Multi-copy quantifiers for single-photon states

Single-photon states are basic resources for hybrid quantum technology with non-Gaussian states of light. Accelerating quantum technology is already able to produce high-quality single-photon states. These states can be used for hybrid quantum information processing, based on a nonclassical phase-space interference represented by negativity of a Wigner function. Therefore, new quantifiers, capable of evaluating such high-quality single-photon states, are required. We propose and analyze quantifiers which process multiple estimates of single-photon state’s statistics. The quantifiers simulate basic capability of single photons to conditionally bunch into a single mode and form a Fock state. This state exhibits complex nonclassical phase-space interference effects making its Wigner function negative in multiple areas. The quantifiers directly evaluate a presence of the multiple negativities corresponding to the Fock state. We verify applicability of the quantifiers by using them to single-photon states from recent experiments. The quantifiers can be further extended to also test indistinguishability of single-photon states. It allows to verify quantum interference of light from single-photon emitters more sensitively than in the traditional Hong-Ou-Mandel test. Besides quantum optics, the multi-copy quantifiers can be also applied to experiments with atomic memories and mechanical oscillators.

By following the physical procedure for generation of the Fock state |n〉 from the single photon states [25][26][27]32 , a conditional output state ρˆo ut can be computed. This state is achievable from n independent statistics of single-photon states ρˆe st i , , = … i n 1, , . The computed state ρˆo ut is used to determine a necessary condition for the Fock state |n〉 capability. Two kinds of multi-copy quantifiers have been previously proposed. The first kind only tests an ideal photon bunching at an unconditioned output 34 . The second kind measures fidelity of the computed conditional state ρˆo ut with the Fock state |n〉 35 . Nevertheless, none of the previous quantifiers evaluate relevant nonclassical features of the state ρˆo ut in phase space. This computed state ρˆo ut exhibits the complex nonclassical phase-space interference which is represented by multiple negative areas of a continuous-variable Wigner function 28 . Therefore, fidelity, averaging over phase-space, is not suitable. On the other hand, the negativities of the Wigner function are a very sensitive feature revealing a complexity of the phase-space interference. The negativities of the Wigner function are an important characteristic of the non-Gaussian states required for hybrid information processing 1,2 . Moreover, this interference is required for advanced quantum computation and simulations with oscillators 16,17 . Multiple negative areas in phase space have been recently used for very precise evaluation of non-Gaussian states in the quantum cavity electrodynamics 36 , superconducting circuits 37 , and trapped ions experiments 38 .
Here, we propose and analyze single-photon phase-space quantifier based on multiple estimates of photon statistics produced by a single-photon state. The first part of the quantifier composes of computing a conditional output Wigner function from the independent estimates of the photon statistics. The computational procedure fully corresponds to the physical conditional protocol of a Fock state preparation from single-photon states [25][26][27]32 schematically depicted in Fig. 1. In the second part, the computed output Wigner function is evaluated. In the case of ideal single-photon states, the output Wigner function fully corresponds to the Fock state |n〉. Negative areas of this Wigner function form concentric annuli. The number of the annuli relates to the number n of the photons in the Fock state. The importance of multiple Wigner function's negativities motivates us to define the quantifier in the following manner. The number of negative annuli of the conditional output state ρˆo ut computed from n copies ρˆe st i , is determined. If the number of negative annuli corresponds to the Fock state |n〉, the single-photon source exhibits a Fock-state |n〉 capability. It means that the single-photon states produced by the source can, in principle, be used to generate quantum non-Gaussian state with negativities of Wigner function corresponding to the Fock-state |n〉. The higher number of the concentric negative annuli in Wigner function demonstrates the higher level of the challenging continuous-variable nonclassical interference in phase space. To determine the number of negative annuli of the output Wigner function, a number of roots of a single-variable polynomial has to be only found. Coefficients of the polynomial depend on the input photon statistics. The quantifier determining the Fock state |n〉 capability is operational and deterministic.
First, we apply the quantifier to idealized single-photon sources without multi-photon contributions. This determines sufficient single-photon fraction to reach the Fock state |n〉 capability. Second, we perform a Monte-Carlo simulation with realistic photon statistics of single-photon states estimated from recent experiments 39 . The multi-photon contributions of the single-photon states and deviations in between them affect the Fock state |n〉 capability. The Monte-Carlo simulation demonstrates sensitivity of the quantifier to the realistic experimental imperfections. Our result offers an operational quantifier sensitively evaluating light from high-quality single-photon emitters with visible negativity of Wigner function, for which the currently used quantifiers are not sufficient. This basic level of analysis evaluates only the statistics of light from single-photon sources. It assumes an ideal interference of all states. In order to test also an ability to interfere, we propose an extension of the quantifier to an advanced level. The advanced quantifier performs the first layer of the bunching  The largest deviation of fit function from the computed data is 9 × 10 −4 .
procedure, depicted in Fig. 1, in a laboratory. This quantifier then processes estimated conditional photon statistics after conditional two-copy interference experiments. As a result, we obtain the Fock state |n〉 capability involving jointly statistics and indistinguishability of two realistic single-photon state copies. This nonclassical interference test is extremely sensitive to statistics of photons and their indistinguishability. It is more sensitive than the commonly used Hong-Ou-Mandel interference experiment with photon counters 24 . The method can be further extended by involving more complex multi-copy interference effects [40][41][42] . The proposed quantifier can be applied to all existing experiments in quantum optics capable to estimate photon-number distribution by either homodyne detection or multi-channel detectors. Moreover, the quantifiers can be applied to other physical platforms, where light is used to read out states of atoms 43 , solid-state systems 44 and mechanical oscillators 45 .

Results
Processing of n estimated density matrices. Single-photon sources typically produce phase-insensitive states diagonal in the Fock state basis. Therefore, the multi-copy quantifier uses n independent copies of the most likely estimate of the phase-insensitive density matrix ρ = ∑ = p m m The density matrices can be reliably obtained by modern quantum estimation methods based on homodyne tomography [46][47][48][49] . Alternatively, photon-number statistics can be achieved from multi-channel photon detectors 50 and photon-number resolving detectors 51 . Although, uncertainties of the density matrix's elements are not incorporated directly, they will be used later to investigate propagation of errors through the quantifier by the Monte-Carlo simulation, which is described in Methods. Now we derive an expression for the conditional output state density matrix. To this end, we apply the virtual physical procedure, depicted in Fig. 1, to the estimates ρˆe st i , . The estimate of the multi-mode state is the input of the quantifier procedure, where â i and ˆ † a i are annihilation and creation operators of i-th mode. To simplify the derivation, we exchange the product and the summation of annihilation and creation operators. By multiplying the tensor product, the expression for the estimate of the overall input state can be rewritten as where represents vacuum state in all modes. The estimated state ρˆi n is now arranged in a form suitable for direct application of the virtual procedure depicted in Fig. 1. The first step of the procedure corresponds to a complex interference of the input modes in a linear optics network represented by the unitary transformation Û . It acts on the photon operators of individual modes in the following way where U i,j is n × n matrix describing complex probability amplitudes of single-photon propagation through the linear network. Note, the output annihilation operators are only linear combinations of the input annihilation operators. They are not mixed with the creation operators. Therefore, the linear transformation is energy conserving. Moreover, it is a passive transformation, which cannot generate nonclassicality from classical states. It only represents a generalized rotation in overall phase space of n modes. Therefore, it does not increase nonclassical effects caused by the phase-space interference.
In the second step, all energy is conditionally concentrated into a single output by the virtual vacuum measurements, depicted in Fig. 1. Therefore, we swap to the Schrödinger picture to describe the measurement process. In the Schrödinger picture, the unitary transformation Û changes the estimate ρˆi n of the overall input state to an effective estimated state ρ ρ =ˆˆ † U U eff i n . The conditional procedure concentrating light into a single mode is implemented by n − 1 measurements projecting the n-mode state ρˆe ff on a vacuum states The state in the output mode of the conditional procedure reads Normalization of the density matrix can be omitted, since we are going to be only interested in negative areas of Wigner function. An appearance of its negative values is independent of normalization, which only scales absolute values of Wigner function. This measurement conditionally merges all photons from input modes to the remaining output mode labeled by 1. All terms that include > a i 1 and > † a i 1 vanish due to the vacuum projective measurement. The output state in mode 1 is proportional to the density matrix This output state can be further simplified to as there is only one output mode. Note, the factor ∑m ( )! i emerges from applying annihilation and creation operators to the vacuum state. All photons are merged in the single output mode. Photon statistics of the output mode strongly depends on input statistics p m i and transition probabilities | | U j,1 2 . However, the output photon statistics does not depend on phases of the probability amplitudes U j, 1 . It means that no phase interference effect appears between single photons propagating from different inputs to the output. This phase insensitivity was already observed for bunching of pure single photons into a single mode 27 . The third step includes universality of the quantifier, i.e. an independence of any prior knowledge about the estimated statistics p m i . To quantify all copies of the estimated quantum states without any preference, we use a balanced linear optics network with probabilities j, 1 2 By considering it, the formula for the output state density matrix simplifies to This formula can be used to calculate ρˆo ut , respectively, its photon number distribution from n independent estimates of matrix elements p m i . The fourth step is evaluation of a characteristic feature of ρˆo ut capable of sensitively quantifying quantum non-Gaussian states. The estimated output state ρˆo ut can be characterized by various nonclassical properties, which are important for advanced quantum processing 16,17 . Currently, the highest nonclassical benchmark for non-Gaussian states is the appearance of Wigner function's interference fringes, which lead to negative values of the Wigner function [36][37][38] . Therefore, the number of negative annuli of the Wigner function corresponding to the Fock state |n〉 is a natural and reasonable choice for the quantifier criterion. The state ρˆo ut has the Wigner function   Number of the polynomial's positive roots is used as the quantifier. The quantifier polynomial (12) has a finite order in the argument z = x 2 > 0. The order of the polynomial is given by the maximum over all sums ∑ = m i n i 1 . This maximum is determined by nM, where n is the number of input state copies and M is the maximal photon number considered in the input state estimates. The operational quantifier is deterministic. However, it computes outcome of the conditional physical transformation. In the Wigner function formalism, this conditional transformation is described by a convolution of the n input Wigner functions W est,i (x, p) rotated in phase space by the transformation (4). The convolution is with a positive Gaussian kernel due to the projections on vacuum.
Scientific RepoRts | 7: 1484 | DOI:10.1038/s41598-017-01333-y Therefore, it cannot generate more negative annuli than the number n of the input single-photon state's copies. The conditional process only physically combine negative values of Wigner functions in phase space.
The last, fifth step, compose of finding roots of the quantifier polynomial (12). This polynomial equation can be solved numerically or graphically, as it is demonstrated below. For n copies of ideal single photon with = p 1 m i , for all m i = 1, the polynomial (12) has n positive roots. Therefore, the Wigner function W n (x, p) has n/2 negative concentric annuli for even n. For odd n, it has (n − 1)/2 negative concentric annuli and one negative disk in origin. We claim that n single-photon states with general p m i have the Fock state |n〉 capability if numbers of negative disks and annuli correspond to the Fock state |n〉. A realistic single-photon state has the Fock state |n〉 capability only if the polynomial (12) has n positive roots. In the following subsections, we first apply the quantifier to idealized high-quality single-photon states. Then, in order to demonstrate practicality of the quantifier, we use it to evaluate estimates of single-photon states from recent experiments. where z > 0. Its roots can be found numerically or graphically for the given η and n. For η = 1, the output corresponds to the ideal Fock state |n〉 with ⌊ ⌋ n/2 negative annuli and n positive roots of the polynomial P n (z). For η < 1, Wigner function oscillations become suppressed, however, the number of negative annuli is preserved, as it is visible from Fig. 2 (LEFT). As the attenuation parameter η decreases, the annulus with the largest diameter narrows until the corresponding two roots of the polynomial P n (z) became equal and the negative annulus disappears. This follows from the nature of polynomial's roots. The largest value of η < 1, for which two roots of P n (z) are equal, determines a threshold value. For this value of η, the annulus with the largest diameter vanishes. Two roots of a polynomial are equal when its discriminant Δ is equal to zero. Therefore, the threshold value of η is determined by the largest root of the equation The time demanding part of the graphical method is computation of the discriminant for the polynomial. For our purpose, computing the threshold value for up to n = 56 is sufficient and it was achievable in reasonable time. For n = 2, a sufficient condition for the Fock state |2〉 capability of single-photon state is η > 2/3. The sufficient η for the Fock state |n〉 capability using n copies of states gradually increases, as it is depicted in Fig. 1. n-Fock state capability for realistic single-photon states. Now, we consider a density matrices estimated from a single-photon experiment 39 . Two different density matrices are considered ρ = .
± .  . The density matrix ρˆe st (1) has the same mean photon number 〈n〉 = 0.838 as the density matrix ρ η= . 0 838 without multi-photon contributions. In the insight in Fig. 2 (RIGHT), it can be seen that the width of the negative annulus with the largest radius is distinctively smaller for the realistic single-photon state than for the attenuated single-photon state. Also the negative peaks are deeper for the attenuated single-photon state. It illustrates a destructive impact of multi-photon contributions on the Fock state |n〉 capability. As the graphical analysis shows, the quantifier conclusively recognizes the appearance of the negative annulus with the largest radius.
For the realistic single-photon state ρˆ( 1) , the values of the quadrature x, for which the cut through the output Wigner function W out (x, 0) has negative values, is depicted against the number n of the input modes in Fig. 3. In the picture, one can see that the mean Wigner function W out (x, 0) exhibits only three negative annuli for eight input states. However, the Fock state |8〉 has four negative annuli. The output Wigner function still has full three negative annuli for seven input states. Therefore, the realistic state ρˆe st (1) has only the Fock state |7〉 capability. The attenuated single-photon state ρ η= . 0 838 with the same mean photon number, but without multi-photon contributions, has the Fock state |25〉 capability. This large difference arises from the small fraction of the multi-photon contribution present in realistic state ρˆe st (1) . The realistic state ρˆe st (2) with mean photon number 〈n〉 = 0.903 exhibits full seven negative annuli for fourteen input state copies (see Fig. 3). Therefore, the state has at least the Fock state |14〉 capability. The attenuated single-photon state with the same mean photon number, but without multi-photon contributions, exhibits an arbitrary Fock state |n〉 capability. It was shown up to numerical limit in the previous analysis (see Fig. 1). Our analysis is restricted by at maximum fourteen input state copies due to time limitations of our program. The limitations of the programs are discussed in Methods.
Advanced quantifiers. The previously introduced computational quantifier does not examine the ability of the realistic single-photon states to interfere with each other. It is only composed of data processing on the estimated statistics of individual modes. Therefore, we propose an advanced hybrid quantifier containing an experimental part implemented before the computational quantifier (see Fig. 4). The first level of the bunching procedure is performed in a laboratory. Two copies of single-photon state interfere on a balanced beam splitter. Then a projection on vacuum is conditionally induced on one mode. The tomogram of the resulting state is measured and its photon statistics is estimated. By repeating this measurement n/2 times, the first level of the bunching procedure is executed and n/2 estimated density matrices are generated. In the extension to the purely computational quantifier, the ability of single-photon pair to interfere is tested. The remaining levels of the bunching procedure are performed by computer. The estimated density matrices ρˆe st,i , where = … i n 1, 2, /2, are operated by data processing as in the case of the computational quantifier. As a result, the Fock state |n〉 capability is determined. The advanced quantifier tests a quality of the single-photon state beyond the examination provided by the purely computational quantifier. The ability to interfere is additionally tested in the experimental part of the hybrid quantifier. Recently, important experimental tests of a two-photon interference with homodyne detectors 32,52 have paved a way for such advanced quantifiers. The next step is an extension to quantifiers employing a three-and a four-photon interference. The extension for challenging phase-space interference tests can be straightforwardly implemented by following the presented methodology. It brings a possibility to build a hierarchy of very sensitive criteria for single-photon sources. From our expertise 53 , they will be much more sensitive than the currently used Hong-Ou-Mandel type of criteria with photon detectors.

Outlook.
We have presented new approach to evaluate high-quality single-photon sources by using multi-copy quantifiers. Principles of the methodology were demonstrated on the idealized case of attenuated single-photon states. Moreover, we used this approach to evaluate realistic single-photon statistics estimated from recent experiment.
The theoretical analysis of the concept of the Fock state |n〉 capability can be further extended. The multi-photon contributions have crucial effects on the Fock state |n〉 capability as it was demonstrated in this manuscript. In practice, the multi-photon contributions are typically small and it is hard to theoretically model them. However, they crucially reduce the Fock state capability. Therefore, further theoretical investigation depends on the experimental tests of the quantifier. The experimental verification will provide information about the predicted influence of multi-photon contributions, a stability of the single-photon generation and a quality of detection on the proposed quantifier. The Fock state |n〉 capability of single-photon states will depend on a quality of the density matrix estimation. Furthermore, an ability to recognize the regular phase-space interference with minima of the Wigner function reaching negative values will be crucial. Although absolute values of the negative peaks are small, they are sufficiently large in comparison to the largest negative peak close to origin. Therefore, it is very important to experimentally test the methodology on multiple independent photon number distributions. Moreover, an efficiency of the method needs to be examined in practice for different single-photon sources. All these experimental insides will focus the further theoretical development. are complements to unity securing that the input states are normalized. For the realistic state ρˆe st (2) , the matrix elements are generated analogously with appropriate mean values and standard deviations. The random sample of thirty output Wigner functions is computed in order to determine mean values and standard deviations of the output Wigner function W out (x, p). The values of the quadrature x, for which the cut through the mean output Wigner function W out (x, 0) exhibits negative values, are computed numerically. From the number of x intervals in which the cut is negative, the number of negative annuli is determined hence the circular symmetry of the Wigner function in phase space.

Monte
The computation of the output Wigner function W out (x, 0) is restricted by ability to perform the summation in the formula (11). The maximal achieved number of input copies for the quantifier is fourteen, which is sufficient to demonstrate the principals of the quantifier. The examination of the Fock state |n〉 capability of realistic Scientific RepoRts | 7: 1484 | DOI:10.1038/s41598-017-01333-y single-photon states with mean photon number close to one is a problem for super computers. An example of such a state is the realistic single-photon state ρˆe st,2 . These states exhibit capability of large Fock states. If we consider up to two-photon contributions in the input density matrices, the number of summands in the formula (11) grows like 3 n with the number of input modes n. Therefore, the computational time grows exponentially with n.