Quantum limits of localisation microscopy

Localisation microscopy of multiple weak, incoherent point sources with possibly different intensities in one spatial dimension is equivalent to estimating the amplitudes of a classical mixture of coherent states of a simple harmonic oscillator. This enables us to bound the multi-parameter covariance matrix for an unbiased estimator for the locations in terms of the quantum Fisher information matrix, which we obtained analytically. In the regime of arbitrarily small separations we find it to be no more than rank two—implying that no more than two independent parameters can be estimated irrespective of the number of point sources. We use the eigenvalues of the classical and quantum Fisher information matrices to compare the performance of spatial-mode demultiplexing and direct imaging in localisation microscopy with respect to the quantum limits.


Introduction
Precisely locating multiple single emitters is a key challenge in fluorescence microscopy. The process of estimating these locations depends on the quality of the image obtained by the microscope. One of the major limitations to the image quality, known since Abbe and Rayleigh, lies in spatially resolving objects substantially smaller than half the wavelength of the light involved [1]. Known as the Rayleigh limit or diffraction limit, it is a consequence of the diffraction of light due to its wave nature.
Over the last couple of decades, ways to circumvent the Rayleigh limit in far-field fluorescence microscopy have been invented [2]. Confocal methods such as STED, RESOLFT, and SSIM [3][4][5][6] use patterned illumination to spatially modulate the fluorescence pattern of emitters within a diffraction-limited region such that not all of them emit simultaneously, thereby achieving sub-Rayleigh resolution. Other far-field methods such as PALM, fPALM and STORM [7][8][9] temporally modulate the fluorescence pattern of emitters with weak laser pulses stochastically such that only a low density of emitters are active within the Rayleigh limit at one time. Repeating the process many times, images with sub-Rayleigh resolution are reconstructed from the measured positions of individual emitters. These techniques, with resolution of tens of nanometers, have provided insights into biological processes at the cellular scale that were hitherto unattainable [10].
Though immensely powerful and impressive, none of these methods seek to extract all the information available in the emitted light field. As in conventional fluorescence microscopy these techniques use 'direct imaging'-intensity measurements on the image plane-to extract information from the incident light. That there is indeed more information in the light field to be extracted was shown by Tsang et al [11]. Using methods from classical and quantum estimation theory, it was shown theoretically that two arbitrarily close incoherent point sources may be resolved, and that this may be achieved in practice using a spatial-mode demultiplexing (SPADE) measurement. In the few years since, theoretical studies have considered different source arrangements or parameters of interest [12][13][14][15][16][17] in one as well as in two and three spatial dimensions [18][19][20][21].
Realistic imaging scenarios typically involve more than two point sources or even extended objects. It has been shown that an extended one-dimensional object much smaller than the Rayleigh limit described only in terms of its centroid and effective radius can be approximated by a two-level quantum system [15]. Theoretical optimality of certain measurement techniques in estimating this effective radius size has also been established in one and two spatial dimensions [17,35,36]. Order-of-magnitude bounds on the precision of estimating the normalised moments of extended sources smaller the Rayleigh limit have also been obtained [36,37].
In this paper, we provide an analytical lower bound on an unbiased estimatorʼs covariance (mean square error) matrix for localisation microscopy-simultaneously estimating the locations of N incoherent, weak point sources of unequal but known intensities in one spatial dimension. The bound is provided by the the quantum Fisher information matrix (QFIM). For a Gaussian point spread function (PSF), we first describe the light field on the image plane as a classical mixture of coherent states. We use this to derive the QFIM analytically. In the limit of the point sources approaching a single point, we find its rank to be no more than two. As the inverse of the QFIM lower bounds the covariance matrix, our result implies that no more than two independent parameters can be estimated in localisation microscopy in the limit of arbitrarily small separations. In this limit, we provide a mathematical explanation for our observation in terms of an approximation of the light field involving only the first two Hermite-Gauss modes. Finally, we compare performance of conventional direct imaging and the recently proposed SPADE [11] in localisation microscopy with the quantum bounds we obtain. In the limit of the point sources approaching a single point, we find the classical Fisher information matrices (CFIMs) for both these detection systems to be rank one. Furthermore, in the sub-Rayleigh limit, SPADE does not attain the quantum limit for localisation microscopy. For the subset of parameters where scalings may be optimal, we find SPADE to be short of the quantum limit in absolute precision. This paper is organised as follows: in section 2 we provide a quantum mechanical description of localisation microscopy. The appropriate framework to study the quantum limits of the localisation problem is quantum estimation theory, the toolbox of which is described in section 3, leading to the definition of the QFIM. In section 4 we provide an analytic expression of the QFIM for localisation microscopy, our main technical result. We then draw conclusions about its rank and its implications for localisation microscopy. We end in section 5 with further insights and discussions about the sinc PSF and the potential of detection systems attaining the quantum limits of localisation microscopy.

Quantum description of localisation microscopy
We consider localisation microscopy-the problem of estimating the locations of N incoherent point sources or emitters located in a one-dimensional spatial configuration as in figure 1. As we assume them to be weak, such that on average no photons arrive on the image place within a coherence time with probability (1−ò), where ò = 1 and one photon arrives with probability ò. We also assume the optical field on the image plane to be quasi-monochromatic and paraxial [11]. The quantum state of this optical field is then where we have neglected terms of second and higher orders in ò and r = ñá vac vac vac | |is the vacuum state and ρ is the one-photon state. The one-photon density matrix on the object plane is an incoherent mixture of position eigenstates An imaging system maps c , x † the creation operator producing one photon in the position x on the object plane, to the corresponding image plane operator c î † [13] ò where χ i is the position on the source on the object plane and y x PSF ( ) is the PSF. On the image plane this becomes as follows from equation (2).
( ) is free of any Rayleigh limit as it transmits all spatial frequencies from the object to the image plane. In practice, a Gaussian PSF where NA is the numerical aperture of the imaging system is a good approximation for quasimonochromatic paraxial light [11,38] and also allows us to obtain analytical results. For such a PSF, the state of equation ( The above is a quantum optical rendition of localisation microscopy-a classical optics problem. It enables us to harness the mathematical formalism associated with coherent states and provides a basis that spans the space of the quantum state as well as its derivative. The latter is an essential ingredient of deriving the QFIM analytically in section 4.1. We also hope that this description will provide insights into the quantum limits to localisation microscopy in the presence of shot noise and assist in designing detection systems that attain these quantum limits.

Quantum estimation theory
Localisation has long been treated as an estimation problem with the unknown locations of the sources being the parameters to be estimated [40,41]. In our formulation, the limits to the localisation of the point sources are the same as estimating the amplitudes a of the coherent states in equation (7). Let these estimates be a a º ĩ {˜}. The precision of our estimate is then given by the covariance (or mean square error) matrix defined as å a a a a a a = - where a p z ( | ) is the probability distribution of the collected data labelled by, for instance, the pixel z on the image plane. a Cov[ ] is a positive symmetric matrix whose ith diagonal element denotes the variance of an estimator of α i given the data collected. The (i, j)th off-diagonal element denote the covariance in the estimation of α i and α j .
Given the data collected, the maximum amount of information that can be extracted from it to obtain the most precise estimate of the locations is given by the Cramér-Rao bound [42]. For an unbiased estimator, this bound is given by where M is the number of coherence times over which the data is collected, making Mò the total photon count. This inequality is saturable but generally only in the asymptotic limit of many repetitions [43]. r P a  , z ( )is the CFIM whose elements are given by [44] å a a a r a a ) results from detecting the light on the image plane using a specific detection system Π z . Fluorescence microscopy typically employs intensity detectors P = z ñá = n n n , 0, 1, , z {| | }  at each pixel z, known as direct imaging. It is then evident that the CFIM depends on the detection system used, and not surprising that it determines the amount of information that can be extracted from the light field at the image plane.
To identify the quantum limit on the precision of localisation microscopy, the CFIM must be maximised over all possible physically allowed detection systems. This set is given by positive operator-valued measures (POVMs) [45] and the maximisation is bounded as [46,47].
by the QFIM. Its matrix elements are given by with L μ being the symmetric logarithmic derivative (SLD) corresponding to the parameter α μ . The SLD is determined by the Lyapunov equation The quantum limit to localisation microscopy is thus given by The QFIM depends only on the light field on the image plane and determines the maximum amount of information that can be extracted from it using detection systems allowed by quantum mechanics. Deriving an analytical expression for r a ( )for state in equation (7) is our main result, which we present in the next section. A practical issue following the identification of the quantum limit is its attainability. For cases where a single parameter is unknown then a measurement can be found to satisfy the equality of equation (11), which involves projecting onto the eigenstates of the SLD [44,48]. However this strategy does not generalise to multiple parameters, as in localisation microscopy, in general.
For multi-parameter estimation the attainability is tantamount to saturating the second inequality in equation (14). A necessary condition for the saturability of any scalar form of equation (14) is the satisfaction of weak commutativity [49,50] Moreover, through the quantum theory of asymptotic normality [51], this condition becomes sufficient with the application of collective measurements over multiple copies of r a [49,50]. Any scalar function of the covariances can be bounded by the inverse of QFIM with the lower bound following from the spectral decomposition of QFIM. To that end, calculating the eigenvalues of the QFIM and their scaling is of importance for the multi-parameter estimation. For localisation microscopy, r a as in equation (7) as well as its derivative are real matrices. Thereby, L μ , L ν are also real and the above condition is always satisfied 3 . The quantum limit for localisation microscopy is therefore attainable, at least in principle, although collective measurements over multiple copies [49,50] of the light field on the image plane may be required.
Alternative parameterisations of the system-where the new parameters a¢ are functions of the old parameters a-can be dealt with by a transformation of the QFIM. Given the transformation matrix B with 3 Since the density matrix and its derivatives are real-valued in the orthonormal f ñ k {| }basis, equation (13) is a system of equations with real coefficients. Hence L ν must be real as well, and so r a m n L L is real-valued. We thank Ben Wang for bringing this to our attention. elements a a = ¶ ¶ ¢ B ij i j , the QFIM of the transformed parameters is [44] ¢ provided the transformation is non-singular. This can be used to recast our results in terms of, for instance, the moments of the point source distribution.

Results
We now present our main result-the analytical expression of the QFIM for localisation microscopy. This expression allows us to conclude that the QFIM is a rank two matrix as a  0 i . Equation (14) then implies that the eigenvalues of a Cov[ ] remains finite for no more than two independent parameters. Thus, no more than two independent parameters can be estimated from the entire set a as a  0.
We lack a fully satisfactory physical explanation for this restriction on the number of estimable parameters, but provide an explanation involving only the first two HG modes for α i =1.

Analytical expression of QFIM
The state in equation (7) can be expressed in the basis of a a ñ ñ a , denotes a diagonal matrix. Although the basis used in equation (17) is nonorthogonal this representation can still be used to evaluate the QFIM [52]. The coherent states a ñ i {| }are linearly independent and span the support of the state in equation (7). The support of the derivative is spanned by a ñ , which are also linearly independent. The Grammian matrix ¡ = A A, 1 9 ( ) † whose elements consist of the scalar products between the basis vectors a a á ñ where ∂ j denotes the derivative with respect to α j and (E j ) kl =δ jk δ jl . Similarly, the SLD L A i can be written in the generic form The Lyapunov equation equation (13) can be now rewritten as and the QFIM elements from equation (12) as Using the Tracy-Singh block kronecker product  and the block column 'vecb' operator [53] defined as ( ) is the column vectorisation of a matrix and X ( |its transpose. Equation (24) can be blockwise vectorised to with  being the identity matrix. Using the matrix identity [53] the QFIM elements from equation (25) can be re-expressed as which is the outstanding quantity to be determined.
We now recast equations (27) and (29) as Putting it all together, we obtain Note that the inverse ¡ -1 always exists since ϒ is the Grammian matrix of linearly independent vectors. The elements of ϒ −1 can be found using the formula of blockwise inversion (see appendix B).
Noticing that G dd j | ) does not contribute in equation (29), equation (32) can be reduced to where  is a 3N 2 ×3N 2 invertible matrix unless α i =α j for some i, j, which is a singular case for which the rank of the density matrix reduces. Hence the unique solution to equation (35) is where the block matrices that compose the - 1 can be found by using the formulas for blockwise inversion (see appendix B). Substituting equation (36) into (29) )is an N 2 ×N 2 matrix and ϒ αα is the inverse of the submatrix of ϒ which exists, as it is the Grammian matrix of linearly independent vectors a ñ i {| }. Equation (37) is an analytic expression for the QFIM elements for localisation microscopy and our main result. Figure 2 shows the elements of the QFIM for the localisation microscopy of three point sources. We choose them to be equidistant, that is, (α 1 , α 2 , α 3 )=(x, 2x, 3x) and w 1 =w 2 =w 3 =1/3 for illustration purposes. Note the non-zero off-diagonal elements evidencing correlations in the precision around and below the Rayleigh limit of x∼1.
While the diagonal elements are all non-vanishing, more crucially as  x 0 the diagonal and off-diagonal elements combine to make the QFIM singular. This is revealed by a closer analysis of the QFIM matrix as in figure 3 which shows that only two of its eigenvalues remain non-zero as the sources approach each other. This is in spite of all the diagonals elements of the QFIM remaining non-zero even as  x 0, as figure 2 shows. This behaviour of only two non-zero eigenvalues also holds for other values of N. We have explicitly checked this for N=4, ..., 10 as well as when the sources are not equally spaced. In figure B1 in appendix B we plot the eigenvalues of the QFIM for N=4, 5 as further examples. In the case of different relative intensities the results are the same except of the limiting case of one extremely bright source where the rank of the QFIM is approximately one ( figure B2 in appendix B).
Since the QFIM has rank two as  x 0, its inverse is ill-defined except on a two-dimensional subspace. This implies that the N×N covariance matrix for localisation microscopy, as per equation (14), will also be unbounded except on a two-dimensional subspace. Thus, no more than two independent parameters can be estimated in localisation microscopy as the point sources approach each other.
In other words, the rank-deficient nature of the QFIM shows that a form of the Rayleigh limit resurfaces for any N>2. This had been suggested by previous works based on order-of-magnitude bounds for the diagonal elements on the CFIM [36] or uppers bounds on the diagonal elements of the QFIM [37]. Our analytical expression for the full QFIM-its diagonal and off-diagonal elements for any N-shows that this rank two behaviour is truly quantum mechanical in origin. Furthermore, knowing the full QFIM matrix allows us to uncover the nature in which N−2 of the eigenvalues approach zero. We return to the behaviour in which this rank deficiency or Rayleigh limit emerges in section 5.

Why rank two?
We now provide an explanation for the rank deficiency of the QFIM in the regime of small separations which can be seen as the re-emergence of the Rayleigh limit. To that end, we expresses the state in equation (7) in terms of the real-valued displacement operator a = a a In the limit of very small separations (a 1 i  ), the displacements are approximately a a a = + - where  is the identity operator and the displacement α i is real. Up to the second order in α i , the normalised quantum state of the light field on the image place is then Equation (40) describes the state of two-level quantum system-the two levels being the first two HG modes. A similar approximation which described the state relative to a PSF centred at a fixed reference point was used in reference [15] to estimate the centroid and the effective radius of a distribution of incoherent point sources. We now consider the more general problem of estimating the location of N point sources.
The QFIM for a (see appendix C) is ) . The QFIM Q is an N×N matrix, which is a product of three matrices of dimensions N×2, 2×2 and 2×N. Since  AB A B rank min rank , rank ( ) { ( ) ( )}, and the matrix  has rank 2, the QFIM Q has rank no more than two. Although a two-level quantum system has the potential of estimating three real parameters, localisation microscopy in this limit can estimate only two as the two-level system possesses a real density matrix 4 . This is another way of arguing that as the point sources get closer, the light field on the image plane has Figure 3. The eigenvalues of the QFIM matrix for 3 sources with equal intensities. The sources are separating from each other at equal distances, (α 1 , α 2 , α 3 )=(x, 2x, 3x). 4 As the localisation parameters a are real, enough information to estimate only two parameters. A physical reason for this observation would be highly desirable.

Discussion
Our analytical expression for the QFIM for localisation microscopy has enabled us to show that as point sources get closer, no more than two independent parameters can be estimated. A rank-deficient QFIM occurs when the quantum state does not contain enough information to permit the estimation some of the parameters or combinations thereof. The parameters that can be estimated correspond to the non-zero eigenvalues of the QFIM. Without additional knowledge of the source distribution this restricts us to estimating functions of the first two moments   f , 1 2 ( ) only deep in the sub-Rayleigh limit. As equation (42) shows, when all {α j } are unknown as in localisation microscopy, there is vanishing information about any single α i itself. This is in contrast to the scalar QFI r a  ii [ ( )] for α i which is non zero, but assumes that all the other {α j } are known. The manner in which the eigenvalues of the QFIM tend to zero is of interest in the search for optimal detection systems for localisation microscopy. Numerical fitting in figure 4 shows the vanishing eigenvalues of the QFIM approach zero polynomially. The degree of the polynomial is given by = m - where μ is the order the eigenvalue when arranged in descending order and⌊·⌋is the floor function. These scalings are now extracted from the elements of the full QFIM of the localisation parameters a-rather than from bounds on estimating the various moments independently as in previous works [36,37].
Unlike the latter, we can now compare the absolute performance of detection systems for localisation microscopy relative to its quantum limit. Indeed, while figure 5 shows the 2nth eigenvalue of the QFIM closely parallel to the nth eigenvalue of the CFIM for SPADE [11], there is a large gap in the absolute terms. This could be due to the sub-optimality of SPADE for estimating the N 2 ⌊ ⌋parameters it is sensitive to 5 . Similar scalings were observed with detection using superpositions of the conventional SPADE basis [35][36][37] that are sensitive to the other half of the moments. For reference over a range of separations, figure E1 in appendix E shows the eigenvalues of the CFIM for SPADE as well as direct imaging. Note that for both, the CFIM tends towards a rank one matrix.
Finally, although our analytical result is derived with a Gaussian PSF, we expect the rank deficiency of the QFIM to be present in a more general family of PSFs. To that end, figure 6 shows the numerically obtained eigenvalues of the QFIM for three equidistant point sources of equal intensities under a sinc PSF (see appendix D) defined as This PSF is the exact form for diffraction through a sharp one-dimensional slit which in its principal peak is wellapproximated as Gaussian. An approximation involving the first two spherical Bessel modes as in section 4.2 can be performed for a sinc PSF as well, leading to similar insights. A proof of this rank deficiency for arbitrary PSFs and a physical explanation remains an open question.
To conclude, we have obtained several insights into the quantum limits of localisation microscopy via an analytical expression for the QFIM. In particular, the behaviour of the eigenvalues of the QFIM deep in the sub-Rayleigh limit revealed that only two parameters are eventually estimable. It also enabled us to compare the performance of known detection systems relative to the quantum limit in absolute terms, a question left open in the literature [34]. The gap identified by us should motivate the search for detection systems, ideally on a single copy of the light field on the image plane, seeking to reduce or eliminate it.  The density matrix is written in terms of the kets y ñ i | , which are expressed in the position space as in equation (3). We assume a normalised Gaussian PSF of the form The kets y ñ i | can be expressed in terms of the complete HG modes as where f ñ q | are the HG modes, which can be expressed in the position space as [11] ò f ps which has the same mathematical form as the coherent states with f ñ q {| }forming the Fock basis [39]. The state in equation (3) can be also written in terms of the displacement operators a = a - e i a a i ( ) where a ( ) is the displacement operator. The derivative of each coherent state with respect to its real amplitude α is given by

Appendix B. Analytic results for N sources
The Tracy-Singh product [53,54] defined for matrices A and B subdivided into blocks A ij and B kl is  11  12   21  22   11  11  11  12  12  11  12  12   11  21  11  22  12  21  12  22   21  11  21  12  22  11  22  12   21  21  21  22  22  21  22 22 Using the above definition, the matrix of equation (31) is found to be r r where the elements of ϒ −1 can be found using the formula of blockwise inversion: The elements of - 1 will be given by the formulas Finally, to complement the discussion in the main text,we present some further examples of the QFIM eigenvalues for N=4, 5 sources and in figure B2 we present the eigenvalues of the QFIM for 3 sources in the case of unequal weights (relative intensities)figure B2.  Since α i are strictly positive, except one that can be zero, this sum of positive terms cannot be equal to zero. Therefore, this statement is always false. Thus, the equation (C.5) becomes    = 1 0 1 0 0 ( ) ( ) , which means that the two eigenvalues can never be zero and the QFIM will be rank 2. In this way the fact that the specific basis is not complete does not affect our calculations.
For the numerical calculations we have to truncate our state in the appropriate amount of modes. From figures 6 and D1, we can see that our conclusions do not change with the use of a non-Gaussian PSF.

Appendix E. Eigenvalues of the CFIM for SPADE and direct imaging
Finally, we present the eigenvalues of the CFIM for SPADE and direct imaging fir a large range of separations.