Testing for entanglement with periodic coarse-graining
Continuous variables systems find valuable applications in quantum information processing. To deal with an infinite-dimensional Hilbert space, one in general has to handle large numbers of discretized measurements in tasks such as entanglement detection. Here we employ the continuous transverse spatial variables of photon pairs to experimentally demonstrate novel entanglement criteria based on a periodic structure of coarse-grained measurements. The periodization of the measurements allows for an efficient evaluation of entanglement using spatial masks acting as mode analyzers over the entire transverse field distribution of the photons and without the need to reconstruct the probability densities of the conjugate continuous variables. Our experimental results demonstrate the utility of the derived criteria with a success rate in entanglement detection of relative to studied cases.
March 6, 2018
The efficient preparation and manipulation of high-dimensional quantum systems allow for the processing of large amounts of quantum information. In addition to advantages, such as increased transmission rates, a number of interesting fundamental aspects related to quantum entanglement Horodecki et al. (2009); Gühne and Tóth (2009), non-locality Brunner et al. (2014) and contextuality Amaral and Cunha (2018) only become evident with the use of quantum systems of high-dimension.
The transverse spatial modes of single or entangled photons constitute an interesting experimental platform for the study of high-dimensional quantum systems Walborn et al. (2010). In principle, spatial degrees of freedom (DOF) are described by an infinite set of modes: the transverse position or momentum modes provide the continuous variable (CV) description of the spatial DOF, while infinite-dimensional discrete basis can also be explored by using, for instance, the orbital angular momentum Allen et al. (1992); Padgett (2017), and radial modes Karimi et al. (2014); Zhou et al. (2017). In the latter approach, a finite dimensional space can be achieved by simply isolating a subset of the entire set of spatial modes. For these -dimensional states, entanglement detection Walborn et al. (2004); Leach et al. (2009); Dada et al. (2011); Salakhutdinov et al. (2012); Giovannini et al. (2013); Hiesmayr and Löffler (2013); Krenn et al. (2014) is performed with tools developed for finite dimensional quantum systems Spengler et al. (2012); Erker et al. (2017).
In the CV regime, on the other hand, the observation of spatial entanglement typically requires the measurement of the joint distributions for the position and momenta of the photons Edgar et al. (2012); Moreau et al. (2012, 2014); Defienne et al. (2018), from which criteria devoted to CV systems are used to test for non-separability Duan et al. (2000); Mancini et al. (2002); Shchukin and Vogel (2005), EPR Reid (1989); Reid et al. (2009); Walborn et al. (2011) correlations and “steering” Wiseman et al. (2007). Albeit fundamentally continuous, real-world experiments are subject to the coarse-graining imposed by the detector resolution Rudnicki et al. (2012), as well as the limited detector range Ray and van Enk (2013). Both of these issues can lead to false positives concerning detection of quantum correlations Tasca et al. (2013); Ray and van Enk (2013). Even though typical CV entanglement and EPR criteria have been adapted for coarse-grained measurements Tasca et al. (2013); Schneeloch et al. (2013), the experimental assessment of the position and momentum correlations typically require a large number of measurements.
A number of techniques have been applied in effort to reduce the number of measurements necessary to identify entanglement in these systems, while maintaining the full range of detection events. For instance, Howland et al. Howland and Howell (2013) have shown that the use of compressive sensing techniques allows for the reconstruction of the SPDC joint detection probabilities with an efficiency improvement over a raster scanning procedure with equivalent resolution. In this case, nevertheless, the identification of entanglement is still bound to the application of the typical CV entanglement criteria Adesso and Illuminati (2007), based–for example–on the evaluation of the moments Duan et al. (2000); Mancini et al. (2002); Shchukin and Vogel (2005) or entropy Walborn et al. (2009); Saboia et al. (2011) of the reconstructed distributions.
In the present contribution, we develop and test experimentally convenient entanglement criteria based on periodic coarse-grained measurements. We start by establishing an uncertainty relation (UR) for the localization of a single quantum particle in a periodic array of position and momentum bins. The developed UR is expressed in terms of the cross-correlation function between a periodic analyzer and the distributions for the position and momentum variables. We then use this UR to build entanglement criteria devoted to bipartite CV quantum systems and apply them to test for spatial entanglement of photon pairs from spontaneous parametric down-conversion (SPDC). In this optical setup, the developed criteria are experimentally accessible via the joint transmission of the photons through periodic apertures playing the role of spatial-mode analyzers. This measurement strategy using periodic spatial mask analyzers acting over the entire transverse field structure of the photons enables approximately uniform single-photon and higher coincidence detection rates than the traditional binning with single apertures, thus yielding better signal-to-noise ratio. In addition, the periodicity of the spatial masks used in position () and momentum () measurements work as free parameters that can be independently tuned to optimize the entanglement detection. In our experiment, we tested different combinations of spatial mask geometries, achieving a success rate in entanglement detection of about .
This paper is structured as follows: in sections II and III, we provide the theoretical background necessary for the development of our entanglement criteria, which is derived in section IV. Our experimental scheme and measurements with the periodic spatial masks are described in section V, and the analysis of our experimental data with the derived entanglement criteria is presented in section VI. We provide concluding remarks in section VII.
ii.1 Optical Fourier Transform
The paraxial propagation of a monochromatic single-photon through a lens system implementing an optical Fourier transform is illustrated in Fig. 1. The propagation direction is along the positive -axis and we denote the input transverse field distribution at the front focal plane of the lens. We thus consider an input pure single-photon whose quantum state in position representation is
where is the transverse coordinate at the input plane. We also introduce the function as the field distribution at the back focal plane of the lens (the Fourier plane). The transformation connecting these two wavefunctions is
where represents the transverse spatial coordinate at the Fourier plane and is a constant related to the optical system: is the wavelength of the photon and the focal length of the lens. It is straightforward to recognize that the Fourier transformed field distribution maps the transverse structure of the input photon in momentum representation, :
Eq. (3) implicitly assumes (we set ) the relation between the transverse momentum component of the photon at the input plane and the spatial coordinate at the Fourier plane.
ii.2 Periodic Spatial Mask Analyzers
We consider spatial mask analyzers given by periodic aperture functions whose transmittance obey , so that assumes either the value of or across the whole transverse plane. Over a single transverse dimension, such apertures are described by periodic square waves with two independent parameters: the periodicity ’’ and an extra spatial parameter ’’ that we name the bin width. Explicitly, we define the periodic spatial mask analyzer as
For a given choice of and , the periodic aperture function is uniquely specified by (4), provided that the mask’s origin is fixed. In general, we allow extra displacement parameters representing positioning degrees of freedom setting the origin. It will be useful to consider the Fourier series expansion of (4):
where , and
are the coefficients of the Fourier expansion.
In Fig. 1 we illustrate the periodic spatial mask analyzer over the single transverse degree-of-freedom (). We thus work with a single pair of conjugate variables satisfying the commutation relation . The periodic analyzer is inserted in the path of the photon, either in the front or back focal plane of the lens, and we allow distinct parameters ’’ and ’’ for the apertures used to probe the position (, ) and momentum (, ) distributions. We name it a symmetric arrangement whenever , but in general any combination of periodic analyzers is allowed – note that since the transverse momentum variable is mapped to the Fourier coordinate by means of the scaling factor , the individual choice of the periodicity and bin width is irrelevant in a symmetric arrangement as long as the ratio is preserved. We further define and as the physical spatial parameters (units of length) of the analyzer used in the Fourier plane of the lens, whereas and express the associated quantities in momentum domain (units of inverse length). From now on, unless specified, we adopt the pair and in our analysis.
Our figure of merit for the development of the UR and entanglement criteria is the photon transmission probability through the periodic analyzer. Denoting and , these transmission probabilities are
where and represent the marginal probability distributions along the relevant degree of freedom. In Eqs. (7), the parameters and describe the transverse displacement of the spatial mask analyzer. The transmission probabilities (7) can be understood as the cross-correlation function between the probability density of the photon and the spatial mask analyzer. This cross-correlation provides the probability that the photon is transmitted as a function of the analyzer’s transverse position. In other words, the spatial mask (4) acts as a filter that is used to analyze the spatial structure of the photon field in position (7a) or momentum (7b) domain. It is worth mentioning that since the considered analyzer is an amplitude mask, no phase-sensitive measurements are necessary in the characterization of (7). This is in contrast with mode analyzers based on spiral Beijersbergen et al. (1994); Oemrawsingh et al. (2004, 2005) or multi-sector Pors et al. (2008, 2011); Giovannini et al. (2012) phase masks that have been utilised in measurements of multi-dimensional orbital-angular-momentum entanglement.
Iii Uncertainty Relation based on periodic analyzers
As it is known, the Fourier transform (2) implies that the photon’s transverse field distribution cannot be arbitrarily well localized in both focal planes of the lens. In this paper we explore this complementarity to build an uncertainty relation based on the transmission probabilities of the photon through periodic apertures. To this end, it is convenient to use the Fourier decomposition of the periodic analyzer (5) to write the transmission probabilities (7) in the following form:
where and stand for the characteristic functions associated with the probability densities and , respectively. Using the uncertainty relation for characteristic functions reported in Ref. Rudnicki et al. (2016), it is straightforward to show (see Appendix A) that the cross-correlation functions (7) satisfy the inequality:
and the function reads
Note that . Moreover, the notation in (9) involving the moduli of the probability distributions (real, non-negative) is technically speaking unnecessary. Eq. 9 is presented in that manner in order to ease the manipulations of apparently complex formulas (8).
The upper bound is a function of the periodic analyzer parameters used to probe the field distribution in position and momentum domain. Since the functional dependence of the Fourier coefficients (6) is only on the ratio , the bound function (10) turns into a simpler form whenever working with a symmetric arrangement [see Eq. (20)]. To shortly summarise, the cross-correlation between the periodic analyzer and the transverse field structure of a single photon measured in complementary domains is subject to the uncertainty relation given in (9). This uncertainty relation constitutes the first theoretical result of the paper.
iii.1 Coarse Graining and Faithful Sampling
Real-world measurements of a CV degree-of-freedom are inevitably subjected to the coarse graining imposed by detection resolution Rudnicki et al. (2012). Here the resolution limitation appears due to the necessarily discretised transverse displacements in the experimental characterisation of the cross-correlation functions (7) – instead of its (unfeasible) sampling over a continuum of displacements . Realistically, the experimentalist holds discretised distributions and by employing a finite set of displacements with scanning resolution and , respectively. To sample the full periodicity of the analyzer, say in position domain, a number of transmission probability measurements are required (). Here we argue that the minimal resolution expected for a “faithful” sampling is such that : a displacement step greater than the analyzer bin width is unable to sample the entire field distribution. Interestingly, when using an analyzer geometry with an integer , the discretised cross-correlation function obtained in the limiting case corresponds to the periodic coarse-graining Tasca et al. (2018) of the single-photon detection probability distribution. In this case, the set of analyzer displacements corresponds to mutually exclusive outcomes of a -dimensional measurement with .
Iv Entanglement criteria
In this section we aim to derive experimentally convenient entanglement criteria along the lines discussed in the previous section, i.e., based on transmission probabilities through periodic analyzers acting as spatial filters in position and momentum domains. We consider bipartite entanglement present in the spatial degrees-of-freedom of a photon pair. In this scenario, the relevant transmission probability is that of the photon pair, serving as an indicative of the spatial correlations.
We name the quantum state associated with the transverse spatial structure of photons and , and restrict ourselves to the analysis of their spatial structure in a single transverse degree-of-freedom . The generalization to both transverse dimensions is straightforward. Following the definitions of the previous section, we consider the detection probabilities in the conjugate planes of lenses placed in the path of the photons and . Explicitly, we define the joint probability distributions:
As before, we look at the marginals along the relevant degree of freedom that is probed by the analyzer’s periodic aperture:
The distributions defined in Eqs. (14) describe spatial correlations of the photons in position and momentum domain, respectively. The probabilities that both photons are transmitted through identical analyzers placed in their propagation path are indicative of the shape and strength of the correlations in question. Our entanglement criteria shall thus be based on such transmission probabilities as quantifiers of the position and momentum correlations shared by the two-photon state . The relevant quantities are clearly given by the two-variable counterparts of the cross-correlation functions (7):
In the two-photon case, the joint distributions and are probed by the product of the individual periodic apertures (16) each photon is subjected to. Obviously, the maximal amount of information on the two-photon correlations is achieved by complete characterization of the cross-correlation functions given in Eqs. (15). This complete characterization requires the evaluation of the joint transmissions for all possible values of the transverse displacement of the analyzers used for photons and . Nevertheless, as we shall now show, the use of periodic apertures allows for entanglement verification with the evaluation of the cross-correlation functions (15) along single directions in phase-space.
We begin by defining as the joint transmission probabilities obtained by displacing the apertures for photons and along the same () or opposite () directions over the full periodicity of the apertures. In other words, encode the cross-correlation function (15a) along the diagonal or anti-diagonal direction of the parameter space given by the individual displacements of the analyzers. Analogously, we define a similar quantity in the momentum domain as . We further introduce:
as the averages of and , respectively. In Fig. 2(b) we illustrate the transverse displacement procedure required for the evaluation of the average joint transmission probabilities (17). We are now in position to establish our second theoretical result, namely the entanglement criteria. The sum of the average transmission probabilities
Proof: As the starting point we observe that the left hand side of (18) is a linear (thus convex) functional of the quantum state, so one can restrict further discussion to separable pure states . In this special case, the joint transmission probabilities factorize: . The single-photon probabilities and , evaluated for and respectively, are given by Eq. (7a). The same factorization scheme applies to the momentum domain.
In the second step we use the basic arithmetic–geometric mean inequality, , to separate the and parts of the joint transmission probabilities. More explicitly:
and similarly for the momentum term. These initial steps are similar to those of Spengler et al Spengler et al. (2012), where entanglement criteria for discrete systems in terms of mutual predictabilities have been derived.
In fact, the sign combination for the part and the labeling of subsystems (pure quantum states) both stop playing a role from now on. The sign ambiguity is removed by the symmetric integrals with respect to while the second assertion is true because in the sum we find pairs of probabilities to be bounded by (9) independently for each subsystem. This observation concludes the proof.
Due to the symmetry of integrals in (17), the same bound as (18) applies to the even sign combination . This is not at all surprising, as it is a generic feature of other criteria for continuous variables like those for variances Duan et al. (2000); Mancini et al. (2002) or entropies Walborn et al. (2009); Saboia et al. (2011); Tasca et al. (2013). However, as in the case of the variance and entropic-based inequalities, for the same sign these are true uncertainty relations necessarily obeyed by all bipartite quantum states. At the same time, the corresponding inequalities with different signs need to be satisfied only by separable states. They are also maximally violated by the EPR state, given by normalized versions of and .
Even though the grid-geometry provided by the periodic analyzers chosen does not allow for an immediate validation of the first assertion (UR for the same signs), one can easily verify the second one. By a direct calculation one can see that for the EPR state and . The sum of both quantities is equal to the upper bound in Eq. (10). Thus, for adequate choice of parameters the EPR state violates inequality (18).
V Experimental scheme
In the experimental part of our investigations we use an SPDC source to generate photon pairs entangled in their transverse spatial variables. As explained in the previous section, we probe their transverse position and momentum correlations using spatial mask analyzers defined by periodic aperture functions.
Our experimental setup is sketched in Fig. 2(a). We adjust the SPDC source, pumped by a nm He-Cd laser, to emit frequency-degenerate down-converted photons at nm in a collinear configuration. The down-converted photons are split by a : beam splitter and directed to spatial light modulators (SLM) that are programmed to display the periodic apertures (4) in the horizontal direction , as illustrated in Fig. 2(b). A switchable lens system placed before the beam splitter is used to produce either the image or the Fourier transform of the SPDC source on to the SLMs. The imaging system is characterised by an optical magnification of , while the optical Fourier transform system has an effective focal length mm. The optical magnifications both for image plane (IP) and far-field (FF) configurations were chosen in order to maximize the spatial resolution of our measurements while keeping the down-converted beams enclosed within the SLM panel. Our SLMs (HOLOEYE PLUTO) have an active area of mmmm with a high-resolution array of pixels of width m. The photons reflected from the SLMs are lens-coupled to multi-mode optical fibres connected to single-photon avalanche diodes (SPAD). The spatial masks displayed on the SLMs work as aperture functions, reflecting the photons incident upon the white stripes (where ) and discarding the photons incident upon the black region (where ). In our experiment, the joint transmission of the photons through the analyzers are measured as coincidence counts over a sampling time of s. To convert the photo-detection rates to transmission probabilities we also record the overall joint detection rates. This measurement procedure is realised independently for each configuration (position and momentum) and analyzer geometry tested in our experiment.
To maximise the joint transmission through the periodic analyzers, we perform a calibration procedure to match the relative displacement of the analyzers with the symmetry of the EPR-like correlations displayed by the SPDC photons – position correlation or momentum anti-correlation Howell et al. (2004). According to the periodic aperture definition (4), maximal coincidence counts in position measurements are obtained using a relative displacement of the analyzers given by (mod ), whereas momentum measurements require (mod ). The result of this calibration can be visualised in Fig. 3, where we plot the measured transmission probability of the photons as a function of analyzer’s displacement. All displayed measurements were acquired using an analyzer geometry given by , and each set of data points (plotted in different colours) represent five different displacements of analyzer . Note that in the symmetric case, for which , the upper bound (10) derived in this paper simplifies to the form
In the measurements presented in Fig. 3 (a) we used a periodicity of mm ( pixels) to define the apertures displayed on the SLMs. This SLM periodicity led to analyzer periodicities –related to the position and momentum variables at the source– of mmmm and mmmm. In this case the joint transmission of the photons through the periodic apertures were independent of the analyzer displacement. In other words, these periodicities lead to bin widths and that are too small to capture the spatial correlations of the photons. On the other hand, the joint transmission measurements plotted in 3 (b) present peaks that depend on the displacement of analyzer A. This type of conditional displacment is somewhat typical in experiments that exploit the EPR-type correlation of the photons. In this latter case, an SLM periodicity times larger was used.
Over the scanning distance of one analyzer period, one coincidence peak is expected for each set of data points, the position of which depends on the displacement of the analyzer and the symmetry of the correlations displayed at the detection plane (position or momentum). As seen in plots (b) of Fig. 3, the transmission peak for position measurements occurs whenever , whilst the peak transmission for momentum occurs for (since in these measurements). The strength of the spatial correlations is inferred from the visibility of the peaks, whose shape is approximately a triangular function resembling the auto-correlation function of a slit of width or : ideal EPR correlations would generate perfect triangular peaks reaching the maximum transmission probability of . Since the five different displacements used for analyzer are with , we can use the peak values of the transmission probabilities to calculate the average joint transmission probabilities (17) involved in the entanglement criteria (18):
where is the number of samples ( for the current measurements). Indeed, the upper bound (20) of inequality (18) calculated for the analyzer parameters used in these measurements [Fig. 3-(b)] limits the sum of the average transmission probabilities in position and momentum domain to . Our measurements provide averages of and for position and momentum, respectively, leading to a violation of the criteria (18),
by over standard deviations. The presented errors are calculated assuming Poissonian statistics for the measured coincidence counts.
Vi Experimental Results
Prompted by the successful experimental verification of spatial entanglement exemplified in (23), we now examine the effectiveness of the derived entanglement criteria (18) for a wider set of periodic analyzer geometries. In the symmetric case, as studied in the last section, the function (20) upper-bounding the sum of the average transmission probabilities depends solely on the product of the analyzer periodicities () and on the ratio . The red solid line in Fig. 4 displays for as a function of . Rather than a simple shape, presents an intricate functional dependence on , including an oscillatory structure for (see inset). Interestingly, assumes its maximal possible value of [see Eq. (10)] whenever for all , shown as the upper dashed line for in Fig. 4. This is in full agreement with the fact that compatible observables do exist in periodic coarse-graining structures with these particular combinations of periodicities in position and momentum domains Tasca et al. (2018); Busch and Lahti (1986); Reiter and Thirring (1989). Obviously, no useful entanglement tests exist in this case. On the other hand, the lower dashed line indicates the value that represents the average joint transmission achieved with perfect correlation in one domain () and the absence of correlation in the conjugate domain (). This construction of maximal correlations in only one basis is typical from entanglement criteria for quantum systems of finite dimension Spengler et al. (2012). As seen from the inset, the lowest values attained for do not exactly reach this minimum in general (we find that does saturate to this minimum for ), but lie slightly above.
Our experimental data are plotted as blue dots in Fig. 4. Using the SPDC setup described in the last section, we recorded a series of transmission probability measurements while varying the analyzer periodicities displayed on the SLMs from mm up to mm. The dark-blue data points with error bars correspond to position and momentum measurements taken with matching SLM periodicities (in IP and FF configurations), whilst the light-blue dots represent the sum of the average transmission probabilities for all other combination of analyzer periodicities. As expected, both and increase as a function of the analyzer periodicity, leading to violations of the entanglement criteria for sufficiently large periods. Also, we obtain when using the smallest SLM periodicity of mm. This is an evidence that the correlation width of the photons Schneeloch and Howell (2016) is greater than (or of the order of) the associated bin width, which at the SLM correspond to m (with ). Evidently, the EPR-like spatial correlations of the photons are not sharp enough to allow for entanglement verification with the product lying in the range shown in the inset of Fig. 4.
The optimal analyzer geometry leading to successful entanglement verification is highly dependent on the spatial correlations; both the periodicity and the bin width can be adjusted to optimise entanglement detection. For the EPR-like spatial correlations generated in our experiment, the most relevant parameter to maximise the joint transmission is the bin width of the periodic analyzer. To show this, we compare and measured for four different analyzer geometries ( and ) as a function of the bin width. Since the maximal value allowed also depends on , we normalise the measured quantities with those associated with ideal EPR correlations: . As seen in plots (a) and (b) of Fig. 5, the same trend is observed for the average joint transmissions as a function of the bin width, regardless of the parameter . It is also possible to recognise that the far-field correlations in our setup are stronger than those in the image plane of the source (note that the plots abscissae are the physical bin widths of the apertures displayed on the SLMs). Since the largest periodicity applied to the SLMs (mm) was the same for all analyzer geometries, the largest used bin width is smaller for larger .
Each data point in plot (c) of Fig. 5 represents an entanglement test calculated from the average transmission probabilities displayed in plots (a) and (b). For these tests, we mixed only position and momentum data with the same , thus corresponding to a symmetric arrangement of analyzers. Entanglement is successfully detected according to our criteria for every data point lying above the red horizontal line representing . An optimal range of analyzer parameters leading to the greatest violations of inequality (18) is centred around . This product value corresponds, for example, to position and momentum data taken with the same SLM bin width of mm. These bin widths are wide enough to allow a large joint transmission of the correlated photons, but not so large as to unnecessarily increase via its dependence on the product . As a consequence, the success rate of entanglement detection as well as the largest achieved violation decrease with . The success rate of entanglement tests in our experiment is shown in plot (d) of Fig. 5. These success rates were calculated from entanglement tests for each and tests for each . The largest SLM periodicity used in these tests was the same for all . The lower part of the error bars represent violations by less than one standard deviation [the error as in Eq. (23)], whilst the upper part accounts for unsuccessful tests by less than one standard deviation. In total, we were able to detect entanglement in out of tests, thus achieving an overall success rate of about .
We have demonstrated that the entanglement present in a bipartite continuous variable system can be detected using a special kind of coarse-graining measurement using periodic sampling projectors. We illustrate the usefulness of the method with an experiment using twin photons from parametric down-conversion entangled in transverse spatial variables. The results have shown that the method works very well and can be easily extended to other CV systems. In comparison with the traditional method of binning the measurement variable, our method presents the advantage of working with higher signal to noise ratio. This advantage comes from the periodicity of the used analyzers that allows the light to be transmitted through more than one slit, while in the standard binning only one slit is allowed. Moreover, in order to achieve enough resolution, traditional binning usually requires very thin slits. In contrast with this requirement, our method was shown to prove entanglement even using slit widths that would be considered very large for the standard binning procedure.
The reported criteria may be a convenient and robust test of entanglement present in non-trivial CV states that itself present periodic phase space structures Neves et al. (2005); Carvalho et al. (2012); Barros et al. (2017). It is worth noting that our results are applicable to any quantum continuous variable, and can be used even if a complete CV measurement has been made, as it is possible to post-process the measured distributions into the necessary periodic binning structure.
Acknowledgements.D.S.T would like to thank D. Giovannini, M. Krenn, G. H. Aguilar, O. J. Farias and F. de Melo for stimulating discussions. This research was funded by the Brazilian funding agencies FAPERJ, CAPES, CNPq and the National Institute for Science and Technology - Quantum Information. Ł.R. acknowledges financial support by grant number 2014/13/D/ST2/01886 of the National Science Center, Poland. M.J.P. and R.S.A acknowledge financial support from the UK EPSRC under a Programme Grant, COAM award number EP/I012451/1.
Appendix A Concise proof of the uncertainty relation for periodic analyzers
The formula (8a) based on the Fourier expansion of the spatial mask function trivially leads to ():
The integral with respect to present in (9) shall only involve the last factor, namely . Since we integrate over the whole period, the integration provides . With this result we can perform the sum with respect to and obtain
The same line of reasoning applies to the second term associated with the momentum variable. As a result, we find that the left hand side of (9) is equal to
In the last step, we maximize the above quantity with respect to both characteristic functions, independently for each value of . The optimization procedure is subject to the constraint
which is a particular instance of the uncertainty relation for the characteristic functions derived in Rudnicki et al. (2016). Also, from the sole definition of the characteristic function, we know that and . In mathematical terms, we are thus maximizing the function with respect to . All are real and non-negative, and we have the constraints: , and with . Since the above inequalities define a convex set, the maximum is obtained at the edge points. If then the maximizing edge is and , while when we shall pick up and . As a result (considering both cases together) we obtain the first desired inequality (exact formula for ).
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