Quantum non-Gaussian Depth of Single-Photon States
Ivo Straka, Ana Predojević, Tobias Huber, Lukáš Lachman, Lorenz Butschek, Martina Miková, Michal Mičuda, Glenn S. Solomon, Gregor Weihs, Miroslav Ježek, Radim Filip
QQuantum non-Gaussian Depth of Single-Photon States
Ivo Straka, ∗ Ana Predojevi´c, Tobias Huber, Luk´aˇs Lachman, Lorenz Butschek, MartinaMikov´a, Michal Miˇcuda, Glenn S. Solomon, Gregor Weihs, Miroslav Jeˇzek, and Radim Filip Department of Optics, Palack´y University, 17. listopadu 1192/12, 771 46 Olomouc, Czech Republic Institut f¨ur Experimentalphysik, Universit¨at Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria Joint Quantum Institute, National Institute of Standards and Technology,and University of Maryland, Gaithersburg, Maryland 20849, USA
We introduce and experimentally explore the concept of quantum non-Gaussian depth of single-photon states with a positive Wigner function. The depth measures the robustness of a single-photonstate against optical losses. The directly witnessed quantum non-Gaussianity withstands significantattenuation, exhibiting a depth of 18 dB, while the nonclassicality remains unchanged. Quantumnon-Gaussian depth is an experimentally approachable quantity that is much more robust than thenegativity of the Wigner function. Furthermore, we use it to reveal significant differences betweenotherwise strongly nonclassical single-photon sources.
PACS numbers: 42.50.Ar, 42.50.Dv, 03.65.Ta
Introduction.—
Single-photon states are important re-sources used in quantum computation and informationprocessing [1]. Furthermore, they are direct evidence ofthe quantum nature of light [2]. This nature is manifestedvia various quantum features of the single-photon state.Some of these features rely on distinguishing the single-photon state from statistical mixtures of certain quan-tum states, such as coherent or Gaussian states. Suchconvex sets then serve to define the respective quantumfeatures of a single photon. Specifically, nonclassicalitymeans the state is inexpressible as a statistical mixtureof classical coherent states [3, 4]. In addition to this, thestate may be inexpressible as a mixture of pure Gaussianstates, thus exhibiting quantum non-Gaussianity [5, 6](note the difference from classical non-Gaussianity [7, 8]).Even further, the state can have a negative Wigner func-tion [9, 10]. These quantum features define three setsof quantum states, each being a subset of the previousone (see Fig. 1). Depending on its qualities, a realisticsingle-photon source will produce a state that can belongto any of these subsets. In such a case, the necessary andsufficient condition for nonclassicality is represented byan infinite hierarchy of criteria [11, 12].Here we perform a conceptual evaluation of quantumfeatures of light. We use the important fact that any op-tical application inevitably includes losses. We do not,however, examine the role of losses in a specific appli-cation. Instead, we aim to test how well an experimen-tally generated single-photon state keeps the propertiesof an ideal single photon. In this protocol-independentapproach, the endurance of quantum features with re-spect to losses becomes imperative [13–16].The nonclassicality depth is defined as the maximumattenuation of a nonclassical state, at which the state isstill able to preserve the nonclassicality [17, 18]. To de-termine this depth based on its definition, one requireshomodyne tomography [10] and quantum estimation ofthe entire density matrix of all emitted modes of light [19]. In practice, such a measurement is only feasible for afixed low number of photons in a few well-defined modes.However, many experimentally generated single-photonstates exhibit a complex multimode structure. Moreover,they are influenced by multiphoton contributions, whichare systematically generated and/or coupled from the en-vironment as noise. Although these contributions alonemight not destroy the quantum features of the generated
FIG. 1: The classification of quantum state sets used in ourdiscussion, where a ⊂ b ⊂ c ⊂ d . a is the set of stateswith negative Wigner function. All states in the set b areguaranteed to be quantum non-Gaussian. Likewise, all statesin the set c are nonclassical. The set d contains all states ingeneral. Equivalently, all classical states are contained in thecomplement ¯ c and all Gaussian mixtures are in ¯ b . The bordersof b and c therefore represent NC and QNG witnesses.The points and non-solid lines represent various realisticquantum states and their respective paths under attenuation.These quantum states can approach vacuum inside differentsets. An ideal single-photon state ( P | (cid:105)(cid:104) | + P | (cid:105)(cid:104) | , reddotted line) is an extremal case of an infinitely robust QNGstate. Other realistic states may exhibit infinite NC depth orleave nonclassical states altogether (dashed lines). The greendot-dashed line represents realistic single-photon states withpositive Wigner function, as generated experimentally. a r X i v : . [ qu a n t - ph ] N ov state, they will affect the state and can substantially limitthe depth of the respective feature. In Ref. [6], increaseddetection noise was simulated and the results show thatit can indeed have a destructive effect on quantum non-Gaussianity. Therefore, noise is a major limiting factorand multiphoton contributions need to be taken into ac-count. Unfortunately, such contributions are commonlyhard to fully estimate and characterize.As alternatives to full quantum state estimation, di-rectly measurable witnesses can detect the quantum fea-tures of multimode states. The witnesses of nonclassi-cality (NC) [3, 4] and quantum non-Gaussianity (QNG)[5, 6, 20–23] allow one to experimentally determine alower bound on their respective depths. In this Let-ter, we show a direct measurement of the lower boundon the depths of both NC and QNG for three differentsources of single-photon states: two are based on sponta-neous parametric down-conversion in a nonlinear crystal,and one is a single quantum dot. Using this approach,we compare the quantum features of these very differ-ent single-photon sources. Our measurements show anextreme robustness of NC for single-photon states withpositive Wigner function. Our results further prove thatQNG is a robust resource for future quantum applica-tions of single-photon states, as opposed to the fragilenegativity of the Wigner function. We demonstrate pre-serving the QNG of a single-photon state for up to 18 dBof attenuation. NC and QNG witnesses.—
Both NC and QNG of aquantum state ρ can be recognized using the criteria de-rived in Refs. [5, 6, 24], which are defined using the prob-abilities P = (cid:104) | ρ | (cid:105) and P = (cid:104) | ρ | (cid:105) . Reference [24]shows that a sufficient condition for quantum nonclassicalstates is that P > − P ln P . Since the error probabilityof multiphoton contributions P = 1 − P − P , the NCand QNG criteria can be rewritten in terms of P and P . Complete knowledge of the photon statistics is notrequired, because P and P are sufficiently informativeparameters describing the main statistical properties oflight emitted by a single-photon source. These param-eters can be efficiently estimated [6] using an autocor-relation measurement [4, 25]. An alternative approach,directly extending the anticorrelation parameter [25] tra-ditionally used for the characterization of NC, has beenrecently proposed in Refs. [24, 26]. Depth of quantum features.—
The lower bound on thedepth of a quantum feature of a quantum state can be op-erationally defined as the maximal attenuation at whichthat quantum feature is still detectable. We proposephysical variable attenuation in front of an autocorrela-tion measurement and application of the criteria for NCand QNG on the measured data, which is discussed indetail in [6, 24] and plotted in Fig. 3. The attenuation ofthe generated state transforms the photon-number statis-tics P n to P (cid:48) n = (cid:80) ∞ m = n (cid:0) mn (cid:1) T n (1 − T ) m − n P m , where T isthe variable transmittance of the attenuator. Because P n before the attenuation cannot be estimated by the auto-correlation measurement, we implement the attenuationexperimentally to determine the depth.Let us assume an ideal single-photon state ˜ ρ = η | (cid:105)(cid:104) | + (1 − η ) | (cid:105)(cid:104) | , influenced solely by attenuation.It is straightforward to show that such a state exhibits infinite NC and QNG depths for any η >
0, while thenegativity of the Wigner function vanishes for η ≤ . ρ . Robust QNG as a single-photon benchmark.—
For asingle-photon source of sufficiently high quality, the gen-erated states can exhibit a remarkably large QNG depthdespite their multimode background noise, as has beenvery recently predicted in Ref. [26]. Then, these realisticstates are highly robust single-photon states, approach-ing the ideal state ˜ ρ . Such states are typically generatedby high-quality single-photon sources, where the gener-ated state can be very well approximated by a mixture ρ ≈ (1 − P − P ) | (cid:105)(cid:104) | + P | (cid:105)(cid:104) | + P | (cid:105)(cid:104) | , (1)where P (cid:28) P and we do not distinguish between pho-tons in different modes. Using the parametrization of P and P , the criterion for NC is approximately given by P < P , whereas the criterion for QNG can be ap-proximated by P < P [26]. After the attenuation,we neglect the transfer from the state | (cid:105)(cid:104) | to | (cid:105)(cid:104) | ,simplifying our description. Explicitly, we use a lowerbound T P on P (cid:48) ≥ T P , which is safe from false QNGwitnessing, and P (cid:48) = T P . We obtain an approxi-mative attenuated state ρ (cid:48) ≈ (1 − T P − T P ) | (cid:105)(cid:104) | + T P | (cid:105)(cid:104) | + T P | (cid:105)(cid:104) | .The depth of ρ (cid:48) strongly depends on the choice of thequantum feature. Under the approximation P (cid:28) P ,the negativity of the Wigner function demands T > (2 P ) − , which is very challenging to fulfill for single-photon sources. On the other hand, if the NC con-dition P < P is satisfied before the attenuation,then P (cid:48) < P (cid:48) is fulfilled after the attenuation aswell. Therefore, the NC withstands any attenuation with T >
0, which means that the state exhibits infinite NCdepth. In contrast, QNG is observable for attenuatortransmittances [26]
T > P P , P (cid:28) P . (2)Note that an arbitrarily small multiphoton contribution P makes the QNG depth finite. If P is substantiallylarger than P , the QNG depth can still be very large,even though the state has positive Wigner function. Ourgoal is to experimentally find such QNG states, whichcan be very good resources for quantum technology. Experimental schemes.—
In our work we used threedifferent systems to generate single-photon states. Ofthese, two were based on spontaneous parametric down-conversion (SPDC) in a nonlinear crystal. The thirdsystem was an InAs/GaAs single quantum dot. Thefirst SPDC source contained a 2-mm-thick type-II BBOcrystal that was used in a collinear configuration andwas operated in the continuous-wave (cw) regime. Here,the pump power was 90 mW while its wavelength was405 nm. Correlated photons were spectrally filtered toa bandwidth of 2.7 nm [6]. The second SPDC sourceproduced entangled photon pairs. It contained a 15-mm-long type-II ppKTP nonlinear crystal embedded ina Sagnac-type interferometer loop [27]. This source waspumped by a 2-ps pulsed laser light of 404 nm wave-length and 80 µ W power per loop direction. The quan-tum dot sample contained low density self-assembledInAs/GaAs quantum dots embedded in a planar micro-cavity. The excitation light was derived from a tunableTi:sapphire laser that could be operated in picosecond-pulsed (82 MHz repetition rate) or continuous-wave mode[28]. Here, we generated two data sets, one in resonanttwo-photon excitation using the pulsed mode and theother in above-band continuous-wave mode.The measurement scheme was a triggered autocorre-lation shown in Fig. 2. Variable attenuation was intro-duced by moving a blade in the beam. Data acquisitionwas carried out by a time-tagging module, which storedarrival times of every detection event. The trigger de-tector conditioned the detections in the signal arm: anydetection within a coincidence time window from a trig-ger detection was considered a coincidence. From these,we measured the probabilities p , p , p , which are es-timators of P (cid:48) , P (cid:48) , P (cid:48) [6]. These parameters allowed usto construct the witnesses for NC and QNG states. Experimental results.—
In Fig. 3, we compare the mea-surement results obtained from all three single-photon
FIG. 2: The autocorrelation measurement scheme. The her-alded state in the signal arm is attenuated, then p and p are measured [6]. sources. Here, the nonclassicality witness p < p is shown as a solid black line while the QNG witness p < p as a solid blue line. For each single-photonsource, we show the results obtained under systemati-cally varied attenuation, given in units of 10 log (1 − T )dB. Additionally, we give a theoretical model of the in-duced losses (dot-dashed lines). These models served usto evaluate the theoretical value of the QNG depth foreach source, as given by Eq. (2). In addition, we ex-perimentally confirmed the QNG character of the statessubjected to a certain maximum attenuation that is theexperimentally proven QNG depth. Since it is challeng-ing to experimentally attenuate the state until it is placedexactly on the border of Gaussian mixtures, the provenQNG depth is always lower than the theoretical predic-tion.For the pulsed SPDC source (cyan triangles) we esti-mated the QNG depth to be 14.5 dB and measured 10.8 (cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236) (cid:224)(cid:224)(cid:224)(cid:224) (cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:72) p (cid:76) L og (cid:72) p (cid:43) (cid:76) FIG. 3: Estimated probabilities of heralded single-photonstates, each series representing various attenuations of a par-ticular state. Full diamonds denote the cw SPDC source: or-ange —coincidence window 2 ns; red—low pump, coincidencewindow 2 ns. Cyan triangles denote the pulsed SPDC source.Square markers denote the quantum dot: purple squares—above-band excitation, cw pump; blue squares—resonant ex-citation, pulsed pump. The dot-dashed lines represent theo-retical prediction of attenuation from the initial point. Thedashed red line is the limit of dark counts for the red atten-uation data. There are two lower bounds for p ; the solidblack line is a bound for classical states and the blue line isfor Gaussian mixtures. Quantum states below these lines areNC or QNG, respectively. Error bars are determined by errorpropagation from event count errors [6]. Horizontal error barsare smaller than plot points. dB. The cw SPDC source measured with 2-ns coincidencewindow (orange diamonds) yields a theoretical depth of19.6 dB and a proven depth of 17.9 dB. Red diamondsstand for weakly pumped cw SPDC. The expected depthis 31.8 dB while the measured value is 18 dB. The stategenerated by a quantum dot excited above-band (purplesquares) shows only nonclassicality and cannot be wellcompared to SPDC states. With resonant pulsed excita-tion (blue square), the quantum dot state exhibits QNGcharacter and the theoretical depth is 5.6 dB. Empty bluesquares show additional quantum dot states measuredwith different collection efficiencies. In order to measurethe QNG depth for the quantum dot, the measurementtime would exceed the stability of the system. Measurable depth of NC and QNG.—
As predicted inthe discussion above, Fig. 3 shows that the NC depthis robust for all single-photon sources. Both the the-oretical models and the directions of the experimentalpoints are parallel to the NC border. In particular, wedemonstrated that even with 2 orders of magnitude ofattenuation, the data points do not exhibit any trendof approaching the NC border. At very high attenua-tion, dark counts eventually limit the signal-to-noise ra-tio. In addition, the required long integration times leadto systematic errors caused by instabilities. Both effectslimit the level of attenuation above which no informa-tion about the original states can be obtained anymore.This places the NC depth of the single-photon sourcesbeyond measurement. In contrast, the border of QNGcan be experimentally reached. Intuitively, this can beunderstood as follows: the QNG border gives a cubicrelation between p and p , while attenuation behavesquadratically. This makes the QNG depth a measur-able feature for single-photon states, and, consequently,a convenient benchmark for single-photon sources. Fur-thermore, QNG shows high robustness for SPDC sources,which proves that the generated states can be consideredhigh-quality single-photon states defined in Eq. (1). Experimental optimization of SPDC sources.—
Opti-mization of the QNG depth can be achieved via cer-tain experimental parameters, depending on the source.These parameters include pump power, the width of thecoincidence window, SPDC efficiency and losses. In-evitable losses in the experimental setup decrease theQNG depth for all types of sources. Optimization of thecoupling or collection efficiencies is therefore essential, aswell as high quantum efficiency of the detectors.The impact of the other parameters on the QNG depthis not straightforward to see. Previously, the effects ofsome experimental factors were examined using a QNGwitness quantification ∆ W [6]. The witness, however,intrinsically differs from the QNG depth.Generally, QNG depth increases with lower pumppower and conversion efficiency. For the cw pump, thecoincidence window has an optimum width depending onthe detector time resolution. For the pulsed pump, the coincidence window is upper bounded by repetition rateand lower bounded by the photon lifetime and detectortime resolution. Moreover, when considering a cw sourceand a comparable pulsed source, the cw source intrinsi-cally yields higher QNG depth. More detailed discussioncan be found in the Appendix [29]. The quantum dot source.—
Quantum dots generatefundamentally different states of light than SPDC. Theyrely on formation of an electron-hole pair and subsequentrecombination that results in photon emission. Specifi-cally, the recombination of the biexciton gives rise to twospectrally distinct photons emitted in a time-ordered cas-cade. In our measurements, the first photon of the cas-cade serves as a trigger for the second photon. If we con-sider resonant excitation by a picosecond laser, only thetransition between the vacuum state and a single biex-citon is possible. The decay time of the biexciton is 2orders of magnitude longer than the pump pulse. There-fore, the probability to systematically generate a multi-photon state by a single pulse is very low. This is anextremely valuable asset, which potentially makes quan-tum dots much closer to an ideal single-photon sourcethan SPDC.In practice, however, there is always some backgroundnoise present in the measurement, that is responsible forthe p contribution. The quantum dot state in Fig. 3(blue square) shows that this noise is stronger than thenoise of an attenuated SPDC single photon (red dia-mond) operated in the cw regime. The QNG depthcan be improved by increasing the collection efficiency[30, 31]. As a result, one can expect an increase in p with p remaining constant. The three blue-square points inFig. 3 show measured states with various degrees of effi-ciency. If the collection efficiency improves by a factor of9, the quantum dot would yield states with higher QNGdepth than the cw SPDC state. The results presented inRef. [30] indicate that by embedding the quantum dot ina micropillar cavity, one can reach a factor of 16 improve-ment. For such collection efficiency, the QNG depth mayexceed 40 dB and surpass the QNG depth of the SPDC. Conclusion.—
We experimentally verified high QNGdepths of various single-photon states. This is in strongcontrast to the fragility of the Wigner function negativ-ity; therefore, our results demonstrate that QNG is arobust quantum resource.It can be seen that SPDC produces single-photonstates with extremely robust QNG depth. The dataprove that with commonly used single-photon sources,quantum non-Gaussianity can be preserved after prop-agating the photon through 8 km of fiber, assuming 4dB/km losses for the wavelength of 0.8 µ m. For similarsources at telecom wavelength, the range is about 180km.When compared to a quantum dot, SPDC can gener-ate much more robust states at present, but its noise isfundamentally unavoidable. Further improvement of thetechnical aspects of quantum dot sources could lead tosingle-photon states more robust than those generated bySPDC.This research has been supported by the Czech ScienceFoundation (13-20319S). The research leading to theseresults has received funding from the European UnionSeventh Framework Programme under Grant AgreementNo. 308803 (project BRISQ2). L.L. thanks the Czech-Japan bilateral project LH13248 of the Ministry of Edu-cation, Youth, and Sports of Czech Republic. M. Mikov´aacknowledges the support of Palack´y University (IGA-PˇrF-2014-008). A.P. acknowledges the support of theUniversity of Innsbruck, given through Young ResearcherAward. Additionally, the work at the University of Inns-bruck was partially supported by the European ResearchCouncil, project ”EnSeNa” (257531). G.S.S. acknowl-edges partial support through the NSF Physics FrontierCenter at the Joint Quantum Institute (PFC@JQI). APPENDIXThe effects of experimental parameters
The two SPDC sources we used were operated in dif-ferent regimes – with pulsed and CW pumping. Never-theless, the photodistribution of the generated two-modestate before losses is the same: P ( m, n ) = δ mn (1 − g ) g n .The temporal width of the state is given by the coinci-dence window τ . P ( m, n ) is the probability of generating m and n photons in the two respective modes, δ mn is theKronecker delta and g is the gain, which is proportionalto pump power and SPDC efficiency. The gain is respon-sible for the systematically generated component in theheralded single-photon state, since p ∝ g for g (cid:28) p < p and thedepth is thus estimated as T min = p p , g needs to beminimal. It is usually well under experimental control,but cannot be decreased arbitrarily due to practical lim-itations.For typical CW-pumped sources, the detected state istemporally multi-mode, which effectively means p ∝ τ ,where τ is much larger than the temporal bandwidth ofthe biphotons. It follows that the coincidence windowneeds to be optimized, too. Its width is lower-boundedby the resolution time of the detectors. If the coinci-dence window is reduced below that limit, it effectivelyintroduces a loss to the state and decreases the QNGdepth. On the other hand, if the coincidence window isexcessively large, the higher p contribution decreasesthe QNG depth as well. Therefore, there is an optimumcoincidence window that maximizes the QNG depth.The coincidence window plays a minor role in thepulsed regime. There, it has no effect on p , assumingthe coincidence window is not shorter than the lifetimeof the photons and not longer than the delay between pulses. Thus, there is a fixed pump energy contribut-ing in each coincidence window. Analogously to the CWregime, if the coincidence window is shorter than thedetector resolution time, the effective loss decreases theQNG depth. Therefore, as long as the coincidence win-dow remains within the aforementioned limits, it has noeffect on the QNG depth. Comparing CW and pulsed pumping regime forSPDC
Let us consider two cases: a CW and a pulsed single-photon source with an identical frequency of heraldedstate generation – a heralding rate. Let both sources haveidentical average pump power ¯ S , the same overall conver-sion efficiency, number of modes and effective losses inthe set-up. In the low-gain approximation g (cid:28)
1, suchtwo sources would have similar heralding rate and p , butdifferent p . Namely, the p CW2+ ≈ µ ¯ Sτ and p pul2+ ≈ µ ¯ Sν ,where ν is the repetition rate of the pulsed pump, τ isthe width of the coincidence window, and µ a commonproportionality constant. Since the QNG depth of theheralded state is given by T min = p p , the ratio of theminimum transmittances T CWmin /T pulmin ≈ τ ν . The coin-cidence window can be minimized to the limit of thedetector resolution, typically ∼ − s. The repetitionrate of the pump laser is not a very flexible parameter( ∼ s − ) and cannot be routinely adjusted. That givesa significant difference in the QNG depth in favor of CW-pumped SPDC sources, assuming similar heralding rates.Furthermore, the CW source often has significantly moremodes than the pulsed, which can effectively lead to anadditional factor of ∼ in T CWmin /T pulmin . There are waysfor pulsed sources to effectively increase the repetitionrate and decrease the p contribution [32]. However,for repetition rates approaching detector resolution, thepulsed source would approach the CW source, but wouldnever yield a larger QNG depth. ∗ Electronic address: [email protected][1] P. Kok and B. W. Lovett,
Introduction to OpticalQuantum Information Processing (Cambridge UniversityPress, Cambridge, UK, 2010).[2] A. Einstein, Ann. Phys. , 132 (1905).[3] R. J. Glauber, Phys. Rev. , 2766 (1963).[4] H. J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev.Lett. , 691 (1977).[5] R. Filip and L. Miˇsta, Jr., Phys. Rev. Lett. , 200401(2011).[6] M. Jeˇzek, I. Straka, M. Miˇcuda, M. Duˇsek, J. Fiur´aˇsek,and R. Filip, Phys. Rev. Lett. , 213602 (2011).[7] M. G. Genoni, M. G. A. Paris and K. Banaszek, Phys.Rev. A , 042327 (2007). [8] M. G. Genoni, M. G. A. Paris, and K. Banaszek, Phys.Rev. A , 060303(R) (2008).[9] E. P. Wigner, Phys. Rev. , 749 (1932).[10] A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J.Mlynek, and S. Schiller, Phys. Rev. Lett , 050402(2001).[11] T. Richter and W. Vogel, Phys. Rev. Lett. , 283601(2002).[12] T. Kiesel, W. Vogel, V. Parigi, A. Zavatta, and M.Bellini, Phys. Rev. A , 021804(R) (2008).[13] R. Ursin et al., Nature Phys. , 481 (2007).[14] A. Fedrizzi, R. Ursin, T. Herbst, M. Nespoli, R. Prevedel,T. Scheidl, F. Tiefenbacher, T. Jennewein, and A.Zeilinger, Nature Phys. , 389 (2009).[15] E. Bimbard, R. Boddeda, N. Vitrant, A. Grankin, V.Parigi, J. Stanojevic, A. Ourjoumtsev, and P. Grangier,Phys. Rev. Lett. , 033601 (2014).[16] J.-i. Yoshikawa, K. Makino, S. Kurata, P. van Loock, andA. Furusawa, Phys. Rev. X , 041028 (2013).[17] C. T. Lee, Phys. Rev. A , R2775 (1991).[18] C. T. Lee, Phys. Rev. A , 6586 (1992).[19] J. ˇReh´aˇcek and M. Paris, Quantum State Estimation (Springer-Verlag, Berlin, 2004).[20] M. Jeˇzek, A. Tipsmark, R. Dong, J. Fiur´aˇsek, L. Miˇsta,Jr., R. Filip, and U. L. Andersen, Phys. Rev. A ,043813 (2012).[21] A. Predojevi´c, M. Jeˇzek, T. Huber, H. Jayakumar, T.Kauten, G. S. Solomon, R. Filip and G. Weihs, Opt. Express , 4789 (2014).[22] M. G. Genoni, M. L. Palma, T. Tufarelli, S. Olivares, M.S. Kim, and M. G. A. Paris, Phys. Rev. A , 062104(2013).[23] H. Song, K. B. Kuntz, and E. H. Huntington, New J.Phys. , 023042 (2013).[24] R. Filip and L. Lachman, Phys. Rev. A , 043827(2013).[25] P. Grangier, G. Roger, and A. Aspect, Europhys. Lett. , 173 (1986).[26] L. Lachman and R. Filip, Phys. Rev. A , 063841(2013).[27] A. Predojevi´c, S. Grabher, and G. Weihs, Opt. Express , 25 022 (2012).[28] H. Jayakumar, A. Predojevi´c, T. Huber, T. Kauten, G.S. Solomon, and G. Weihs, Phys. Rev. Lett. , 135505(2013).[29] See the Appendix for details on experimental parametersand cw-pulsed comparison.[30] A. Dousse, J. Suffczy´nski, A. Beveratos, O. Krebs, A.Lemaˆıtre, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart,Nature (London) , 217 (2010).[31] O. Gazzano, S. M. de Vasconcellos, C. Arnold, A. Nowak,E. Galopin, I. Sagnes, L. Lanco, A. Lemaˆıtre, and P.Senellart, Nat. Commun. , 1425 (2013).[32] M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G.White, Opt. Express19