The basic idea behind Rayleigh's criterion on resolving two incoherent optical point sources is that the overlap between the spatial modes from different sources would reduce the estimation precision for the locations of the sources, dubbed Rayleigh's curse. We generalize the concept of Rayleigh's curse to the abstract problems of quantum parameter estimation with incoherent sources. To manifest the effect of Rayleigh's curse on quantum parameter estimation, we define the curse matrix in terms of quantum Fisher information and introduce the global and local immunity to the curse accordingly. We further derive the expression for the curse matrix and give the necessary and sufficient condition on the immunity to Rayleigh's curse. For estimating the one-dimensional location parameters with a common initial state, we demonstrate that the global immunity to the curse on quantum Fisher information is impossible for more than two sources.
Conventional incoherent imaging based on measuring the spatial intensity distribution in the image plane faces the resolution hurdle described by the Rayleigh diffraction criterion. Here, we demonstrate theoretically using the concept of the Fisher information that quadrature statistics measured by means of array homodyne detection enables estimation of the distance between two incoherent point sources well below the Rayleigh limit for sufficiently high signal-to-noise ratio. This capability is attributed to the availability of spatial coherence information between individual detector pixels a…
Rayleigh's criterion states that it becomes essentially difficult to resolve two incoherent optical point sources separated by a distance below the width of point spread functions (PSF), namely in the subdiffraction limit. Recently, researchers have achieved superresolution for two incoherent point sources with equal strengths using a new type of measurement technique, surpassing Rayleigh's criterion. However, situations where more than two point sources needed to be resolved have not been fully investigated. Here we prove that for any incoherent sources with arbitrary strengths, a one- or two…
Coherence distillation is a basic information-theoretic task in the resource theory of coherence. In this paper, we present the necessary and sufficient conditions under which a mixed state can be distilled into a pure coherent state via stochastic incoherent operations (sIOs). With the help of this result, we further show the following: (i) Any $2$-dimensional coherent state is distillable via sIOs if and only if it is a pure coherent state; (ii) a state $ρ$ is n-distillable via sIOs if and only if it is 1-distillable; and (iii) the set of distillable states via stochastic maximally incoheren…
Rayleigh's criterion for resolving two incoherent point sources has been the most influential measure of optical imaging resolution for over a century. In the context of statistical image processing, violation of the criterion is especially detrimental to the estimation of the separation between the sources, and modern farfield superresolution techniques rely on suppressing the emission of close sources to enhance the localization precision. Using quantum optics, quantum metrology, and statistical analysis, here we show that, even if two close incoherent sources emit simultaneously, measuremen…
We analyze the fundamental resolution of incoherent optical point sources from the perspective of a quantum detection problem: deciding whether the optical field on the image plane is generated by one source or two weaker sources with arbitrary separation. We investigate the detection performances of two measurement methods recently proposed by us to enhance the estimation of the separation. For the detection problem, we show that the method of binary spatial-mode demultiplexing is quantum-optimal for all values of separations, while the method of image-inversion interferometry is near-optimal…
It has been argued that, for a spatially invariant imaging system, the information one can gain about the separation of two incoherent point sources decays quadratically to zero with decreasing separation, an effect termed Rayleigh's curse. Contrary to this belief, we identify a class of point spread functions with a linear information decrease. Moreover, we show that any well-behaved symmetric point spread function can be converted into such a form with a simple nonabsorbing signum filter. We experimentally demonstrate significant superresolution capabilities based on this idea.
Considerable work has recently been directed toward developing resource theories of quantum coherence. In most approaches, a state is said to possess quantum coherence if it is not diagonal in some specified basis. In this letter we establish a criterion of physical consistency for any resource theory in terms of physical implementation of the free operations, and we show that all currently proposed basis-dependent theories of coherence fail to satisfy this criterion. We further characterize the physically consistent resource theory of coherence and find its operational power to be quite limit…
We obtain the multiple-parameter quantum Cramér-Rao bound for estimating the transverse Cartesian components of the centroid and separation of two incoherent optical point sources using an imaging system with finite spatial bandwidth. Under quite general and realistic assumptions on the point-spread function of the imaging system, and for weak source strengths, we show that the Cramér-Rao bounds for the $x$ and $y$ components of the separation are independent of the values of those components, which may be well below the conventional Rayleigh resolution limit. We also propose two linear optics…
We devise a systematic method to determine the Fisher information required for resolving two incoherent point sources with a diffraction-limited linear imaging device. The resulting Cramér-Rao bound gives the lowest variance achievable for an unbiased estimator. When only intensity in the image plane is recorded, this bound diverges as the separation between the sources tends to zero, an effect which has been dubbed as Rayleigh's curse. However, this curse can be lifted using suitable coherent measurements. In particular, we determine a class of measurements that can be easily implemented as p…
No imaging apparatus can produce perfect images: spatial resolution is limited by the Rayleigh diffraction bound that is a consequence of the imager's finite spatial extent. We show some N-photon strategies that permit resolution of details that are smaller than this bound, attaining either a 1/sqrt(N) enhancement (standard quantum limit) or a 1/N enhancement (Heisenberg limit) over standard techniques. In the incoherent imaging regime, the methods presented are loss resistant, because they can be implemented with classical-state light sources. Our results may be of importance in many applicat…
Using a semiclassical model of photodetection with Poissonian noise and insights from quantum metrology, we prove that linear optics and photon counting can optimally estimate the separation between two incoherent point sources without regard to Rayleigh's criterion. The model is applicable to weak thermal or fluorescent sources as well as lasers.
The manifestation of the wave nature of light through diffraction imposes limits on the resolution of optical imaging. For over a century, the Abbe-Rayleigh criterion has been utilized to assess the spatial resolution limits of optical instruments. Recently, there has been an enormous impetus in overcoming the Abbe-Rayleigh resolution limit by projecting target light beams onto spatial modes. These conventional schemes for superresolution rely on a series of spatial projective measurements to pick up phase information that is used to boost the spatial resolution of optical systems. Unfortunate…
We establish the multiparameter quantum Cramér-Rao bound for simultaneously estimating the centroid, the separation, and the relative intensities of two incoherent optical point sources using alinear imaging system. For equally bright sources, the Cramér-Rao bound is independent of the source separation, which confirms that the Rayleigh resolution limit is just an artifact of the conventional direct imaging and can be overcome with an adequate strategy. For the general case of unequally bright sources, the amount of information one can gain about the separation falls to zero, but we show that …
We establish a simple method to assess the quantum Fisher information required for resolving two incoherent point sources with an imaging system. The resulting Cramér-Rao bound shows that the standard Rayleigh limit can be surpassed by suitable coherent measurements. We explicitly find these optimal strategies and present a realization for Gaussian and slit apertures. This involves a projection onto the optimal bases that is accomplished with digital holographic techniques and is compared with a CCD position measurement. Our experimental results unequivocally confirm unprecedented sub-Rayleigh…
Can one certify the preparation of a coherent, many-body quantum state by measurements with bounded accuracy in the presence of noise and decoherence? Here, we introduce a criterion to assess the fragility of large-scale quantum states which is based on the distinguishability of orthogonal states after the action of very small amounts of noise. States which do not pass this criterion are called asymptotically incertifiable. We show that, if a coherent quantum state is asymptotically incertifiable, there exists an incoherent mixture (with entropy at least log 2) which is experimentally indistin…
A novel interferometric method - SLIVER (Super Localization by Image inVERsion interferometry) - is proposed for estimating the separation of two incoherent point sources with a mean squared error that does not deteriorate as the sources are brought closer. The essential component of the interferometer is an image inversion device that inverts the field in the transverse plane about the optical axis, assumed to pass through the centroid of the sources. The performance of the device is analyzed using the Cramér-Rao bound applied to the statistics of spatially-unresolved photon counting using ph…
I propose classical and quantum limits to the statistical resolution of two incoherent optical point sources from the perspective of minimax parameter estimation. Unlike earlier results based on the Cramér-Rao bound, the limits proposed here, based on the worst-case error criterion and a Bayesian version of the Cramér-Rao bound, are valid for any biased or unbiased estimator and obey photon-number scalings that are consistent with the behaviors of actual estimators. These results prove that, from the minimax perspective, the spatial-mode demultiplexing (SPADE) measurement scheme recently propo…
We compute analytically the maximal rates of distillation of quantum coherence under strictly incoherent operations (SIO) and physically incoherent operations (PIO), showing that they coincide for all states, and providing a complete description of the phenomenon of bound coherence. In particular, we establish a simple, analytically computable necessary and sufficient criterion for the asymptotic distillability under SIO and PIO. We use this result to show that almost every quantum state is undistillable --- only pure states as well as states whose density matrix contains a rank-one submatrix …
A pair of quantum observables diagonal in the same "incoherent" basis can be measured jointly, so some coherence is obviously required for measurement incompatibility. Here we first observe that coherence in a single observable is linked to the diagonal elements of any observable jointly measurable with it, leading to a general criterion for the coherence needed for incompatibility. Specialising to the case where the second observable is incoherent (diagonal), we develop a concrete method for solving incompatibility problems, tractable even in large systems by analytical bounds, without resort…
We analyze the fundamental quantum limit of the resolution of an optical imaging system from the perspective of the detection problem of deciding whether the optical field in the image plane is generated by one incoherent on-axis source with brightness $ε$ or by two $ε/2$-brightness incoherent sources that are symmetrically disposed about the optical axis. Using the exact thermal-state model of the field, we derive the quantum Chernoff bound for the detection problem, which specifies the optimum rate of decay of the error probability with increasing number of collected photons that is allowed …
Any imaging device such as a microscope or telescope has a resolution limit, a minimum separation it can resolve between two objects or sources; this limit is typically defined by "Rayleigh's criterion", although in recent years there have been a number of high-profile techniques demonstrating that Rayleigh's limit can be surpassed under particular sets of conditions. Quantum information and quantum metrology have given us new ways to approach measurement ; a new proposal inspired by these ideas has now re-examined the problem of trying to estimate the separation between two poorly resolved po…
Quantum coherence and quantum entanglement represent two fundamental features of non-classical systems that can each be characterized within an operational resource theory. In this paper, we unify the resource theories of entanglement and coherence by studying their combined behavior in the operational setting of local incoherent operations and classical communication (LIOCC). Specifically we analyze the coherence and entanglement trade-offs in the tasks of state formation and resource distillation. For pure states we identify the minimum coherence-entanglement resources needed to generate a g…
Recent work considered the ultimate (quantum) limit of the precision of estimating the distance between two point objects. It was shown that the performance gap between the quantum limit and that of ideal continuum image-plane direct detection is the largest for highly sub-Rayleigh separation of the objects, and that a pre-detection mode sorting could attain the quantum limit. Here we extend this to a more general problem of estimating the length of an incoherently radiating extended (line) object. We find, as expected by the Rayleigh criterion, the Fisher information (FI) per integrated photo…
Coherence, the superposition of orthogonal quantum states, is indispensable in various quantum processes. Inspired by the polynomial invariant for classifying and quantifying entanglement, we first define polynomial coherence measure and systematically investigate its properties. Except for the qubit case, we show that there is no polynomial coherence measure satisfying the criterion that its value takes zero if and only if for incoherent states. Then, we release this strict criterion and obtain a necessary condition for polynomial coherence measure. Furthermore, we give a typical example of p…
We investigate the localization of two incoherent point sources with arbitrary angular and axial separations in the paraxial approximation. By using quantum metrology techniques, we show that a simultaneous estimation of the two separations is achievable by a single quantum measurement, with a precision saturating the ultimate limit stemming from the quantum Cramér-Rao bound. Such a precision is not degraded in the sub-wavelength regime, thus overcoming the traditional limitations of classical direct imaging derived from Rayleigh's criterion. Our results are qualitatively independent of the po…
Imaging systems measuring intensity in the far field succumb to Rayleigh's curse, a resolution limitation dictated by the finite aperture of the optical system. Many proof-of-principle and some two-dimensional imaging experiments have shown that, by using spatial mode demultiplexing (SPADE), the field information collected is maximal, and thus, the resolution increases beyond the Rayleigh criterion. Hitherto, the SPADE approaches are based on resolving the lateral splitting of a Gaussian wavefunction. Here, we consider the case in which the light field originates from a bi-photon source, i.e. …
In this work we investigate the quantum information theoretical limits to several tasks related to lifetime estimation and discrimination of a two-level spontaneous optical emitter. We focus in particular on the model problem of resolving two mutually incoherent exponential decays with highly overlapping temporal probability profiles. Mirroring recent work on quantum-inspired super-resolution of point emitters, we find that direct lifetime measurement suffers from an analogue of "Rayleigh's Curse" when the time constants of the two decay channels approach one another. We propose alternative me…
The wave-particle duality of light introduces two fundamental problems to imaging, namely, the diffraction limit and the photon shot noise. Quantum information theory can tackle them both in one holistic formalism: model the light as a quantum object, consider any quantum measurement, and pick the one that gives the best statistics. While Helstrom pioneered the theory half a century ago and first applied it to incoherent imaging, it was not until recently that the approach offered a genuine surprise on the age-old topic by predicting a new class of superior imaging methods. For the resolution …
Estimating the angular separation between two incoherently radiating monochromatic point sources is a canonical toy problem to quantify spatial resolution in imaging. In recent work, Tsang {\em et al.} showed, using a Fisher Information analysis, that Rayleigh's resolution limit is just an artifact of the conventional wisdom of intensity measurement in the image plane. They showed that the optimal sensitivity of estimating the angle is only a function of the total photons collected during the camera's integration time but entirely independent of the angular separation itself no matter how smal…
