parameters that constitute an im-age. Optimal metrology for an object of known param-eterisation can be performed [20, 29, 32, 33] but foran object with an unknown spatial configuration it isnot obvious what parameters we need to estimate. Weshow that the spatial configuration of a thermally ra-diating source is determined by spatial correlations inthe far-field. In this paper we show how to optimallyestimate these parameters, which completely determinethe state, therefore mapping the problem of imaging tostate estimation.The density matrix of blackbody radiation is con-structed from tensor products of many independentthermal k -modes [22]: ρ = ⊗ k ρ k , (1) where the tensor product runs over all k of the form k = ∆ k ( x, y, z ) where x, y, z are non-negative integers.Note we coarse grain k -space rather than working witha continuum of modes. Physically, we motivate thiscoarse graining by appealing to the finite size of anyrealisable detector.The spatial properties of the source are usually un-known, and without knowledge of them we typicallycannot make accurate temperature measurements. Asimple example helps to clarify the problem: considerthe estimation of a star’s temperature. If we simplymeasure the photon count at a single frequency in thefar-field, we cannot distinguish between hotter sourcesand those with a larger angular size since an increasein either parameter will lead to a larger photon count.Therefore, if we wish to measure the temperature ofthe source, we must either know the angular size ofthe source or attempt to estimate it by measuring morethan one frequency mode.If we are interested in the temperature of the sourcealone, then the radiating area—or more precisely thesolid angle of the source—is a nuisance parameter. Re-gardless of whether we are interested in the solid angleof the source, it is necessary to estimate it in order toprovide optimal estimates of the temperature. The con-verse is also generally true. However, as we will show,it is not possible to make measurements of arbitraryspatial properties of the source by measuring spectral modes alone. In fact, it is necessary to consider spa-tially separated modes, where measurements of the cor-relations provide information about the spatial config-uration of the source.A blackbody emits at all frequencies with an inten-sity determined by the Planck distribution. This dis-tribution can be considered as an infinite number ofindependent spectral modes with a spectral width in-versely proportional to the time over which the stateis observed [22]. It is therefore quite straightforwardto determine the statistics of these independent modesand calculate the quantum Fisher information to deter-mine the optimal measurements for simultaneous tem-perature and solid angle estimates. Considering multi-ple spatial modes is somewhat more cumbersome as thespatial modes are not generally independent. Here, byappealing to the fundamental quantum mechanical de-scription of blackbody radiation, we demonstrate thatit is the correlations between modes that convey the in-formation about the spatial configuration of the sourceto the far-field.It has been stated before that the optimal mea-surement for temperature estimates is photon numbercounting, [11, 27]. However, these results rely upon theassumption that we can measure a single thermal modein the far-field, an assumption that we claim cannot besatisfied without knowledge of the exact spatial prop-erties of the source. This is due to the transverse co-herence area of the far-field radiation being determinedby the spatial properties of the source [22], therefore toguarantee we are measuring exactly one mode, our de-tectors must be designed with these properties in mind.In this paper, the state ρ we consider is defined in Fig. 1.The state describes the radiation occupying a small vol-ume A ρ × cτ in the far-field of a black body source,where A ρ is the transverse area of the state and τ is theobservation time.An important assumption is that the transverse areaof the state A ρ is much smaller than the coherence areaof the source. The effect is that we observe a fraction ofeach spectral mode, therefore ensuring that the trans-verse coherence is approximately equal to one acrossthe entire area A ρ . This allows us to coarse grain inthe transverse direction, ignoring any effects due to thefinite transverse coherence area of the radiation, withina single spatial mode.In Sec. 2 we review the Gaussian state formalism, giv-ing the definition of these states and establishing thenotation we will use throughout this paper. Sec. 3 in-troduces the quantum Fisher information and gives anoutline of the general principles of quantum metrology.In Sec. 4 we determine the density matrix for a black-body state observed in the far-field and determine theoptimal measurements for the parameters that govern Accepted in
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Quantum2017-07-12, click title to verify lack body R ρcτA ρ Figure 1: The definition of our state ρ . We assume that thetransverse area over which the state is measured, A ρ , is muchsmaller than the coherence area of the radiation. We also as-sume that the time over which the state is measured, τ , definesthe spectral width ∆ ν = τ − . these states. Unsurprisingly these turn out to be pho-ton counting. In Sec. 5 we consider the effect of addinga secondary spatial mode, transversely separated fromthe first. We find it important to consider the coherencebetween the spatial modes, and in fact this becomes theessential parameter to estimate if we wish to glean spa-tial information about the source. We consider how tooptimally obtain this spatial information and discoverthat the optimal observables do not commute. There-fore we are forced to find a measurement that can beconsidered optimal whilst being restricted by the inher-ent uncertainty of incompatible observables. A preciselyoptimal separable POVM is found that depends uponthe values of the parameters. We also find a POVMwhich is independent of the parameter values and yet isclose to optimal, which is of great practical importance. Gaussian states are defined to be states with a Gaus-sian characteristic function, which for an n -modestate defined by the bosonic mode operators a =(ˆ a , ˆ a † , . . . , ˆ a n , ˆ a † n ) T , has the form [7] χ ( ξ ) = Tr[ ρ e − a T Ω ξ ] = exp (cid:18)
12 (Ω ξ ) T ΣΩ ξ + ξ Ω µ (cid:19) , (2) where ξ = ( ξ , ξ ∗ , . . . , ξ n , ξ ∗ n ) T , Ω = ⊕ ni =1 iσ y , σ y is thePauli y matrix, and µ and Σ are the first and secondmoments defined by µ α = h a α i , (3)Σ αβ = 12 (cid:0) h λ α λ β i + h λ β λ α i (cid:1) , (4) and λ α = a α − µ α . Often the quantities we wish tocalculate can be given in closed form as a function of these moments. Due to the central limit theorem, Gaus-sian states are frequently encountered in systems wherethere are a large number of randomly fluctuating influ-ences [14, 15]. In classical parameter estimation, the Cram´er-Raobound states that the covariance matrix of unbiased es-timates of the parameters, θ = ( θ , . . . , θ d ) T , is boundedby the inverse of the Fisher information matrix [17].That is, Σ θ = h (ˆ θ − h ˆ θ i )(ˆ θ − h ˆ θ i ) T i ≥ I − C , (5) where the hat over the θ signifies that we are concernedwith estimates of the parameters, and not the param-eters themselves (and should not be confused with aquantum mechanical operator), and I C is the classicalFisher information matrix: I C = X x ( ∇ θ p (x | θ )) ( ∇ θ p (x | θ )) T p (x | θ ) , (6) where ∇ θ = ( ∂∂θ , . . . , ∂∂θ d ) T . The sum in Eq. (6) is per-formed over all possible outcomes, x , of the conditionalprobability distribution p (x | θ ) .The matrix inequality of Eq. (5) should be under-stood as implying that the matrix Σ θ − I − C is positivesemi-definite. It follows that the inequality Tr[ G Σ θ ] ≥ Tr[ G I − C ] , (7) also holds for any positive definite weight matrix G .The weight matrix G allows us to assign a relative im-portance to different parameters.The bound in Eq. (5) expresses a limitation of clas-sical data processing. It states that estimates of aset of parameters, θ , are constrained in their vari-ance by a quantity that is entirely determined by theprobability distribution from which we sample data, { x } . In quantum metrology, we consider how the co-variance matrix of our estimates is constrained if thestate, ρ ( θ ) , is measured by some self-adjoint operator, ˆ X = P x x ˆΠ x , giving rise to the probability distribution p (x | θ ) = Tr[ ρ ( θ ) ˆΠ x ] . We assume that we are capableof measuring any self-adjoint operator, which allows usto reduce the variance by changing p (x | θ ) . Finding theobservable that minimises the variance is therefore theobjective of quantum metrology.In analogy with classical metrology, we can define aquantum Fisher information matrix, I Q , which leadsto a quantum version of the Cram´er-Rao bound. Fora single parameter, the quantum Fisher information is Accepted in
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Quantum2017-07-12, click title to verify iven by the expectation of the square of L , the so-calledsymmetric logarithmic derivative (SLD): I Q = Tr[ ρ L ] . (8) The SLD is implicitly defined by ∂ θ ρ = ρ L + L ρ . (9) In the multi-parameter case, this generalises to a quan-tum Fisher information matrix with elements [28] I Q = Re (cid:16) Tr[ ρ LL T ] (cid:17) , (10) where we have defined L = ( L , . . . , L d ) T , which is thevector of SLD operators, and L i is defined in analogyto the single parameter case ( ρ L i + L i ρ ) = ∂ θ i ρ . Forbrevity, we write ∂ i = ∂ θ i throughout the remainder ofthe paper.In [7] Gao and Lee derived a closed form for the QFIof a Gaussian state in terms of the first and secondmoments, µ and Σ . For blackbody states µ = 0 andthe expression found in [7] reduces to I Q = 12 M − αβ,γκ (cid:16) ∇ θ Σ γκ (cid:17)(cid:16) ∇ θ Σ αβ (cid:17) T , (11) where the matrix M = Σ ⊗ Σ+ Ω ⊗ Ω , and the summa-tion convention is used for greek indices. Gao and Leealso provide an expression for the SLD, which is givenin terms of M : L i = 12 M − γκ,αβ (cid:0) ∂ i Σ αβ (cid:1) ( a γ a κ − Σ γκ ) , (12) which we will make use of later to calculate the SLDsfor blackbody states.Eq. (10) gives the so-called SLD form of the quantumFisher information. A quantum version of Eq. (5) canbe determined, which, for a single parameter, is knownto be attainable, at least in the asymptotic sense [3],by which we mean that it is possible to find an esti-mator where the variance of estimates tends towardsthe inverse of the QFI as the number of independentmeasurements tends to infinity.To attain the bound requires finding a measurementfor which the classical Fisher information is equal tothe quantum Fisher information. For a single param-eter, the optimal measurement is related to the SLD.However, in general the SLD will depend on the ex-act value of the parameter, which is assumed unknownbefore any measurements are made. The conventionalmethod to circumvent this issue is to perform a sub-optimal measurement first to obtain an initial estimateof the parameter, then use this estimate to measure anapproximately optimal operator, preferably adaptively changing the measurement as more information is ob-tained about the parameters true value [2, 3, 9, 10].For multiple parameters, the bound is not always at-tainable because the optimal observables may not com-mute and therefore may not be simultaneously measur-able [12]. This introduces a new problem, independentto the problem of the optimal measurements dependingon the true values of the parameters. Even if the SLDsdo not commute, the quantum Cram´er-Rao bound isasymptotically attainable if the following condition issatisfied [30]: Tr[ ρ [ L i , L j ]] = 0 . (13) However, to achieve the quantum Cram´er-Rao boundin this context assumes the use of a collective measure-ment on multiple independent copies of the state. Al-though this is interesting, the implementation of sucha measurement may well prove difficult. In Sec. 5 weencounter a problem with non-commuting SLDs and at-tempt to find the optimal separable measurement forour system.
In this section we show how the quantum Fisher in-formation can be calculated for temperature estimatesof far-field blackbody radiation. We also find the op-timal estimators for both temperature and solid angleand show that the quantum Cram´er-Rao bound is at-tainable since the SLDs commute.Blackbody radiation exhibits a Gaussian characteris-tic function and can therefore be analysed with the pre-ceding formalism by Gao and Lee. In addition, black-body states also have zero first moment, i.e., µ α = 0 for all α . We assume that we observe the radiationfrom a blackbody state in the far-field, paraxial regime.Considering only a single spectral mode centred on thefrequency ν , we find that the state has covariance ma-trix Σ = (cid:18) h n ν i + 12 (cid:19) σ x , (14) where σ x is the Pauli x operator. The covariance matrixdepends only on h n ν i and we can therefore, estimate atmost a single parameter.From Planck’s law, we can calculate the average num-ber of photons for the arrangement given in Fig. 1, h n ν i = ν A S A ρ πc R βhν − ν κ h n th i , (15) where A S is the radiating area of the source, h n th i =[exp( βhν ) − − is the average number of photons per Accepted in
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Quantum2017-07-12, click title to verify hermal mode, and κ contains all of the parameters re-garding the arrangement of the source and its relationto the state ρ . We can also interpret ν κ as the num-ber of modes of frequency ν that are captured by thestate ρ . As mentioned above, we assume throughoutthat ν κ (cid:28) .Given that we do not know the value of κ beforehand,the only way we can ensure that this condition is sat-isfied is by using appropriate values for the parametersthat we can control, A ρ and ν , the frequency we chooseto observe. For example, using the area of a single CCDpixel as A ρ , ∼ − m , we find that this condition isequivalent to cot ϑ (cid:29) αν , where ϑ is the angular sizeof the source and α = 10 − s − . This condition shouldeasily be satisfied in the far-field regime for frequenciesup to X-rays.Identifying θ = T and θ = κ we find that the QFImatrix for the spectral mode centred on ν is given by I ( ν ) Q = 1 h n ν i + h n ν i (cid:18) ( ∂ h n ν i ) ∂ h n ν i ∂ h n ν i ∂ h n ν i ∂ h n ν i ( ∂ h n ν i ) (cid:19) = 1 h n ν i + h n ν i ∇ θ h n ν i ( ∇ θ h n ν i ) T , (16) which is a rank 1 projector and therefore rank deficient.It is thus a singular matrix that does not have a well-defined inverse. Consequently, it is not possible to es-timate both parameters with finite precision. We con-clude that conforming to our expectation, we cannotfind optimal unbiased estimators for both parametersfrom a single mode. However, if for some reason we doknow one of the parameters precisely, we can make anestimate of the other. The SLD in that case is given by L i = ∂ i h n ν ih n ν i + h n ν i ˆ a † ˆ a − h n ν i ∂ i h n ν ih n ν i + h n ν i , (17) and the quantum Cram´er-Rao bound for the single pa-rameter θ i is h (∆ θ i ) i ≥ [ h n ν i + h n ν i ] / ( ∂ i h n ν i ) . Thisresult was obtained in Ref. [11] and shows that the opti-mal measurement for temperature estimation is photoncounting measurements. This follows from the form ofthe SLD in Eq. (17) and the observation that the opti-mal estimator for θ i is given by [28] X = L i I Q . (18) Measurements of X are therefore achieved by measur-ing in the basis of the SLD L i . Since this is the num-ber basis we can perform photon number counting andpostprocess the outcomes to find the value of θ i .When we are ignorant of both parameters, which willgenerally be the case, we are required to make use of atleast two spectral modes. The covariance matrix then becomes Σ = M ⊕ i =1 (cid:18) h n ν i i + 12 (cid:19) σ x , (19) where M is the total number of spectral modes we con-sider. Since each spectral mode is independent, the QFImatrix is given by I Q = M X i =1 I ( ν i ) Q , (20) where I ( ν i ) Q are given by matrices of the form Eq. (16) .Despite the matrices I ( ν i ) Q being singular, the matrix I Q is non-singular as long as M ≥ . The SLD for θ i isgiven by L i = M X j =1 ( ∂ i h n ν j i )( h n ν j i − ˆ n ν j ) h n ν j i + h n ν j i . (21) From Eq. (21) we see that [ L , L ] = 0 and therefore itis possible to measure in the eigenbasis of both SLDssimultaneously. Therefore, photon counting in each ofthe M independent spectral modes provides an optimalestimate of both parameters simultaneously. In practicethis is typically how stellar temperatures are measured,although often more sophisticated models are used toallow for deviations away from exact black body be-haviour.We stress that photon counting must be performed onat least two spectral modes in order to obtain estimatesfor both parameters. The variance for the parameter θ i is h (∆ θ i ) i ≥ P Ml =1 (cid:2) C ( ν l ) (cid:3) ii P Ml,k =1 (cid:16)h I ( ν l ) Q i h I ( ν k ) Q i − h I ( ν l ) Q i h I ( ν k ) Q i (cid:17) , (22) where C ( ν l ) is the cofactor matrix of I ( ν l ) Q , and I ( ν l ) Q is I ( ν l ) Q = ( ∇ θ h n ν l i )( ∇ θ h n ν l i ) T h n ν l i + h n ν l i . (23) Assuming that we are interested in estimating T andtreating κ as a nuisance parameter, we plot in Fig. 2 thevariance in estimates of T for two spectral modes as afunction of the frequency of each mode. In Fig. 2 we seethat there is a well defined optimum for the frequencies ν and ν , the exact value of which we would expect todepend upon the precise values of the parameters to beestimated.Numerically we find that the optimal values of ν and ν are independent of κ and linearly dependent Accepted in
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Quantum2017-07-12, click title to verify × Hz)0123 f r e q u e n c y o f m o d e ( × H z ) Figure 2: The natural logarithm of the variance of T (in unitsof Kelvin) as a function of the frequencies ν and ν . Theexact location of the optimal frequencies is dependent on thevalue of T . In this plot we have used T = 10000 K and κ = 10 − s , which ensures that N (cid:28) over the range offrequencies plotted. It is not surprising that the values on thebar are so large when we consider that the variance quotedis for a single round of measurement and we are using onlytwo spectral modes. Also, at 10000 K the average numberof photons per measurement is always less than 0.0003. Thewhite crosses show the location of the minimum. on T , approximately ν (min)1 = (1 . × Hz K − ) T , ν (min)2 = (1 . × Hz K − ) T . This is reminiscentof Wien’s displacement law, which gives the peak fre-quency of blackbody radiation as ν (max) = (5 . × Hz K − ) T . In the absence of an initial estimateof T , the best we can do is to perform photon count-ing for two frequencies and adjust these correspondinglyas information is obtained about the temperature. Ifwe happen to know of an a priori distribution for thetemperature (for example, when estimating the temper-ature of stars an a priori distribution can be obtainedusing observational data about the relative frequency ofstellar temperatures), we can choose the initial frequen-cies based upon minimisation of the averaged variance,that is min ν , ν Z d T p ( T ) [ I − Q ] , (24) where p ( T ) is our prior distribution over T .Generally we may want to estimate both parameters.A higher precision can be achieved in our estimates byincreasing the number of spectral modes observed. As-suming that we are only capable of photon countingin M spectral modes simultaneously, an optimal mea-surement can be found given these limited resources. Black body
R ρ x
Figure 3: The addition of a secondary spatial mode, the modesare transversely separated by a distance x = | x − x | . Thestate ρ is now a composite system consisting of two spatiallyseparated modes. We again assume that the transverse areaof each spatial mode is less than the coherence area of theradiation, which means that each mode captures a fraction ofa spectral mode. The van Cittert-Zernike theorem allows usto calculate the far-field complex degree of coherence betweenthe two modes, γ ( x , x ) = γ . Defining ν = ( ν , . . . , ν M ) T , an optimal measurementis given by min ν Tr[ G I − Q ( ν )] . (25) The solution of this minimisation will depend on theactual values of the parameters T , and κ , and the chosenweight matrix G . Therefore, without prior knowledgeabout the parameters, an optimal measurement for agiven weight matrix is one that minimises min ν Z d κ d T p ( κ, T )Tr[ G I − Q ( ν )] , (26) where p ( κ, T ) is a prior distribution over the parame-ters, which will depend strongly upon the exact appli-cation.We are now naturally led to the question: what mea-surement can we perform to determine the spatial con-figuration of the source? Most single parameter prob-lems can be treated by estimating κ , which can then beused to determine this single parameter. For exampleif the source is simply planar circular, then the estima-tion of the parameter κ ∝ A S /R = 2 πr /R = 2 πϑ isequivalent to estimating the angular radius ϑ . In gen-eral however, since the QFI depends only upon the aver-age number of photons in each frequency mode, which inturn depends only on the source temperature and κ , wecannot estimate general spatial properties of the source,no matter how many spectral modes we measure.Since the spatial properties are all contained withinthe parameter κ , any attempt to estimate more than asingle spatial parameter results in a singular QFI ma-trix. In order to allow the estimation of arbitrary spa-tial properties, we need to consider the effect of addingadditional spatial modes, which are spatially separatedfrom the first mode, see Fig. 3. Accepted in
Quantum Estimation of Spatial Parameters
In the previous section we showed how to optimally es-timate the temperature of black body sources. In thissection we show that the addition of spatial modes al-lows for the determination of the spatial properties ofthe source. We compare different POVMs, finding theoptimal separable POVM, as well as a POVM which isclose to optimal for a large range of parameters. Thisnear optimal POVM has the advantage that its imple-mentation does not require knowledge of the parame-ters, and therefore it can be implemented at the outsetand without the need to adaptively change the POVM.In order to compare various schemes we use as a quan-tifier the cost function V M ( I Q ) = Tr[ I Q I − C ( M )] , (27) where I Q is the quantum Fisher information and I − C is the inverse of the classical Fisher information matrixfor a given POVM, M . In general any positive matrixcan be used as the weight matrix in the cost function.The reason for this particular choice of weight matrix isthat it maximises the average fidelity between the esti-mated state and the actual state [2] and is therefore thenatural choice for state estimation. We would expectthat a measurement which is simultaneously optimalfor all parameters should achieve V ( I Q ) = d , where d is the number of parameters. However, when the SLDsdo not commute and we restrict ourselves to separablemeasurements, we find V ( I Q ) > d . The state ρ now describes the composite system of twospatially separated volumes, see Fig. 3. The addition ofa secondary spatial mode changes the covariance matrixin the following way: Σ = M ⊕ i =1 c ( i )1 b i c ( i )1 b ∗ i b ∗ i c ( i )2 b i c ( i )2 , (28) where b i = h ˆ a ( i ) † ˆ a ( i )1 i , c ( i ) j = h ˆ a ( i ) † j ˆ a ( i ) j i + 1 / , and ˆ a ( i ) j isthe annihilation operator in the spatial mode j and thespectral mode i . To determine the elements b i and b ∗ i wefirst note that the complex degree of coherence betweenthe two spatial modes is defined as γ ( i )12 = h ˆ a ( i ) † ˆ a ( i )1 i h h n ( i )1 ih n ( i )2 i i , (29) which allows us to write b i = [ h n ( i )1 ih n ( i )2 i ] γ ( i )21 and b ∗ i = [ h n ( i )1 ih n ( i )2 i ] γ ( i )12 . We now make use of the van Cittert-Zernike theorem, which states that the far-fieldcomplex degree of coherence of a spatially incoherentsource is proportional to the Fourier transform of theintensity distribution [22]. Therefore the correlationsbetween the two modes provide information about theFourier transform of the source distribution.In Section 4 we saw that the variables h n ( i ) j i canconvey only information about the temperature of thesource and κ . In contrast, the variables γ ( i )12 convey de-tailed information about the distribution of the source.Since the Fourier transform is an injective mapping,knowledge about the Fourier transform over the entirefar-field plane can be used to exactly reconstruct thesource intensity distribution. Restricting the detectionarea of the state will result in incomplete informationabout the Fourier transform and therefore limits theresolution of a reconstructed intensity distribution. Inthis paper we restrict our attention to the optimal esti-mation of the state consisting of two spatial modes forsimplicity.We make an additional simplification and assumethat the average photon number in both spatial modesis approximately equal, h n ( i )1 i = h n ( i )2 i . We expect thisassumption to hold in the far-field of an isotropic emit-ter. For brevity we will also drop the superscript ( i ) since each frequency mode is independent and thereforecan be optimised separately. Under this assumption we calculate the SLDs for theparameters θ = ( h n i , | γ | , φ ) T , where γ = | γ | exp( iφ ) ,which are found to be of the form L j = P j ˆ n tot + Q j ˆ a † ˆ a + Q ∗ j ˆ a † ˆ a + R j , (30) where ˆ n tot = ˆ n + ˆ n and explicit expressions for P j , Q j , and R j are given in App. A. As we show in App. A,the commutators for these operators are [ L , L ] = 0 , [ L , L ] = 0 , [ L , L ] = 0 and therefore we cannot finda simultaneous eigenbasis for all three operators. Thisrules out the possibility of simultaneously measuringin the eigenbases of the SLDs to achieve a simultane-ously optimal estimate of each parameter. Since theydo however satisfy the condition Eq. (13) (see App. A),it is possible that a collective measurement exists whichattains the QCRB asymptotically [8].In this paper we will not consider collective measure-ments due to the immense technical obstacles. Insteadwe consider only the class of measurements which areseparable. To find the optimal POVM, we first deter-mine the operators X = I − Q L , where we have defined X = ( X , X , X ) T and L = ( L , L , L ) T . This is themulti-parameter extension of Eq. (18) and has the de-sirable property that Var( X i ) = [ I − Q ] ii .Determining the commutators of these operators, we Accepted in
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Quantum2017-07-12, click title to verify S ϕ Figure 4: Measurement scheme with variable phase shift ϕ .Choosing different values for the phase shift allows optimalestimation of the parameters h n i , | γ | , and φ . find [ X , X ] = [ X , X ] = 0 , [ X , X ] = 0 (see App. A)and therefore X , X cannot be simultaneously mea-sured. Next we calculate the quantum Fisher informa-tion from the covariance matrix Eq. (28) and determinethe classical Fisher information for measurements of theoperators X , X . We find, I Q = [I Q ] [I Q ] Q ] [I Q ]
00 0 [I Q ] , (31) I C ( X ) = [I Q ] [I Q ] Q ] [I Q ]
00 0 , (32) I C ( X ) = [I Q ] δ δ δ
00 0 [I Q ] , (33) where δ (cid:28) [I Q ] and δ (cid:28) [I Q ] .We notice that a slight asymmetry exists between theoperators X and X . Whilst measurements of X donot provide information about φ (since [ I C ( X )] =0 ), measurements of X do provide a small amount ofinformation about | γ | ( δ , δ = 0 ). In App. B we arguethat the optimal measurement scheme is given by thesolution to the following minimization: min p Tr[ I Q ( p I C ( X ) + (1 − p ) I C ( X )) − ] . (34) This scheme can be interpreted as probabilisticallychoosing to measure X or X with probabilities p and − p respectively. The minimisation in Eq. (34) ensuresthat we choose the optimal weighting with respect to X and X . We refer to this scheme as the weightedmeasurement scheme.Due to the simple quadratic forms of the SLDs, andtherefore also the operators X , we can determine theeigenmodes of X and X . We find that both opera-tors can be measured using the scheme shown in Fig. 4,where the value of ϕ is different for X and X . Tomeasure X requires the choice ϕ = φ and X can beachieved by setting ϕ = φ − π/ . An optimal mea-surement therefore requires switching between the two distinct phase settings. As discussed above, the proba-bility with which each measurement is made should alsobe optimised. The exact value of the probability p willdepend on the exact values of the parameters. Since thisscheme clearly requires knowledge of the parameters weare attempting to measure, it can only be implementedin an adaptive scenario. However, since we now knowthe optimal measurement, we can use this to compareother, simpler schemes, with the aim to find a schemethat is close to optimal but is independent of the pa-rameters. a) 0 0.5 1-0.5-1 | γ | cos φ | γ | s i n φ × − b) 0 0.2 0.4 0.6 0.8 1 | γ | V o p ( I Q ) / V F T ( I Q )( × − ) Figure 5: V op ( I Q ) /V FT ( I Q ) for the measurement scheme in-volving only a Fourier transform and photon counting. a) Wesee that this scheme is suboptimal for the entire parameterrange ≤ | γ | ≤ , ≤ φ < π and in fact never achievesmore than 1.7 per cent of the performance of the optimal es-timator. b) Shows a cut through fig a) along the dashed line.In this plot we set h n i = 0 . . Accepted in
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Quantum2017-07-12, click title to verify e first consider how well the parameters can be es-timated using a Fourier transform and photon countingon the two input modes. This is of interest for opticsbecause it closely approximates the action of lensingand intensity measurements, which is by far the mostcommonly used method in imaging. The photon countprobability distributions for both the state ρ and thestate after the action of the Fourier transform are de-rived in App. C.We find that without the Fourier transform the pho-ton count distribution is independent of the phase φ and therefore cannot be used to estimate φ . However,once the Fourier transform has been applied to the inputmodes all three parameters can be estimated. In Fig. 5we plot the ratio of the cost function for the Fouriertransform scheme, V FT ( I Q ) , to the optimal scheme forthe entire range of the parameters ≤ | γ | ≤ , and ≤ φ < π , which follows from the definition of γ and the restriction ≤ | γ | ≤ . We see that thisscheme performs far below the optimal scheme and istherefore not an efficient measurement for determiningthe spatial configuration of the source. The second scheme we consider is a scheme wherewe randomly select a phase shift, uniformly over therange ϕ ∈ [0 , π ) , then pass the modes through a beam-splitter and measure in the number basis. We call thisscheme the random phase (RP) scheme. In Fig. 6 weplot the ratio V op ( I Q ) /V RP ( I Q ) to compare the randomphase scheme to the optimal measurement scheme. Wefind that the random phase scheme is very close to op-timal for most of the parameter range ≤ | γ | ≤ , ≤ φ < π . The advantage of the random phasescheme is that, in implementing this scheme, we donot require the values of the parameters. This there-fore means that this POVM can be implemented beforewe have acquired any information about the parame-ters. Surprisingly, even without this information, therandom phase scheme performs very efficiently.As discussed above, the truly optimal POVM requiresus to choose measurements of X with probability p and X with probability (1 − p ) . Although this achieves theoptimal performance it requires knowledge of the pa-rameter values and therefore can only be implementedin an adaptive way. We expect that an approximatelyoptimal scheme can be achieved by first making mea-surements using the random phase scheme, and whenthe estimates of the parameters have reached a sufficientprecision the optimal scheme can be used to further en- a) 0 0.5 1-0.5-1 | γ | cos φ | γ | s i n φ | γ | V o p ( I Q ) / V R P ( I Q ) Figure 6: The ratio of V op ( I Q ) to V RP ( I Q ) . The random phasescheme is seen to be close to optimal unless | γ | is close toone. For this figure we randomly sampled 1000 phases ϕ anddetermined the inverse of the classical Fisher information forsuch a set of measurements. Since the exact performance isaffected by the exact values of ϕ we averaged each point forover 400 trials to remove most of the statistical noise. The insetshows the performance over the whole range, which shows thatthe performance of the random phase scheme is independentof φ . hance the measurement precision. In this paper we have derived a complete description ofstates originating in far-field blackbody sources and ob-served over a finite detection area in the far-field. Thisdescription is complete in the sense that it encompassesall parameters that determine the state, including nui-sance parameters that must be estimated in order toestimate the temperature of the blackbody and its spa-tial configuration. We found that it is necessary to at-
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Quantum2017-07-12, click title to verify empt to estimate these nuisance parameters in orderto estimate the temperature and to do so requires pho-ton counting to be performed on at least two spectralmodes.In order to derive our results, it was necessary tomake certain assumptions about the arrangement of thesource and its relation to the state ρ . In the paperwe assumed that the transverse area of the source ismuch less than one coherence area. We demonstratedthat this gave a bound on the frequencies we can mea-sure, − ν (cid:28) cot ϑ , where ϑ is the angular size ofthe source. We also found a linear relationship betweenthe optimal pair of frequencies and the temperature.Putting these results together allows us to find the fol-lowing bound cot ϑ (cid:29) (cid:15) T , where (cid:15) ≈ − K − whichshows that the optimal frequencies for very hot sources( ∼ − K) should fall in the range of acceptablefrequencies for far-field objects.Interestingly, we found that the estimation of generalspatial parameters requires that the states are observedover at least two spatially separated modes. The intro-duction of a second spatial mode introduces a complexparameter which determines the coherence between thespatial modes and can be identified as the Fourier trans-form of the intensity distribution of the source. We thenidentified the optimal separable measurement for the re-sulting three parameter estimation problem.The optimal strategy depends upon the exact valueof the parameters we wish to estimate, and thereforecan only be implemented in an adaptive setting. How-ever, we also identified another strategy, the randomphase scheme, which performs close to optimal for awide range of the parameters (see Fig. 6). The imple-mentation of the random phase scheme is independentof the parameter values and therefore has the advantageof not requiring an adaptive arrangement. We can thensuppose that an asymptotically optimal scheme can beachieved by starting with the random phase scheme, us-ing this to find a reasonable estimate of the parameters,and then implementing the optimal scheme to furtherincrease the precision of the estimates.We also considered the performance of the measure-ment scheme consisting of simply a Fourier transformand photon counting. This is actually a special case ofthe measurement scheme in Fig. 4, with ϕ fixed at aconstant value (namely zero). Our motivation is thatthis scheme most closely resembles a typical imagingscheme, with the role of the Fourier transform beingprovided by a lens and photon counting provided bya CCD or other photosensitive surface. We find thatthis scheme performs poorly over the entire parameterrange.Our results expose simple imaging schemes as farfrom optimal, even in the simplistic two mode settings. We have shown how spatial information is conveyed tothe far-field, and how we can optimally extract this in-formation for the simplest case of two modes. We ex-pect that this work will enable further research intoquantum optimal imaging and helps to answer the twoopen questions: 1) What are the optimal measurementsfor measuring spatial features of radiating sources? 2)How well do standard imaging techniques compare tooptimal schemes? Acknowledgements
The authors would like to thank Mark Howard, MadalinGuta, and Mankei Tsang for helpful discussions relat-ing to the issues addressed in this paper. MEP andPK acknowledge EPSRC for funding via the QuantumCommunications Hub. ETC is supported by the EP-SRC (EP/M024261/1).
A Symmetric Logarithmic Derivatives of h n i , | γ | , and φ Here we give the explicit forms of the variables P j , Q j ,and R j , defined in Eq. (30) , which determine the SLDsfor the three parameters θ = ( h n i , | γ | , φ ) T . They are: P h n i = h n i + 1 h n i [ h n i | γ | − ( h n i − ] (35) Q h n i = | γ | e − iφ h n i [ h n i | γ | − ( h n i − ] (36) R h n i = 2 h n i ( | γ | h n i − h n i − h n i [ h n i | γ | − ( h n i − ] (37) P | γ | = 2 h n i + 1( | γ | −
1) [ h n i | γ | − ( h n i − ] (38) Q | γ | = e − iφ (1 + h n i + | γ | h n i )( | γ | −
1) [ h n i | γ | − ( h n i − ] (39) R | γ | = 2 | γ | (cid:2) h n i ( | γ | − (cid:3) ( A −
1) [ h n i | γ | − ( h n i − ] (40) P φ = 0 (41) Q φ = i | γ | e − iφ (42) R φ = 0 . (43) Taking the commutator of each of the SLDs pairwise,we find [ L h n i , L | γ | ] = 0 , and [ L h n i , L φ ] , [ L | γ | , L φ ] ∝ ˆ n − ˆ n . As we show in the main paper, the observablesthat provide optimal information about each parameterindependently are the set X , given by X i = h I − Q L i i . (44) Accepted in
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Quantum2017-07-12, click title to verify rom Eqs. (35) to (43) we can determine the commu-tators of the X operators. These are [ X h n i , X | γ | ] =[ X h n i , X φ ] = 0 , and [ X | γ | , X φ ] ∝ ˆ n − ˆ n . Therefore wecannot measure X | γ | and X φ simultaneously. B Optimality of Weighted MeasurementScheme
Here we present evidence of the optimality of theweighted measurement scheme presented in Sec. 5. Weappeal to the results of [9] where the authors show
Tr[ I − Q I C ] ≤ D − , (45) where D is the dimension of the Hilbert space. For a d parameter problem, it immediately follows that Tr[ I Q I − C ] ≥ d D − . (46) In our setting, the dimension is infinite. However, sincethe state consists of thermal modes, which typicallyhave very small average photon numbers [22], we cantruncate the state to the 0, 1 photon basis with lowtruncation error. Using this D = 3 approximation (thebasis of | , i , | , i , | , i ) suggests that 4.5 is a goodlower bound on Tr[ I Q I − C ] . However, there is no reasonto believe it is attainable. In the main text, we reportedthat we can achieve just below 5. Our approach is tochoose between the optimal measurements with someprobability. Below we show that for qubits, such anapproach is optimal and saturates the lower bound ofEq. (46) . However, for qutrits and low photon num-ber Gaussian states, our numerical investigation indi-cate that it is not possible to saturate Eq. (46) , thoughwe come very close.The quantity Tr[ I Q I − C ] is invariant under reparame-terisation and we can always work in parameterisationwhere I Q is diagonal [9]. If we make the optimal mea-surement for parameter i , we obtain a classical Fisherinformation I ( i ) C such that [ I ( i ) C ] i,i = [ I Q ] i,i . We may ob-tain information about other parameters and so I ( i ) C ≥ ˜ I ( i ) C , (47) where ˜ I ( i ) C = diag(0 , . . . , [ I Q ] ii , . . . , . For a d param-eter problem, we propose a scheme were each optimalobservable X i is measured on a fraction /d of the sam-ples. Then we obtain I C = X j d I ( j ) C ≥ X d ˜ I ( i ) C = 1 d I Q . (48) Therefore,
Tr[ I Q I − C ] ≤ d Tr[ ] = d . (49) For a qubit, Eq. (46) gives a lower bound of d and sowe see such a scheme is optimal. For a general qutritproblem, we have d ≤ Tr[ I Q I − C ] ≤ d (50) with the actual optimal depending on the exact natureof the problem. For a three parameter problem, we havea lower bound of 4.5, and the above scheme achieves 9or better. In the problem of interest to us, some of ouroptimal measurements commute (see App. A) and sowe expect to do much better than 9.Inspired by the above, to find the exact optimum forour problem we allow for minimisation with respect tothe relative probability that we measure X A and X φ .Denoting the probability that we measure X A as p andusing the approximation δ = δ = 0 , we find min p Tr[ I Q ( p I (2) C + (1 − p ) I (3) C ) − ] =min p p + 11 − p = 5 , (51) where the value of p that minimises this expression is p = 1 / . When δ , δ = 0 , the value of p that minimisesis slightly less than one half and the cost is just lessthan 5.To further support our claim that this scheme is in-deed optimal, we perform a numerical optimisation overa subset of POVMs. In order to reduce the numberof parameters to optimise, we search over the trun-cated Hilbert space of | , i , | , i , | , i . We searchover the set of POVMs with six elements correspond-ing to two sets of three orthogonal components. Toperform the optimisation, we apply two unique threedimensional unitaries to the POVM with elements {| , ih , | , | , ih , | , | , ih , |} , giving two POVMswith three elements each, M and M . A six elementPOVM is constructed by taking M = pM + (1 − p ) M ,for which we calculate the classical Fisher informa-tion. We then search for the minimum with respectto the cost function Tr[ I Q I − C ] , where I C is the clas-sical Fisher information associated with measurementsof this POVM. Performing the optimisation across therange ≤ | γ | ≤ , the results are in good agreementwith the values obtained for the weighted measurementscheme. C Photon Count Probability Distribu-tions
To determine the photon count probability distributionswe first notice that the state with covariance matrix
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Quantum2017-07-12, click title to verify q. (28) can be obtained by the action of the unitary U ( φ ) = 1 √ (cid:18) − e − iφ e − iφ (cid:19) , (52) acting on the mode space of the two mode, thermalstate with covariance matrix (cid:18) x + x + (cid:19) ⊕ (cid:18) x + x + (cid:19) , (53) where x = h n i (1 − | γ | ) , x = h n i (1 + | γ | ) . Since themodes are independent, the photon count probabilitydistribution is simply given by p in ( n , n ) = x n (1 + x ) n +1 x n (1 + x ) n +1 , (54) and the state can be expressed in the number basis as ρ in = ∞ X n ,n =0 p in ( n , n ) | n , n ih n , n | . (55) Making use of the relation | n i i = n − i (ˆ a † i ) n i | i , andtaking the transformation [ˆ a in ] i = U † ij ( φ )[ˆ a out ] j , we findthe photon counting probability distribution is given by p out ( m , m ) = m + m X n =0 p in ( n , m + m − n ) n !( m + m − n )! m ! m !2 m + m × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X j =0 ( − j (cid:18) n j (cid:19)(cid:18) m + m − n m − j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (56) where we have made use of the binomial theorem. Wenotice that this distribution is independent of φ andtherefore cannot be used to obtain estimates of φ .To obtain the distribution after the application of theFourier transform, we note that the discrete 2D-Fouriertransform is given by U (0) , we therefore use the trans-formation [ˆ a in ] i = [( U (0) U ( φ )) † ] ij [ˆ a out ] j to obtain thedistribution p out ( m , m ) = m + m X n =0 p in ( n , m + m − n ) n !( m + m − n )! m ! m !4 m + m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X j =0 ( − j (cid:18) n j (cid:19)(cid:18) m + m − n m − j (cid:19) (1 − e − iφ ) m + n − j (1 + e − iφ ) m − n +2 j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (57) The presence of φ in Eq. (57) means that, after the ap-plication of the Fourier transform, photon counting canbe used to determine φ . The classical Fisher informa-tion for these distributions is simply given by I C = ∞ X m ,m =0 ( ∇ θ p out ( m , m )) ( ∇ θ p out ( m , m )) T p out ( m , m ) , (58) where ∇ θ = ( ∂∂θ , ∂∂θ , . . . , ∂∂θ l ) T . References [1] A. Alonso, S. Arribas, and C. Mart´ınez-Roger.
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