arXiv:1007.2510v2 [quant-ph] 19 Jul 2010
Experimental Demonstration of a Heralded Entanglement
ClaudiaWagenknecht1†, Che-MingLi1,2†, AndreasReingruber1, Xiao-HuiBao1,3, AlexanderGoebel1,
Yu-Ao Chen1,3, Qiang Zhang1, Kai Chen3⋆, and Jian-Wei Pan1,3⋆
1Physikalisches Institut, Ruprecht-Karls-Universit¨ at Heidelberg, Philosophenweg 12, 69120 Hei-
2Department of Physics and National Center for Theoretical Sciences, National Cheng Kung Uni-
versity, Tainan 701, Taiwan
3Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern
Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
†These authors contributed equally to this work.
⋆e-mail: email@example.com; firstname.lastname@example.org
The heralded generation of entangled states is a long-standing goal in quantum informa-
tion processing, since it is indispensable for a number of quantum protocols1,2. Polarization
entangled photon pairs are usually generated through spontaneous parametric down con-
version (SPDC)3whose emission, however, is probabilistic. Their applications are generally
accompanied with post-selection and destructive photon detection. Here, we report a source
of entanglement generated in an event-ready manner by conditioned detection of auxiliary
photons4. This scheme profits from the stable and robust properties of SPDC and requires
only modest experimental efforts. It is flexible and allows to significantly increase the prepa-
ration efficiency by employing beam splitters with different transmission ratios. We have
achieved a fidelity better than 87% and a state preparation efficiency of 45% for the source.
This could offer promising applications in essential photonics-based quantum information
tasks, and particularly enables optical quantum computing by reducing dramatically the
Quantum entanglement is one of the key resources in quantum information and quantum
foundation. Besides its fundamental interest to reveal fascinating aspects of quantum mechanics,
they are also crucial for a variety of quantum information tasks1,2. In particular, photonic entan-
gled states are robust against decoherence, easy to manipulate and show little loss, both in fiber
and free-space transmission, and thus are exceptionally well suitable for long distance quantum
communication and linear optical quantum computing6,7. Consequently, an event-ready source for
entangled photonic states is of great importance, both from the fundamental and the practical point
of views. Entanglement sources based on the probabilistic generation process of SPDC allow for
demonstrations of a number of quantum protocols, but do not permit on-demand applications, de-
terministic quantum computing and significantly limit the efficiency of multi-photon experiments.
Alternative solutions, such as the controlled biexciton emission of a single quantum dot8–10or the
creation of heralded entanglement from atomic ensembles11, face severe experimental disadvan-
tages, such as liquid-helium temperature environment and large-volume setups.
There has been considerable progress towards the demonstration of heralded photonic Bell
pairs. The scheme by Knill, Laflamme and Milburn (KLM)12provides a theoretical breakthrough
as proof that efficient quantum computing is possible with linear optics. Although the KLM
scheme allows the nearly non-probabilistic creation of entanglement, the method they use is still
intrinsically probabilistic. The fact that the KLM scheme uses a single photon source, perfect
photon-number-resolving detectors and moreover requires a large computational capacity makes it
barely accessible experimentally. The proposal of Browne and Rudolph5comprises a significant
advance in achieving experimental implementation by using photonic Bell pairs as the primary
resource and experimentally realistic detectors. Using their proposal, the number of optical op-
erations per logical two-qubit gate reduces to ∼ 100, in contrast to the original KLM scheme,
which would have ∼ 100,000 (refs 5,6,12). Central to such a dramatic improvement is the use of
a heralded entanglement source5,6.
Various ideas based on conditional detection of auxiliary photons or multi-photon interfer-
ence were recently proposed to overcome the probabilistic character of SPDC4,13–18. Following
this line, we demonstrate an experimental realization of a heralded entangled photon source by
adopting the proposal of´Sliwa and Banaszek4. This source provides a substantial advance over
the general methods by using linear optics12,13. In the experiment, we only use commercial thresh-
old single photon counting modules (SPCM) as detectors and passive linear optics. The source is
feasible to support on-demand applications, such as the controlled storage of photonic entangle-
ment in quantum memory19to realize the quantum repeater scheme20. Moreover, it is suitable to
To demonstrate the basic principle of the heralded entangled photon source, we illustrate the
scheme of´Sliwa and Banaszek4in Fig. 1. With an input of SPDC source emitted from modes a,b,
theschemeherald an entangled photonpair in c,dmodes conditionedby triggers offour photonsin
e,f modes. As an input of the optical circuit, three-pair component of the down converted photons
entangled in polarizations is utilized. The quantum state of the three-pair photon term is given by22
y− ˆ a†
where |vac? denotes the vacuum state, and ˆ a†andˆb†are the creation operators of photons in the
modes a and b. Horizontal and vertical polarization are represented by x and y, respectively. The
optical circuit (see the Methods section) transforms |Ψ3? into4:
ThefirsttermofEq.(2)iscomposedofatensorproductoftwostates: thestate|θ?t= ˆ e†
denoting one photon in each of the four trigger modes, and the maximally entangled photon pair
in the output modes
x+ ˆ c†
The normalized state |Γ?tsis a superposition of all states that do not exactly have one photon in
each of the trigger mode ˆ ex, ˆ ey,ˆfx′ andˆfy′. Hence, the scheme for heralded entanglement source
is clearly based on the fact that when detecting a coincidence of four single photons in the trigger
modes (ˆ ex, ˆ ey;ˆfx′,ˆfy′), the two photons in the output modes (ˆ cx,ˆ cy) and (ˆdx,ˆdy) nondestructively
collapse to the maximally entangled state |Φ+?s.
In the experiment, we will generate the three-pair photon states (1) by a photon source (see
the Methods section) and consequently implement the transformation of the linear optical circuit.
A schematic diagram of our experimental set-up is shown in Fig. 2, which is based on the proposal
of Ref. 4.
When taking all of experimental imperfections into account (see the Methods section), it is
crucial to evaluate the performance this source. Therefore, we have measured the state preparation
efficiency and its fidelity, where the efficiency is defined by the number of heralded photon pairs
created from the source per trigger signal. For an ideal case, one trigger signal of a fourfold single-
photon coincidence perfectly heralds one photon pair creation. In our experiment performed with
standard SPCMs, obviously, additional terms yielding triggers will thus result in a reduction of
the preparation efficiency. To overcome this obstacle, we limit their emergence by decreasing the
transmission coefficients of the beamsplitter. In this regime, the probability of transmitting more
than the minimum number of photons to the trigger becomes lower and as such, the danger of
under counting photons in the trigger detectors decreases. However, enhancing the preparation
efficiency in this way will lower the over all preparation rate.
In order to show the relation between the efficiency of state preparation and the transmission
coefficients of the partial reflecting beam splitters, we have chosen BS with three different reflec-
tion/transmission (R/T) ratios: 48.6/51.4, 57.0/43.0 and 68.5/31.5 in the experiment (Fig. 2). In
what follows we will denote them by 50/50, 60/40 and 70/30, respectively, for short. This relation
is shown in Fig. 3. The experimental efficiency can be straightforwardly represented as the follow-
ing relation by the number of triggers nt, the average detection efficiency ηsfor output states, the
number of six-fold coincidences nsamong four trigger modes and two output modes
For each experimental detection efficiencies ηsand R/T ratio: 0.129 (50/50), 0.133 (60/40) and
0.15 (70/30), the average coincidence counts (ns,nt) observed per 10 hours are: (37,9710),
(37,4940) and (14,1347), respectively. As can be seen from Fig. 3, the experimental results are
highly consistent with theoretical estimation (see Supplementary Information):
(1 − ηtT/2)2,
where ηtrepresents the average detection efficiency for trigger photons. Thus, with this setup
we have significantly improved the preparation efficiency in comparison with the one provided
by the standard procedure through SPDC. One can consider single input pulses of UV laser and
output photon pairs of SPDC as trigger signals and output states, respectively. The probability of
generatingoneentangledphotonpairperUV pulsemeans thepreparationefficiency ofthestandard
procedure through SPDC22.
To quantify the entanglement of the output photons and evaluate how the prepared state
is similar to the state |Φ+?s, we have determined the state fidelity by analyzing the polarization
state of the photons in the modes (ˆ c,ˆd) in the three complementary bases: linear (H/V ), diagonal
(+/−), and circular (R/L). For an experimental state ˆ ρ, the fidelity is explicitly defined by
F= Tr(ˆ ρ??Φ+?
4(1 + ?ˆ σxˆ σx? − ?ˆ σyˆ σy? + ?ˆ σzˆ σz?),
where |Φ+?ss?Φ+| =1
4(ˆI+ˆ σxˆ σx−ˆ σyˆ σy+ˆ σzˆ σz), ˆ σz= |H??H|−|V ??V |, ˆ σx= |+??+|−|−??−| and
ˆ σy= |R??R|−|L??L|. Eq. (6) implies that we can obtain the fidelity of the prepared state ˆ ρ by con-
secutively carrying out three local measurements ˆ σxˆ σx, ˆ σyˆ σyand ˆ σzˆ σzon the photons in the output
modes (ˆ cx,ˆ cy) and (ˆdx,ˆdy) (see the Methods section). In the experiment, we only used threshold
SPCMs to perform measurements. The experimental results are shown in Fig. 4. The experimental
integration time for each local measurement, with respect to different reflection/transmission ratio
of the BS, took about: 19 h (50/50), 17 h (60/30) and 36 h (70/30). For all three splitting ratios,
we recorded more than about 50 events of desired six-photon coincidences for each local measure-
ment: ∼ 65 (50/50), ∼ 58 (60/30) and ∼ 62 (70/30). As can be seen from Table 1, the measured
values for the fidelity are sufficient to violate CHSH-type Bell’s inequality23for Werner states by
three standard deviations. Since we only used threshold SPCMs as detectors, the measured coin-
cidences are then affected by unwanted events. In our experiment, the effect of the dark count rate
in the detectors on the six-fold coincidence is rather small. (About the dark count contribution, the
main part is that one detector is triggered by dark counts, and the other five detectors are triggered
by the down conversion photons. Given a three-pair state, the probability of generating a six-fold
coincidence count within any particular coincidence window is about S ∼ η6, whereas the leading
dark count contribution is about Sd∼ η5D, where D = ndt, ndis the average dark count rate of
detector, and t denotes the coincidence window. In our experiment, we have nd ∼ 300 Hz and
t = 12×10−9sec. Then it is clear that the dark count rate in detectors contribute a very small part
of the six-fold coincidences: Sd/S = ndt/η ∼ 2×10−5. Here η = 15% is used for the estimation.)
In conclusion, we have demonstrated a heralded source for photonic entangled states, which
is capable of circumventing the problematic issue of probabilistic nature of SPDC. Such source is
based on the well known technique of type-II SPDC, which is robust, stable and needs only modest
experimental efforts by using standard technical devices. Photon number resolving detectors are
not involved in the setup, and therefore we do not endure the restriction inherent to other schemes
for implementing heralded entanglement sources13,15. To evaluate the performance of our source,
we have measured the fidelity of the output state, and demonstrated the relation between the am-
plitude reflection coefficient of the used beam splitters and the preparation efficiency of the source.
A fidelity better than 87% and a state preparation efficiency of 45% are achieved. For future ap-
plications, the simple optical circuit of our source could be miniaturized by an integrated optics
architecture on a chip using the silica-on-silicon technique24. Using waveguides instead of bulk
optics would be beneficial to stability, performance and scalability21,25. We note that during the
preparation of the manuscript presented here, we learned of a parallel experiment by Barz et al.26.
Optical circuit. The transformation of the optical circuit consists of BS and HWP operations. The
BS operation describes the following transformation of the annihilation operators of the modes ˆ ak
andˆbk(note that we use annihilation operators to denote the corresponding modes): ˆ ak=√Rˆ ck+
√Tˆ ekandˆbk =
√Rˆdk+√Tˆfk, for k = x,y. R (T) is the amplitude reflection (transmission)
coefficient of the BS. For the modesˆfxandˆfy, the transformation of HWP at −22.5◦is defined by:
ˆfx= (ˆfx′ −ˆfy′)/√2 andˆfy= (ˆfx′ +ˆfy′)/√2. The optical circuit is able to prevent false signals
rising from two-pair emission. This is an important feature of the scheme4since the creation
probabilitiesfor two pairs are much larger than for three pairs. Furthermore, contributionsfrom the
higherordertermsofSPDC can belimitedby controllingthecorrespondingcreation probabilities4.
It is also worth noting that for a given three-pair photon state, the probability of creating a heralded
entangled state, i.e., T4R2/2, is controllable by changing the transmission coefficients of the BS,
which can be up to ∼ 0.0114.
Photon source. The required photon pairs are generated by type-II SPDC from a pulsed laser
in a β-Barium-Borate (BBO) crystal. Here, we use a pulsed high-intensity ultraviolet (UV) laser
with a central wavelength of 390 nm, a pulse duration of 180 fs and repetition rate of 76 MHz.
For an average power of 880 mW UV light and after improvements in collection efficiency and
stability of the photon sources, we observe ∼ 80×103photon pairs per second with a visibility of
V = (91 ± 3)% measured in the diagonal (+/−) basis. (The visibility is defined by V = (Nd−
Nud)/(Nd+Nud), were Nd(Nud) denotes the number of two-fold desired (undesired) coincidence
counts. Then there exists a direct connection between visibility and fidelity of a measured state ˆ ρ:
F = Tr(ˆ ρ|Ψ−??Ψ−|) =
4(1 + Vx+ Vy+ Vz), where Vkfor k = x,y,z denotes the visibility of
photon pair in the diagonal, circular, and linear bases, respectively. Here |Ψ−? is the singlet Bell
state.) Then the probability of creating three photon pairs is about 5.7 × 10−5per pulse, which
is ∼ 33 times larger than that of the next leading order term. The estimation of the three-pair
creation probability per pulse is based on the experimental pair generation rate and the theoretical
n-pair creation probability22pn = (n + 1)tanh2nr/cosh4r, where r is a real-valued coupling
coefficient. From the two-fold coincidence measurement result, the experimental pair generation
rate is p′= (80 × 103)/(0.152× 76 × 106) ≈ 4.7%. We assume that p1= p′, r can directly be
derived from p1. Thus the estimated creation probability p3and p4are obtained.
Experimental imperfections. With single photon resolving detectors and 100% detection effi-
ciency, one can see that the three-pair state can provide a maximally entangled photon pair in
the output modes deterministically with a 100% probability, if and only if the remaining photons
give rise to a fourfold coincidence among the four trigger modes. With the widely used standard
SPCM, one cannot discriminate pure single photons from multi-photons which in reality leads to
a significant problem of under counting photons. Accordingly, the trigger detectors can herald a
successful event even though more than two photons from either mode (ˆ ax,ˆ ay) or (ˆbx,ˆby) or both
have been transmitted to the trigger channels. Furthermore, experimentally we were only able to
obtain an average detection efficiency of about η = 15% resulting from limited collection and
detector efficiencies. Here the mean detection efficiency is averaged over the coupling efficiency
of eight fibre couplers and the quantum efficiency of the detectors. In addition to the imperfect de-
tections, there are two other factors that affect the performance of our source: the non-ideal quality
of the initially prepared pairs and the higher-order terms of down-converted photons. For perfectly
created pairs, destructive two-photon interference effects27–29will extinguish the contribution of
two-pair emission to the trigger signal. With an experimental visibility of (91 ± 3)% imperfectly
created states may still give rise to a contribution of two-pair events that leads to the detection of
the auxiliary triggers. In addition, four-pair emission can again contribute to both the triggers and
the output. Although the experimentally estimated creation probability for a four-pair emission is
only ∼ 1.7 × 10−6per pulse and is much smaller than the probability for a three-pair photon state
∼ 5.7 × 10−5per pulse, four-pair contribution can lead to an error of the theoretical estimation of
the expected preparation efficiency of about 4.5%. The four-pair contribution is evaluated in the
same way as the three-pair state, where the limited detection efficiency of the trigger detectors is
considered in the calculation (see Supplementary Information). In Fig. 3, the fluctuations of our
experimental data mainly result from the intrinsic statistics of detector counts, and the stability of
Experimental fidelity F. Every expectation valuefor a correlation function is obtained by making
a local measurement along a specific polarization basis and computing the probability over all the
possible events. For instance, to get the expectation value of RR correlation Tr(ˆ ρ|RR??RR|), we
perform measurementsalongthecircularbasis andthen gettheresultby thenumberofcoincidence
counts of RR over the sum of all coincidence counts of RR, RL, LR and LL. All the other
correlation settings are performed in the same way. The fidelity F can then directly be evaluated.
1. Nielsen, M.A. & Chuang, I. L. Quantum Computation and Quantum Information. (Cambridge
University Press, Cambridge, 2000).
2. Bouwmeester, D., Ekert, A. K. & Zeilinger, A. The Physics of Quantum Information.
(Springer, Berlin, 2000).
3. Kwiat, P. G., Mattle, K., Weinfurter, H., Zeilinger, A., Alexander, V. S. & Shih, Y. H. New
high-intensity source of polarization-entangled photon pairs. Phys. Rev. Lett. 75, 4337-4341
4.´Sliwa, C. & Banaszek, K. Conditional preparation of maximal polarization entanglement.
Phys. Rev. A 67, 030101(R) (2003).
5. Browne, D. E. & Rudolph, T. Resource-efficient linear optical quantum computation. Phys.
Rev. Lett. 95, 010501 (2005).
6. Kok, P. et al. Linear optical quantum computing with photonic qubits. Rev. Mod. Phys. 79,
7. Pan, J.-W., Chen, Z.-B., Zukowski, M., Weinfurter, H. & Zeilinger, A. Multi-photon entangle-
ment and interferometry. Rev. Mod. Phys. Preprint at http://arXiv:0805.2853 (2008).
8. Benson, O., Santori, C., Pelton, M. & Yamamoto, Y. Regulated and Entangled Photons from a
Single Quantum Dot. Phys. Rev. Lett. 84, 2513-2516 (2000).
9. Akopian, N. et al. Entangled Photon Pairs from Semiconductor Quantum Dots. Phys. Rev.
Lett. 96, 130501 (2006).
10. Stevenson, R. M. et al. A semiconductor source of triggered entangled photon pairs. Nature
439, 179-182 (2006).
11. Zhao, B., Chen, Z.-B., Chen, Y.-A., Schmiedmayer, J. & Pan, J.-W. Robust creation of entan-
glement between remote memory qubits. Phys. Rev. Lett. 98, 240502 (2007).
12. Knill, E., Laflamme, R. & Milburn, G. J. A scheme for efficient quantum computation with
linear optics. Nature 409, 46-52 (2001).
13. Kok, P. State Preparation in Quantum Optics., PhD thesis, University of Wales (2000).
14. Kok, P. & Braunstein, S. Limitations on the creation of maximal entanglement. Phys. Rev. A
62, 064301 (2000).
15. Pittman, T.B. et al. Heralded two-photon entanglement from probabilisticquantum logic oper-
ations on multiple parametric down-conversion sources. IEEE J. Sel. Top. Quantum Electron.
9, 1478-1481 (2003).
16. Hnilo, A.A. Three-photon frequency down-conversion as an event-ready source of entangled
states. Phys. Rev. A 71, 033820 (2005).
17. Eisenberg, H.S., Hodelin, J.F., Khoury, G. &Bouwmeester, D. Multiphotonpathentanglement
by nonlocal bunching. Phys. Rev. Lett. 94, 090502 (2005).
18. Walther, P., AspelmeyerM. & Zeilinger, A. Heralded generation of multiphotonentanglement.
Phys. Rev. A 75, 012313 (2007).
19. Chen, Y.-A. et al. J. Memory-built-in quantum teleportation with photonic and atomic qubits.
Nature Physics 4, 103-107 (2008).
20. Briegel, H.-J., D¨ ur, W., Cirac, J. I., & Zoller, P. Q. Quantum repeaters: the role of imperfect
local operations in quantum communication. Phys. Rev. Lett. 81, 5932-5935 (1998).
21. Matthews, J.C.F., Politi, A., Stefanov, A., & O’Brien, J. L. Manipulation of multiphoton en-
tanglement in waveguide quantum circuits. Nature Photonics 3, 346-350 (2009).
22. Kok, P. & Braunstein, S. Postselected versus nonpostselected quantum teleportation using
parametric down-conversion. Phys. Rev. A 61, 042304 (2000).
23. Clauser J., Horne, M., Shimony, A. & Holt, R. Proposed experiment to test local hidden-
variable theories. Phys. Rev. Lett. 23, 880-884 (1969).
24. Politi, A., Cryan, M. J., Rarity, J. G., Yu, S. & O’Brien,J. L. Silica-on-silicon waveguide
quantum circuits. Science 320, 646-649 (2008).
25. Clark, A. S., Fulconis, J., Rarity, J. G., Wadsworth, W. J. & O’Brien, J. L. All-optical-fiber
polarization-based quantum logic gate. Phys. Rev. A 79, 030303(R) (2009).
26. Barz, S., Cronenberg, G., Zeilinger, A. & Walther, P. Heralded generation of entangled photon
pairs. Nature Photon. (in the press).
27. Shih, Y.H. & Alley, C.O. New type of Einstein-Podolsky-Rosen-Bohmexperiment using pairs
of light quanta produced by optical parametric down conversion. Phys. Rev. Lett. 61, 2921-
28. Hong, C. K., Ou, Z. Y. & Mandel, L. Measurement of subpicosecond time intervals between
two photons by interference. Phys. Rev. Lett. 59, 2044-2046 (1987).
29. Lamas-Linares, A., Howell, J., Bouwmeester, D. Stimulated emission of polarization-
entangled photons. Nature 412, 887-890 (2001).
This work was supported by the European Commission through the European Re-
search Council (ERC) Grant and the Specific Targeted Research Projects (STREP) project Hybrid Infor-
mation Processing (HIP), the Chinese Academy of Sciences, the National Fundamental Research Program
of China under grant no. 2006CB921900, and the National Natural Science Foundation of China. C.W.
was additionally supported by the Schlieben- Lange Program of the ESF. The authors are grateful to Dr
Xian-Min Jin for help in improving the figures.
C.W., X.-H.B., Y.-A.C., Q.Z., and J.-W.P. designed the experiment. C.W., C.-
M.L., A.R., A.G., Y.-A.C., and K.C. performed the experiment. C.W., C.-M.L., A.R., X.-H.B., K.C., and
J.-W.P. analyzed the data. C.W., C.-M.L., K.C. and J.-W.P. wrote the paper.
The authors declare that they have no competing financial interests.
Correspondence and requests for materials should be addressed to K.C. and J.W.P.
Figure 1 Schematic setup. The heralded generation of entangled photon pairs is imple-
mented with the optical circuit composed of non-polarizing partial reflecting beam splitters (BS),
a half wave plate (HWP) and two polarizing beam splitters (PBS). The BS split mode ˆ a (ˆb) into
a trigger mode ˆ e (ˆf) and an output mode ˆ c (ˆd). The auxiliary trigger photons are detected in
(ˆfx′,ˆfy′) in the diagonal (+/−) basis and in (ˆ ex, ˆ ey) in the linear (H/V ) basis. The setup will
output an entangled photon pair after successful triggering of the four auxiliary photons.
Figure 2 Experimental setup for event-ready entanglement source. After emission, the
a correction BBO (C BBO) before the photons are directed onto the partial reflecting beamsplitter
(PRBS). To control the additional phase introduced by the PRBS we used a combination of two
quarter-wave plates (QWP) and one HWP. All photons are filtered by narrow bandwidth filters
(∆λ ≈ 3.2 nm) and are monitored by silicon avalanche single-photon detectors. Coincidences are
recorded by a laser clocked FPGA (Field Programmable Gate Array) based coincidence unit.
Figure 3 Efficiency of state preparation. Theoretical and experimental values of prepara-
tion efficiency for the amplitude reflection coefficients R = 0.486,0.570 and 0.685 are depicted.
The error bars are according to Poissonian statistics of counts. The curve is a function graph of
Eq. 5 with an average detection efficiency ηt= 0.1823 for triggers. efftheoryis an increasing func-
tion of R and up to 100%. The quantum efficiency of detectors q used is about 60%. For each
50/50, 60/40, and 70/30 BS ratio, our experimental coupling efficiencies of trigger (pt) and sig-
nal detectors (ps) are as follows, (pt,ps): (27.8%,21.5%), (28.8%,22.2%), and (34.5%,25.0%),
respectively. Note that pq is defined as the detection efficiency η.
Figure 4 Experimental data for fidelity measurements. We have performed a complete 3-
setting local measurements for ˆ σzˆ σz, ˆ σxˆ σxand ˆ σyˆ σy, which corresponding to three complementary
bases of |H?/|V ?, |+?/|−? and |R?/|L?, with |+? = (|H? + |V ?)/√2, |−? = (|H? − |V ?)/√2,
|R? = (|H? + i|V ?)/√2 and |L? = (|H? − i|V ?)/√2. The plots are for three different splitting
ratios R/T of the partial reflecting beamsplitters 50/50 (a), 60/40 (b) and 70/30 (c). The error bars
relate to Poissonian statistics of counts.
?? ?? ??
HH ++ LR HV +- LL VH -+ RR VV -- RL?
?? ?? ??? ? ??? ? ?? ? ??? ? ?
?? ?? ??
HH ++ LR HV +- LL VH -+ RR VV -- RL?
?? ?? ??? ? ??? ? ?? ? ??? ? ?
?? ?? ??
HH ++ LR HV +- LL VH -+ RR VV -- RL?
?? ?? ??? ? ??? ? ?? ? ??? ? ?
Table 1: Experimental fidelity of the entangled output state with respect to the reflection
coefficients R of the beam splitters.
Efficiency of state preparation efftheory
In order to show a clear picture of the theoretical estimation, let us consider first the case
that we have ideal detection efficiency of 100%. Eq. (5) will be naturally derived afterwards. It is
instructive to start with the output state |Ψ′
3? in the following complete form:
3? = α|θ?t
s+ β |ϑ?ts+ γ |ϕ?ts.
For the first term of Eq. (S1), as already indicated in Eq. (2), the appearance of the perfect
four-photon trigger state|θ?twill herald a photon pair in state |Φ+?sin theoutput with a probability
of 100%. The second term β |ϑ?tsrepresents all additional states, which actually yield a trigger
signal in all of the (ˆ ex, ˆ ey) and (ˆfx′ˆfy′) modes generated by more than four photons, but the state
in the output (ˆ cx,ˆ cy) and (ˆdx,ˆdy) contains only a single photon or vacuum. For completeness, the
last term γ|ϕ?tsrepresents states, which do not contribute to the trigger, but contain more than two
photons in the output modes. These states do not affect our experimental results since without
trigger signal their contribution is not recorded. All normalization factors are summarized in α, β
and γ. Therefore, according to the definition of the efficiency of state preparation, we obtain the
state preparation efficiency for perfect detections: efftheory = α2/(α2+ β2) = R2/(1 − T/2)2.
Now we proceed to discuss the effect of imperfect detection. First, we introduce the detection loss
of coupling by replacing the creation operators in the trigger modesˆt
†for t = ex,ey,fx′,fy′ with
√pˆt Download full-text
†+√1 − pˆ˜t†, where p denotes the coupling efficiency for the trigger photons4. The operators
ˆ˜t†describe photons that escape from detections. Then we consider all the components of the state
3? that can becollected intotrigger detectors, e.g., β′p2√1 − pˆd†
we take the efficiency of detector into account. For each considered term, we therefore can obtain
the corresponding probability of giving a herald signal from its probability amplitude and state
vector. Forexample,fortheterm illustratedabovetheprobabilityis β′2q4p4(1−p), whereq denotes
the efficiency of trigger detector. Finally, by the probability of measuring a heralded photon pair
overthe total probabilityof generating a trigger signal, we have efftheory= R2/(1−pqT/2)2. Here
pq is defined as the detection efficiency for trigger photons ηt. For each 50/50, 60/40 and 70/30 BS
ratios, our experimental detection efficiencies achieve: 0.167, 0.173 and 0.207, respectively.