This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
In recent proposals for achieving optical super-resolution, variants of the Quantum Fisher Inform... more In recent proposals for achieving optical super-resolution, variants of the Quantum Fisher Information (QFI) quantify the attainable precision. We find that claims about a strong enhancement of the resolution resulting from coherence effects are questionable because they refer to very small subsets of the data without proper normalization. When the QFI is normalized, accounting for the strength of the signal, there is no advantage of coherent sources over incoherent ones. Our findings have a bearing on further studies of the achievable precision of optical instruments.
Ideal photon-number-resolving detectors form a class of important optical components in quantum o... more Ideal photon-number-resolving detectors form a class of important optical components in quantum optics and quantum information theory. In this article, we theoretically investigate the potential of multiport devices having reconstruction performances approaching that of the Fock-state measurement. By recognizing that all multiport devices are minimally complete, we first provide a general analytical framework to describe the tomographic accuracy (or quality) of these devices. Next, we show that a perfect multiport device with an infinite number of output ports functions as either the Fock-state measurement when photon losses are absent or binomial mixtures of Fock-state measurements when photon losses are present and derive their respective expressions for the tomographic transfer function. This function is the scaled asymptotic mean squared error of the reconstructed photon-number distributions uniformly averaged over all distributions in the probability simplex. We then supply mor...
By projecting onto complex optical mode profiles, it is possible to estimate arbitrarily small se... more By projecting onto complex optical mode profiles, it is possible to estimate arbitrarily small separations between objects with quantum-limited precision, free of uncertainty arising from overlapping intensity profiles. Here we extend these techniques to the time-frequency domain using mode-selective sum-frequency generation with shaped ultrafast pulses. We experimentally resolve temporal and spectral separations between incoherent mixtures of single-photon level signals ten times smaller than their optical bandwidths with a tenfold improvement in precision over the intensity-only Cramér-Rao bound.
International Journal of Quantum Information, 2017
Wigner and Husimi quasi-distributions, owing to their functional regularity, give the two archety... more Wigner and Husimi quasi-distributions, owing to their functional regularity, give the two archetypal and equivalent representations of all observable-parameters in continuous-variable quantum information. Balanced homodyning (HOM) and heterodyning (HET) that correspond to their associated sampling procedures, on the other hand, fare very differently concerning their state or parameter reconstruction accuracies. We present a general theory of a now-known fact that HET can be tomographically more powerful than balanced homodyning to many interesting classes of single-mode quantum states, and discuss the treatment for two-mode sources.
In continuous-variable tomography, with finite data and limited computation resources, reconstruc... more In continuous-variable tomography, with finite data and limited computation resources, reconstruction of a quantum state of light is performed on a finite-dimensional subspace. In principle, the data themselves encode all information about the relevant subspace that physically contains the state. We provide a straightforward and numerically feasible procedure to uniquely determine the appropriate reconstruction subspace by extracting this information directly from the data for any given unknown quantum state of light and measurement scheme. This procedure makes use of the celebrated statistical principle of maximum likelihood, along with other validation tools, to grow an appropriate seed subspace into the optimal reconstruction subspace, much like the nucleation of a seed into a crystal. Apart from using the available measurement data, no other assumptions about the source or preconceived parametric model subspaces are invoked. This ensures that no spurious reconstruction artifacts...
We present an experimental realization of spatial-mode demultiplexing (SPADE) method to obtain th... more We present an experimental realization of spatial-mode demultiplexing (SPADE) method to obtain the superresolution of two incoherent point sources. The digital holography projection onto the Hermite-Gaussian basis is compared with the CCD position measurement.
We present a simple and efficient Bayesian recursive algorithm for the data-pattern scheme for qu... more We present a simple and efficient Bayesian recursive algorithm for the data-pattern scheme for quantum state reconstruction, which is applicable to situations where measurement settings can be controllably varied efficiently. The algorithm predicts the best measurements required to accurately reconstruct the unknown signal state in terms of a fixed set of probe states. In each iterative step, this algorithm seeks the measurement setting that minimizes the variance of the data-pattern estimator, which essentially measures the reconstruction accuracy, with the help of a data-pattern bank that was acquired prior to the signal reconstruction. We show that with this algorithm, it is possible to minimize the number of measurement settings required to obtain a reasonably accurate state estimator by using just the optimal settings and, at the same time, increasing the numerical efficiency of the data-pattern reconstruction.
We introduce an operational and statistically meaningful measure, the quantum tomographic transfe... more We introduce an operational and statistically meaningful measure, the quantum tomographic transfer function, that possesses important physical invariance properties for judging whether a given informationally complete quantum measurement performs better tomographically in quantum-state estimation relative to other informationally complete measurements. This function is independent of the unknown true state of the quantum source, and is directly related to the average optimal tomographic accuracy of an unbiased state estimator for the measurement in the limit of many sampling events. For the experimentally-appealing minimally complete measurements, the transfer function is an extremely simple formula. We also give an explicit expression for this transfer function in terms of an ordered expansion that is readily computable and illustrate its usage with numerical simulations, and its consistency with some known results.
Scheme for optimal spin state estimation is considered in analogy with phase detection in interfe... more Scheme for optimal spin state estimation is considered in analogy with phase detection in interferometry. Recently reported coherent measurements yielding the average fidelity (N +1)/(N +2) for N particle system corresponds to the standard limit of phase resolution 1/ √ N. It provides the bound for incoherent measurements when each particle is detected separately and information is used optimally. For specific states, improvement up to the value 1/N is possible in quantum theory. The best results are obtained combining sequentially coherent measurements on fractional groups of particles.
The purpose of this chapter is to review the basic concepts leading to definition of nonclassical... more The purpose of this chapter is to review the basic concepts leading to definition of nonclassical optical phanomena which have fully quantum origin, have no classical analogues and can be explained only in the framework of quantum theory. Their experimental observations (Chap. 8) represent a complete confirmation of the quantum theory of Nature. The most important nonclassical phenomena are quadrature
In this chapter we review the most important experiments performed in quantum optics supporting q... more In this chapter we review the most important experiments performed in quantum optics supporting quantum theory. We deal with one-photon as well as two-photon experiments, resonance fluorescence experiments, experiments in cavity quantum electrodynamics, experiments with various feedbacks, and experiments with squeezed light and sub-Poisson light generated in various nonlinear optical processes, such as optical parametric processes, particularly frequency down conversion, four-wave mixing, phase conjugation, optical Kerr effect as well as Raman scattering. In particular the process of the interaction of single atoms with light can provide effective tests of quantum behaviour. Some experiments related to violation of classical inequalities, Bell’s inequalities and quantum “nonlocality” were discussed in Chap. 7.
Inspite of its abstract mathematical structure quantum mechanics is above all the pragmatic theor... more Inspite of its abstract mathematical structure quantum mechanics is above all the pragmatic theory which enables to predict the results of the experiments. The weakness of classical physics to explain the behavior of microsystems initialized the revolution in physics at the beginning of this century. The experiments, both “gedanken” or feasible, are playing the primary role in development of quantum
In this chapter we provide the basic properties of the coherent-state technique and its applicati... more In this chapter we provide the basic properties of the coherent-state technique and its application to the interaction of radiation with matter. Greater details about this technique can be found in books by Klauder and Sudarshan (1968), Malkin and Man’ko (1979), Klauder and Skagerstam (1985), Perelomov (1986), Peřina (1991), in reviews by Zhang, Feng and Gilmore (1990), Kim and Noz
Interference is one of the most important phenomena in optical fields representing their wave pro... more Interference is one of the most important phenomena in optical fields representing their wave properties. Such two optical fields can in principle be in constructive or destructive interference according to their mutual phase (value of difference of their simultaneous phases) giving rise to an increase in the intensity (white interference fringes) or to a decrease in the intensity (dark interference
Maximum Likelihood estimation is a versatile tool covering wide range of applications, but its be... more Maximum Likelihood estimation is a versatile tool covering wide range of applications, but its benefits are apparent particularly in the quantum domain. For a given set of measurements, the most likely state is estimated. Though this problem is nonlinear, it can be effectively solved by an iterative algorithm exploiting the convexity of the likelihood functional and the manifold of density matrices. This formulation fully replaces the inverse Radon transformation routinely used for tomographic reconstructions. Moreover, it provides the most efficient estimation strategy saturating the Cramer-Rao lower bound asymptotically. In this sense it exploits the acquired data set in the optimal way and minimizes the artifacts associated with the reconstruction procedure. The idea of maximum likelihood reconstruction is further extended to the estimation of quantum processes, measurements, and discrimination between quantum states. This technique is well suited for future applications in quantum information science due to its ability to quantify very subtle and fragile quantum effects.
The theory of quantum state reconstruction is illustrated here on several examples taken from mod... more The theory of quantum state reconstruction is illustrated here on several examples taken from modern experimental praxis. Maximum-likelihood estimation is applied to experiments on physical systems of increasing complexity, starting with a simple one-dimensional problem of quantum phase estimation, continuing with the absorption and phase neutron tomographies, further discussing quantum tomography of higher-dimensional discrete quantum systems, and closing with the homodyne tomography of an infinite dimensional system-a mode of light. All these experiments nicely demonstrate the utility of present state-of-art techniques for manipulating states of a neutron and internal as well as external states of a photon.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
In recent proposals for achieving optical super-resolution, variants of the Quantum Fisher Inform... more In recent proposals for achieving optical super-resolution, variants of the Quantum Fisher Information (QFI) quantify the attainable precision. We find that claims about a strong enhancement of the resolution resulting from coherence effects are questionable because they refer to very small subsets of the data without proper normalization. When the QFI is normalized, accounting for the strength of the signal, there is no advantage of coherent sources over incoherent ones. Our findings have a bearing on further studies of the achievable precision of optical instruments.
Ideal photon-number-resolving detectors form a class of important optical components in quantum o... more Ideal photon-number-resolving detectors form a class of important optical components in quantum optics and quantum information theory. In this article, we theoretically investigate the potential of multiport devices having reconstruction performances approaching that of the Fock-state measurement. By recognizing that all multiport devices are minimally complete, we first provide a general analytical framework to describe the tomographic accuracy (or quality) of these devices. Next, we show that a perfect multiport device with an infinite number of output ports functions as either the Fock-state measurement when photon losses are absent or binomial mixtures of Fock-state measurements when photon losses are present and derive their respective expressions for the tomographic transfer function. This function is the scaled asymptotic mean squared error of the reconstructed photon-number distributions uniformly averaged over all distributions in the probability simplex. We then supply mor...
By projecting onto complex optical mode profiles, it is possible to estimate arbitrarily small se... more By projecting onto complex optical mode profiles, it is possible to estimate arbitrarily small separations between objects with quantum-limited precision, free of uncertainty arising from overlapping intensity profiles. Here we extend these techniques to the time-frequency domain using mode-selective sum-frequency generation with shaped ultrafast pulses. We experimentally resolve temporal and spectral separations between incoherent mixtures of single-photon level signals ten times smaller than their optical bandwidths with a tenfold improvement in precision over the intensity-only Cramér-Rao bound.
International Journal of Quantum Information, 2017
Wigner and Husimi quasi-distributions, owing to their functional regularity, give the two archety... more Wigner and Husimi quasi-distributions, owing to their functional regularity, give the two archetypal and equivalent representations of all observable-parameters in continuous-variable quantum information. Balanced homodyning (HOM) and heterodyning (HET) that correspond to their associated sampling procedures, on the other hand, fare very differently concerning their state or parameter reconstruction accuracies. We present a general theory of a now-known fact that HET can be tomographically more powerful than balanced homodyning to many interesting classes of single-mode quantum states, and discuss the treatment for two-mode sources.
In continuous-variable tomography, with finite data and limited computation resources, reconstruc... more In continuous-variable tomography, with finite data and limited computation resources, reconstruction of a quantum state of light is performed on a finite-dimensional subspace. In principle, the data themselves encode all information about the relevant subspace that physically contains the state. We provide a straightforward and numerically feasible procedure to uniquely determine the appropriate reconstruction subspace by extracting this information directly from the data for any given unknown quantum state of light and measurement scheme. This procedure makes use of the celebrated statistical principle of maximum likelihood, along with other validation tools, to grow an appropriate seed subspace into the optimal reconstruction subspace, much like the nucleation of a seed into a crystal. Apart from using the available measurement data, no other assumptions about the source or preconceived parametric model subspaces are invoked. This ensures that no spurious reconstruction artifacts...
We present an experimental realization of spatial-mode demultiplexing (SPADE) method to obtain th... more We present an experimental realization of spatial-mode demultiplexing (SPADE) method to obtain the superresolution of two incoherent point sources. The digital holography projection onto the Hermite-Gaussian basis is compared with the CCD position measurement.
We present a simple and efficient Bayesian recursive algorithm for the data-pattern scheme for qu... more We present a simple and efficient Bayesian recursive algorithm for the data-pattern scheme for quantum state reconstruction, which is applicable to situations where measurement settings can be controllably varied efficiently. The algorithm predicts the best measurements required to accurately reconstruct the unknown signal state in terms of a fixed set of probe states. In each iterative step, this algorithm seeks the measurement setting that minimizes the variance of the data-pattern estimator, which essentially measures the reconstruction accuracy, with the help of a data-pattern bank that was acquired prior to the signal reconstruction. We show that with this algorithm, it is possible to minimize the number of measurement settings required to obtain a reasonably accurate state estimator by using just the optimal settings and, at the same time, increasing the numerical efficiency of the data-pattern reconstruction.
We introduce an operational and statistically meaningful measure, the quantum tomographic transfe... more We introduce an operational and statistically meaningful measure, the quantum tomographic transfer function, that possesses important physical invariance properties for judging whether a given informationally complete quantum measurement performs better tomographically in quantum-state estimation relative to other informationally complete measurements. This function is independent of the unknown true state of the quantum source, and is directly related to the average optimal tomographic accuracy of an unbiased state estimator for the measurement in the limit of many sampling events. For the experimentally-appealing minimally complete measurements, the transfer function is an extremely simple formula. We also give an explicit expression for this transfer function in terms of an ordered expansion that is readily computable and illustrate its usage with numerical simulations, and its consistency with some known results.
Scheme for optimal spin state estimation is considered in analogy with phase detection in interfe... more Scheme for optimal spin state estimation is considered in analogy with phase detection in interferometry. Recently reported coherent measurements yielding the average fidelity (N +1)/(N +2) for N particle system corresponds to the standard limit of phase resolution 1/ √ N. It provides the bound for incoherent measurements when each particle is detected separately and information is used optimally. For specific states, improvement up to the value 1/N is possible in quantum theory. The best results are obtained combining sequentially coherent measurements on fractional groups of particles.
The purpose of this chapter is to review the basic concepts leading to definition of nonclassical... more The purpose of this chapter is to review the basic concepts leading to definition of nonclassical optical phanomena which have fully quantum origin, have no classical analogues and can be explained only in the framework of quantum theory. Their experimental observations (Chap. 8) represent a complete confirmation of the quantum theory of Nature. The most important nonclassical phenomena are quadrature
In this chapter we review the most important experiments performed in quantum optics supporting q... more In this chapter we review the most important experiments performed in quantum optics supporting quantum theory. We deal with one-photon as well as two-photon experiments, resonance fluorescence experiments, experiments in cavity quantum electrodynamics, experiments with various feedbacks, and experiments with squeezed light and sub-Poisson light generated in various nonlinear optical processes, such as optical parametric processes, particularly frequency down conversion, four-wave mixing, phase conjugation, optical Kerr effect as well as Raman scattering. In particular the process of the interaction of single atoms with light can provide effective tests of quantum behaviour. Some experiments related to violation of classical inequalities, Bell’s inequalities and quantum “nonlocality” were discussed in Chap. 7.
Inspite of its abstract mathematical structure quantum mechanics is above all the pragmatic theor... more Inspite of its abstract mathematical structure quantum mechanics is above all the pragmatic theory which enables to predict the results of the experiments. The weakness of classical physics to explain the behavior of microsystems initialized the revolution in physics at the beginning of this century. The experiments, both “gedanken” or feasible, are playing the primary role in development of quantum
In this chapter we provide the basic properties of the coherent-state technique and its applicati... more In this chapter we provide the basic properties of the coherent-state technique and its application to the interaction of radiation with matter. Greater details about this technique can be found in books by Klauder and Sudarshan (1968), Malkin and Man’ko (1979), Klauder and Skagerstam (1985), Perelomov (1986), Peřina (1991), in reviews by Zhang, Feng and Gilmore (1990), Kim and Noz
Interference is one of the most important phenomena in optical fields representing their wave pro... more Interference is one of the most important phenomena in optical fields representing their wave properties. Such two optical fields can in principle be in constructive or destructive interference according to their mutual phase (value of difference of their simultaneous phases) giving rise to an increase in the intensity (white interference fringes) or to a decrease in the intensity (dark interference
Maximum Likelihood estimation is a versatile tool covering wide range of applications, but its be... more Maximum Likelihood estimation is a versatile tool covering wide range of applications, but its benefits are apparent particularly in the quantum domain. For a given set of measurements, the most likely state is estimated. Though this problem is nonlinear, it can be effectively solved by an iterative algorithm exploiting the convexity of the likelihood functional and the manifold of density matrices. This formulation fully replaces the inverse Radon transformation routinely used for tomographic reconstructions. Moreover, it provides the most efficient estimation strategy saturating the Cramer-Rao lower bound asymptotically. In this sense it exploits the acquired data set in the optimal way and minimizes the artifacts associated with the reconstruction procedure. The idea of maximum likelihood reconstruction is further extended to the estimation of quantum processes, measurements, and discrimination between quantum states. This technique is well suited for future applications in quantum information science due to its ability to quantify very subtle and fragile quantum effects.
The theory of quantum state reconstruction is illustrated here on several examples taken from mod... more The theory of quantum state reconstruction is illustrated here on several examples taken from modern experimental praxis. Maximum-likelihood estimation is applied to experiments on physical systems of increasing complexity, starting with a simple one-dimensional problem of quantum phase estimation, continuing with the absorption and phase neutron tomographies, further discussing quantum tomography of higher-dimensional discrete quantum systems, and closing with the homodyne tomography of an infinite dimensional system-a mode of light. All these experiments nicely demonstrate the utility of present state-of-art techniques for manipulating states of a neutron and internal as well as external states of a photon.
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