Demonstration of analyzers for multimode photonic time-bin qubits
Jeongwan Jin, Sascha Agne, Jean-Philippe Bourgoin, Yanbao Zhang, Norbert Lütkenhaus, Thomas Jennewein
DDemonstration of analyzers for multimode photonic time-bin qubits
Jeongwan Jin , , ∗ Sascha Agne , , † Jean-Philippe Bourgoin , , YanbaoZhang , , ‡ Norbert L¨utkenhaus , , and Thomas Jennewein , , § Institute for Quantum Computing, University of Waterloo,200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada Department of Physics and Astronomy, University of Waterloo,200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada Quantum Information Science Program, Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada (Dated: July 10, 2018)We demonstrate two approaches for unbalanced interferometers as time-bin qubit analyzers forquantum communication, robust against mode distortions and polarization effects as expected fromfree-space quantum communication systems including wavefront deformations, path fluctuations,pointing errors, and optical elements. Despite strong spatial and temporal distortions of the opticalmode of a time-bin qubit, entangled with a separate polarization qubit, we verify entanglement usingthe Negative Partial Transpose, with the measured visibility of up to 0 . ± .
01. The robustnessof the analyzers is further demonstrated for various angles of incidence up to 0.2 ◦ . The outputof the interferometers is coupled into multimode fiber yielding a high system throughput of 0.74.Therefore, these analyzers are suitable and efficient for quantum communication over multimodeoptical channels. INTRODUCTION
Quantum communication experiments in free space [1–5] are usually based on polarization-encoded photons dueto their robustness against atmospheric turbulence [6].However, the quality of polarization is fundamentallylimited by nonideal steering optics and telescopes, andreference-frame alignment [7, 8]. For instance, photonpolarization typically experiences phase shifts when re-flected off optical surfaces with a nonzero angle of inci-dence (AOI), leading to errors in encoded information.This problem becomes significant when communicatingparties are located on moving platforms, such as air-crafts or satellites, where a signal-tracking system intro-duces fluctuations in the system alignment [8]. More-over, polarization can be changed in a nonunitary man-ner that cannot be corrected when passing through tem-pered glasses and polycarbonates, which are widely usedin vehicles and buildings.Time-bin encoding [9] is an interesting alternative dueto its immunity against polarization drifts. The methodhas been demonstrated with various quantum commu-nication protocols in optical fibers, including plug-and-play [10, 11], differential phase shift [12], and coherentone-way [13] quantum key distribution (QKD) proto-cols, as well as Mach-Zehnder interferometer-based sys-tems [14], quantum teleportation [15], and elements ofquantum repeaters [16]. Despite its versatility, time-binencoding has been implemented only in single-mode opti-cal fibers and is generally considered impractical for free-space channels. The reason is that spatial and temporalmodes of a photon are distorted during the transmissionthrough multimode optical channels such as multimodefibers and turbulent free space [17]. Mode distortionsintroduce path distinguishabilities in unbalanced Michel- son or Mach-Zehnder interferometers, which are typicallyused as time-bin analyzers, hindering single-photon in-terference required for analysis of time-bin states. In ad-dition, telescope pointing errors as well as turbulence-induced angular fluctuations [18] degrade the qualityof interference even further. For example, Ursin etal. [1] reported turbulence-induced AOI errors of up to4 × − degrees over a horizontal 143 km link, and wereported [19] pointing errors of 6 × − degrees on amoving platform. This leads to phase shift and visibil-ity reduction. Spatial filters such as single-mode fiberscan be used to combat those problems, which, however,discards most of the impinging photons [21].Here we investigate two types of unbalanced Michelsoninterferometers for analyzing time-bin qubits encoded onspatially and temporally distorted photons, originally de-veloped for Doppler spectroscopy of stars [22]. Theseso-called field-widened interferometers use imaging op-tics, or carefully chosen refractive indices, to correctAOI-induced phase shifts and visibility reduction, henceachieving a larger field of view than conventional Michel-son interferometers. However, it was not known whethersuch interferometers are capable of analyzing entangle-ment or quantum sates transmitted over multimode op-tical channels. To prove that, we take the followingtwo steps. First, we compare the performance betweenthe conventional and field-widened interferometers usingclassical light. Next, by utilizing quantum entanglement,we demonstrate the viability of the interferometers asmultimode time-bin receivers for quantum applications. MULTIMODE TIME-BIN ANALYZER METHODS
Let us consider an unbalanced Michelson interferome-ter with long and short paths of lengths l L and l S , re- a r X i v : . [ qu a n t - ph ] A p r δ ( α ) α Turbulence
Time-Bin Receiver Ι > Ι > Δ l /2 FIG. 1. (Color online)
Time-bin-based quantum com-munication in a turbulent free-space channel.
Whena time-bin-encoded photon, whose path is deviated by atmo-spheric turbulence as well as telescopes misalignment, entersa time-bin receiver with variable angle of incidence α , a lat-eral offset δ ( α ) occurs between the paths (red and blue line)at the interferometer exit. This is due to the receiver lengthasymmetry and reduces the interference quality, resulting inlower distinguishability of the time-bin states in superpositionbases. Turbulence-induced spatial-mode distortions furtherlower the interference visibility [see text for details]. spectively. While the path-length difference for zero-angle incidence is simply ∆ l = 2( l L − l S ), a nonzeroAOI translates into an angle-dependent path length anda lateral offset as the beam propagates. Using geometri-cal ray tracing through the interferometer, we find thatthe path-length difference is given by∆ l ( α ) = ∆ l (cid:18) α ) + 1 − tan( α )cos( α ) + sin( α ) (cid:19) + δ ( α ) tan (cid:16) α − π (cid:17) , (1)where δ ( α ) = ∆ l tan( α ) / (1 + tan( α )) is the lateral offsetbetween the two rays coming from each path of the inter-ferometer at the output beam splitter [see Fig. 1]. FromEq. (1), we see that a nonzero AOI introduces path dis-tinguishability and rapidly modulates the interferometerphase at the same time. The relative phase between thetwo paths is very sensitive to the AOI, with a predicted π –shift per 2 × − degrees input angle variation. Inorder to quantify interference degradation due to input-angle fluctuations, we compute the interference visibility.Considering a single-mode Gaussian beam with intensity I and a beam width σ at the interferometer input, thevisibility is given by [23] V ( α ) = V exp (cid:32) − (cid:18) ∆ l tan( α ) √ σ (1 + tan( α )) (cid:19) (cid:33) , (2)where V denotes the system visibility at zero AOI.For instance, with σ = 1 .
49 mm and ∆ l = 0 .
60 m,due to Eq. (2), the visibility will drop to 0.70 for α = 0.1 ◦ and V = 0.91. The relationship Eq. (2)is verified experimentally with a single-mode beam[see Fig. 2(a)], generated by a continuous-wave laserat 776 nm. For instance, as shown in Fig. 2(d), theinitial interference visibility of V single0 = 0.91 ± V multi0 = 0.16 ± ◦ . These observationsclearly show that, given the expected angular deviationsreported for free-space quantum channels, it would betechnically very challenging to achieve a reliable, stableand efficient operation of time-bin qubit analyzers usingstandard interferometers.These interference challenges are overcome by uti-lizing relay optics in the long arm of the unbalancedMichelson interferometer (Method 1). The idea is toreverse differences in the evolution of spatial modesover the length ∆ l in the long arm, as shown in Fig.2(c). This effectively guarantees identical wavefrontevolutions in the short and long paths of the interfer-ometer. Consequently, spatial indistinguishability isrestored regardless of spatial mode and AOI of the inputbeam. For verification, we set ∆ l = 0.60 m (2.0 ns) andmeasure interference visibilities by applying voltagesto a piezo mounted on a mirror in the short path,allowing it to change the phase of the interferometerat various AOIs. Having a single-mode beam as aninput, we obtain an interference visibility of V single =0.91 ± V multi =0.89 ± α . The op-tical path difference in the interferometer is given by∆ l = 2( n L l L cosα L − n S l S cosα S ), where n L(S) and α L(S) I n t e r f e r en c e v i s i b ili t y n n Fiber coupler 50/50 Beam splitter LensMirrorFree-space detectorGlass Δ AOI( α )AOI( α ) AOI (degrees ) corrected TQAuncorrected TQA (a) (b) (c) (d) (e) (f) (g) (h) f f f f --- -- l / 2 FIG. 2. (Color online)
Our multimode time-bin qubit analyzers (MM-TQA). (a)
Image of the incident single-modeGaussian beam, captured with a beam-profiling camera (WinCamD-UCD12). (b)
Image of the incident multimode beam,generated by a 1 m-long step-index multimode fiber (Thorlabs M43L01). (c)
Schematic diagram of our MM-TQA usingimaging optics. Temporal separation between paths is set to 2.0 ns (∆ l = 0 .
60 m). f denotes a focal length of the lens. (d)
Interference visibility for the single-mode beam with (green circles) and without (black squares) relay optics. Measuredvisibilities are in good agreement with theoretical prediction of Eq. (2) (dashed black line). (e)
Interference visibility for themultimode beam with (green circles) and without (black squares) relay optics. (f )
Schematic diagram of our MM-TQA withdifferent refractive indices for each path. Temporal separation between paths is set to 0.57 ns (∆ l = 0 .
17 m). (g)
Interferencevisibility for the single-mode beam with (green circles) and without (black squares) glass. Measured visibilities are in goodagreement with theoretical prediction of Eq. (2) (dashed black line). (h)
Interference visibility for the multimode beam with(green circles) and without (black squares) glass. The uncorrected TQA visibilities in (g) and (h) are higher than in (d) and(e), because shorter path-length difference introduces less path distinguishability. Uncertainties are smaller than symbol size. denote refractive index and reflection angle from a mir-ror in path l L(S) , respectively. Using Snell’s law andTaylor’s expansion, the difference is approximated as2( n L l L − n S l S ) − sin α ( l L /n L − l S /n S ) for small angles α L and α S . With a proper choice of refractive indicesfor both paths, we can remove the second term so that∆ l becomes insensitive to AOI, thus restoring indistin-guishability at the interferometer output. In our imple-mentation, we use 118 mm-long glass with the refractiveindex n =1.4825 in the long path and none in the shortpath, providing an optical path-length difference of ∆ l =0.17 m (0.57 ns). Interference visibilities of 0.94 ± ± QUANTUM COMMUNICATION EXPERIMENTS
We demonstrate the viability of our MM-TQAs for usewith quantum signals using the experimental setup de-picted in Fig. 3. Light from a 404 nm continuous-wavelaser with an average power of 6 mW pumps a periodi-cally poled potassium titanyl phosphate crystal inside aSagnac interferometer. This generates polarization en-tangled photon pairs at 842 nm (A) and 776 nm (B) ina form of | ψ (cid:105) = √ ( | H (cid:105) A | V (cid:105) B + | V (cid:105) A | H (cid:105) B ) via type-IIspontaneous parametric down-conversion. Here, | H (cid:105) and | V (cid:105) are the horizontal and vertical polarization states, re-spectively, forming the eigenstates of the computationalbasis. Unused pump photons are removed by band-pass LASER
404 nm 776 nm (B) 842 nm (A)
PQA EPS time c o i n c i den c e Time tagger Multimode Channel Multimode Channel MM-TQA MM-TQA TQC TQC D2 PQA for source visibility D1
Method 1 Method 2
AOI ( α ) AOI ( α ) HWP QWP BPF POL DM FM
M Lens
PBS BS FPC Glass
Si-APD
PPKTP SMF MMF
T1 T1
FIG. 3. (Color online)
Experimental setup.
The polarization-entangled photon-pair source (EPS) is described in the text.For projection measurements, photon A is directed to a polarization-qubit analyzer (PQA), consisting of a quarter-wave plate(QWP), a half-wave plate (HWP), a polarizing beam splitter (PBS), and silicon avalanche photodiodes (Si-APDs). Afterreflection at a dichroic mirror (DM), via a flip mirror (FM), photon B is sent either to a PQA or a time-bin qubit converter(TQC ) followed by a multimode fiber (MMF) and a multimode time-bin qubit analyzer (MM-TQA ). All detectionsignals are sent to a time tagger for data analysis. filters. While photon A is directed to a polarization ana-lyzer (PQA), photon B is sent either to a separate PQAor a time-bin converter (TQC) followed by a multimodechannel and a MM-TQA for various measurements. ThePQAs measure reference entanglement visibility with thesource of polarization entanglement.To convert the polarization state of photon B into atime-bin state, we use an unbalanced interferometer as aTQC [24], whose path-length difference is matched tothe MM-TQA . At the input polarizing beam splitterof the TQCs, a photon is either reflected or transmit-ted into the short or long path, respectively. A fiber-polarization controller (FPC) ensures the faithful map-ping of the vertical (horizontal) polarization onto theearly (late) temporal bin. The inserted quarter-waveplate in each path guides photons to the desired outputport. Leaving the TQCs, photons pass through a polar-izer set to an equal superposition between the polariza-tions, erasing polarization information for each time-binstate at the cost of 50 % transmission loss. This com-pletes the map | V (cid:105) (cid:55)→ | E (cid:105) and | H (cid:105) (cid:55)→ | L (cid:105) , resulting thetwo-photon entangled state in a form of | ψ (cid:105) = 1 √ | H (cid:105) A | E (cid:105) B + e i φ | V (cid:105) A | L (cid:105) B ) , (3)where | E (cid:105) ( | L (cid:105) ) denotes the quantum state in which pho-ton B is in early (late) temporal mode, and φ is a relativephase between the modes introduced during the conver-sion process. Photon B then travels through a 1m-longstep-index multimode fiber, as a multimode channel, dis-torting the spatial mode [see Fig. 4(a)] and temporalmode (measured dispersion is about 50 ps, drasticallyexceeding the photon’s coherence time of 3.2 ps [25]),prior to entering the MM-TQAs. For Method 2, we cal-culate for the glass a dispersion of 5.48 waves/nm and5.21 waves/ ◦ C. In order to minimize dispersion effects,we symmetrize the paths in the time-bin converter andanalyzer. After being analyzed in the MM-TQAs, bothphotons A and B are detected by silicon avalanche photo-diodes and the detection signals are sent to a time taggerand computer for data analysis.
AOI (degrees) E n t ang l e m en t v i s i b ili t y Phase (radians) S i ng l e c oun t s ( k H z ) C o i n c i den c e s ( H z ) N o r m a li z ed c o i n c i den c e s Time (ns) -ZA +ZB +ZA -ZB-ZA -ZB Z Y X ϕ ' +ZA +ZB (d)(a) (b)(c) + ϕ '- ϕ ' FIG. 4. (Color online)
Experimental results for entangled photons analyzed with our MM-TQAs. (a)
Spatial modeof photon A before entering the MM-TQA. The image is captured with an electron multiplier CCD camera (Hamamatsu C9100-13). (b)
Joint detections for the projection +Z A ⊗ ± Z B (dotted green lines) and − Z A ⊗ ± Z B (solid orange lines) as a functionof detection-time difference between the photon A and B. (c) Joint detections for the projection + φ A (cid:48) ⊗ φ B (green squares)and − φ A (cid:48) ⊗ φ B (orange circles) as a function of phase φ A (cid:48) of a polarization qubit using motorized wave plates. Visibilities V ± φ are obtained from sinusoidal fittings. Single counts remain essentially constant as we scan the phase. (d) The measurements(b) and (c) are repeated for different AOIs. Red circles and blue squares are average visibilities obtained with Method 1 and2, respectively. Solid red and dotted blue lines are reference visibilities measured directly with our source of polarization-polarization entanglement prior to the measurement of time-polarization entanglement using Method 1 and 2, respectively. Wemaintain high entanglement visibility (close to source visibility) despite the high multimode nature of incoming photons.
MEASUREMENTS AND RESULTS
The MM-TQA performance is verified by entangle-ment visibility measurements. For the measurements,photons A and B are directed to a polarization and atime-bin qubit analyzer, respectively. Each qubit is firstprojected onto the computational basis, i.e., | ± Z (cid:105)(cid:104)± Z | ,where | + Z A (cid:105) ≡ | H (cid:105) , | − Z A (cid:105) ≡ | V (cid:105) , | + Z B (cid:105) ≡ | E (cid:105) ,and | − Z B (cid:105) ≡ | L (cid:105) . The coincidence counts are usedto calculate correlation visibilities V ± Z , ± Z ≡ (N ± Z ± Z − N ∓ Z ± Z ) / (N ± Z ± Z + N ∓ Z ± Z ), from which we obtain theaverage V Z = ( V +Z , +Z + V − Z , − Z ) /
2. Here, N ij denotesthe joint-detection counts when polarization qubit A isprojected onto | i (cid:105)(cid:104) i | and time-bin qubit B onto | j (cid:105)(cid:104) j | ,where i, j ∈ { +Z , − Z } [see Fig. 4(b)]. The qubits arethen projected onto superposition states, i.e., | ± φ (cid:105)(cid:104)± φ | ,where | ± φ A(B) (cid:105) ≡ √ ( | + Z A(B) (cid:105) ± e i φ A(B) | − Z A(B) (cid:105) ).To measure the visibility, we vary the relative phase be-tween basis states of the polarization qubit. A com-plete scan of the phase along the XY-plane of the Blochsphere is performed [see Fig. 4(c)], yielding the average V φ = ( V + φ + V − φ ) /
2. These allow us to compute an aver-age visibility V avg ≡ V Z / V φ /
3. For a concluding as-sessment of the performance of the MM-TQAs, we com-pare V avg to the source visibility obtained from the orig-inal polarization entanglement. This is done by routingphoton B to a polarization analyzer. For Method 1(2),we measure visibilities of V Z =0.95 ± ± V φ =0.80 ± ± V avg = 0.85 ± ± ± ± ◦ . Note that this angle rangeis already larger than the measured error of our signalpointing system on a moving vehicle [19]. In addition, theMM-TQA is able to recover AOI-induced phase shifts.Without correcting optics, a varying AOI also leads tophase fluctuations in the interferometer [26]. From ourtheoretical model, we anticipate a 5 π shift with an AOIof only 1 × − degrees [see inset of Fig. 5]. To as-sess the phase stability of the MM-TQA with AOI, usingMethod 1, we measure correlation visibilities for AOIschanging from -0.20 ◦ to +0.20 ◦ continuously over 20 sec.The measured visibilities remain almost constant withinexperimental errors [see Fig. 5], showing that the MM-TQA prevents AOI-caused phase fluctuations.The suitability of our MM-TQA for quantum com-munication is further substantiated by examining theCHSH-Bell inequality [27] using Method 1. We searchfor the maximally achievable CHSH-Bell parameter S within the correlation data taken while the mea-surement basis of the time-bin qubits drifted slowly. AOI (degrees)-0.20 -0.10 0 0.10 0.2000.20.40.60.81
Time (seconds)0 5 10 15 20AOI (degrees) C o rr e l a t i o n v i s i b ili t y -1 C o rr e l a t i o n v i s i b ili t y FIG. 5. (Color online)
Phase stability of our MM-TQA
Correlation visibilities V φ A (cid:48) ,φ B (green circles) are measuredusing Method 1 as the AOI is continuously varied from -0.2 ◦ to +0.2 ◦ over 20 sec. The inset shows the calculated visibil-ities without relay optics as a function of AOI. Due to AOI-induced phase fluctuations, the value rapidly changes withAOI and yields an average value of zero (red circle). Thesephase fluctuations are corrected with relay optics, allowing anear constant visibility. VVz
No verified entanglement
FIG. 6. (Color online)
Entanglement verification.
TheNegative Partial Transpose (NPT) criterion [28, 29] is usedto obtain the required entanglement visibilities, certifying thepresence of entanglement in an arbitrary 2 × The observed maximum value of S exp = 2.42 ± S theo = 2.47 ± × DISCUSSION
While theoretically a unit visibility is possible, the per-formance of our current MM-TQAs using imaging opticsand different refractive-indexed paths show V = 0.89 ± ± CONCLUSION
We demonstrated two types of unbalanced inter-ferometers as multimode time-bin qubit analyzers forquantum communication, which are compatible withspatially and temporally distorted photons and robustagainst angle of incidence fluctuations. With opticalinput modes emerging from a multimode optical fiber,the analyzers show an average interference visibility ofup to 0.91 ± ◦ .The viability of the analyzers for quantum communi-cation is substantiated by a measured entanglementvisibility of up to 0.85 ± et al. [30], C. Zeitler et al. [31],and G. Vallone et al. [32]. The groups who authoredRefs. [30, 31] and [32] both developed a time-bin ana-lyzer using an unbalanced Mach-Zehnder interferometertogether with imaging optics, that are conceptually sim-ilar to our Method 1. With their analyzers, the authorsobserved an interference visibility of 0.93 for angular vari-ations up to 8.6 × − degrees [31], and the coherentsuperposition of a laser pulse attenuated to the single-photon level after being reflected from a satellite [32]. ACKNOWLEDGMENTS
The authors would like to thank Jacob Koenig, RolfHorn, Evan Meyer-Scott, and Patrick Coles for useful dis-cussions, and Martin Laforest for lending us the Hama-matsu EM-CCD camera. We gratefully acknowledge sup-ports through the Office of Naval Research (ONR), theCanada Foundation for Innovation (CFI), the OntarioResearch Fund (ORF), the Canadian Institute for Ad-vanced Research (CIFAR), the Natural Sciences and En-gineering Research Council of Canada (NSERC), and In-dustry Canada.
APPENDIX A: ESTIMATION OF CHSH-BELLPARAMETER
The nonclassicality of time-polarization entanglementis bounded with an estimate of the Bell-CHSH inequal-ity violation [27]. Despite the absence of active phasecontrol, required to set measurement bases for the time-bin qubit deterministically, we search for the maximallyobtainable violation using Method 1 by varying the mea-surement basis.We first set the measurement basis for the polariza-tion qubit to A ≡ | Z + X (cid:105)(cid:104)
Z + X | using wave plates,and slowly and continuously change the path-length dif- ference of the MM-TQA by externally heating it. Thisallows us to scan projection measurements for the time-bin qubit in superposition bases. Fig. 7(a) shows co-incidences between a polarization qubit (two detectors,i.e., D1 and D2) and a time-bin qubit (three temporalmodes). Detections in the early/late bin (middle bin)correspond to a projection of the time-bin qubit onto B ≡ | Z (cid:105)(cid:104) Z | ( B ≡ | φ (cid:105)(cid:104) φ | ). Owing to the absence of thesecond output of the MM-TQA, we consider all possibleexpectation values E ( A i , B j ) between any two points intime, i.e., t and t [see Fig. 7(b)], which is defined as E ( A i , B j ) = N ++ij + N −− ij − N + − ij − N − +ij N ++ij + N −− ij + N + − ij + N − +ij . (4)Here, N ij are the coincidence counts for the projections A i ⊗ B j , where i, j ∈{ } and superscript(+ , − ) denotestwo outcomes of the projection measurement. Amongall the computed expectation values, we find the abso-lute maximum expectation value. We then change themeasurement basis for the polarization qubit to A ≡| Z − X (cid:105)(cid:104) Z − X | and repeat the procedure. Finally, wecompute the CHSH-Bell inequality parameter S = | E ( A , B ) − E ( A , B ) + E ( A , B ) + E ( A , B ) | (5)and find the value of S exp = 2.42 ± S = 2. To see whether this valueagrees with the measured visibilities, we model the two-qubit state with noise, described by an asymmetric de-polarization channel, on a time-bin qubit. The outputstate is described by ρ out = (1 − (cid:88) j=X , Y , Z p j ) | ψ (cid:105)(cid:104) ψ | + (cid:88) j=X , Y , Z p j ( I ⊗ j) | ψ (cid:105)(cid:104) ψ | ( I ⊗ j) , (6)where p j (j = X , Y , Z) and j denote the depolarizationprobability and single-qubit Pauli operator, respectively.Here, | ψ (cid:105) is the input state, described in Eq. (3). As-suming unbiased depolarizations in superposition bases,i.e., ( p X = p Y ) ≡ p φ , we calculate expectation values E ( A i , B j ) = Tr ( ρ out A i ⊗ B j ) for given measurementbases. Using the definition of visibility, we further rep-resent the CHSH-Bell parameter S theo = √ V Z + V φ )as a function of entanglement visibilities. We find S theo = 2.47 ± S exp = 2 . ± . APPENDIX B: LONG-TERM PHASE STABILITYOF OUR MM-TQA
Our MM-TQAs are passively stabilized by enclosingthem with black cardboard. In order to assess the
40 80 120 160001002003004005000100200300400500 40 80 120 1600 0 40 80 120 16004080120160 10.80.60.40.20 (a)(c) t (seconds) t ( s e c ond s ) E (A ,B ) 0 40 80 120 16004080120160 0.40.20-0.2-0.4-0.6t (seconds) t ( s e c ond s ) (seconds) t ( s e c ond s ) E (A ,B ) E (A ,B )A B i A B i (i=1,2) (i=1,2) (b) (seconds) t ( s e c ond s ) E (A ,B ) (d) C o i n c i den c e c oun t s ( H z ) Time (seconds) C o i n c i den c e c oun t s ( H z ) Time (seconds)
D1 & early
D2 & early
D1 & late
D2 & late
D1 & middle
D2 & middle
FIG. 7. (Color online)
Estimation of the CHSH-Bell parameter. (a)
Long-dashed yellow and short-dashed light blue linesare coincidences from joint projections (Z + X) ⊗ +Z (early temporal bin), long dash-dotted green and short dash-dotted orangelines from (Z + X) ⊗ − Z (late temporal bin), and solid purple and dotted blue lines from (Z + X) ⊗ φ (middle temporal bin).Dashed black lines are times at which maximal expectation values are extracted. (b) Surface plot of calculated expectationvalues for the projections in (a) between any two points in time. (c)
Long-dashed yellow and short-dashed light blue lines arecoincidences from joint projections (Z − X) ⊗ +Z, long dash-dotted green and short dash-dotted orange lines from (Z − X) ⊗− Z,and solid purple and dotted blue lines from (Z − X) ⊗ φ . Dashed black lines are times at which maximal expectation values areextracted. (d) Surface plot of calculated expectation values for the projection measurements in (c). The Bell-CHSH parameteris calculated using the maximum expectation values. The measurement duration is chosen arbitrarily and yields a violation ofthe inequality of 2.42 ± phase stability of the MM-TQA, using Method 1, weperform joint-projection measurements onto superposi-tion bases more than a half hour. The time-bin qubitis projected onto | φ (cid:105)(cid:104) φ | and the polarization qubit al-ternatively between | φ (cid:48) (cid:105)(cid:104) φ (cid:48) | or | φ (cid:48) + π/ (cid:105)(cid:104) φ (cid:48) + π/ | . Asshown in Fig. 8, the correlation visibility remains alwayshigher than 0.65, which is well above the required valuefor verifying entanglement, given entanglement visibility V Z = 0.95 ± APPENDIX C: ENTANGLEMENTVERIFICATION
The ability to verify effective entanglement is a nec-essary condition for secure QKD [33]. This is espe- cially important in the absence of a complete secu-rity analysis of a QKD implementation, and appliesto prepare-and-measure QKD as well as entanglement-based schemes. We assume that the spontaneous para-metric down-conversion process generates a pair of pho-tons with negligible multiple-photon-pair events. Eachphoton is a polarization qubit and the pair of photonsis potentially entangled. By detecting a photon A in thepair, we can herald the other photon B. After the conver-sion from polarization qubit to time-bin qubit, the pho-ton B is transmitted to the MM-TQA. Suppose that Aliceholds the polarization qubit while Bob holds the time-binqubit. To include conversion and transmission losses ofthe time-bin qubit, we enlarge the dimension of Bob’s sys-tem from 2 to 3 by adding a dimension corresponding to C o rr e l a t i o n v i s i b ili t y V ϕ A', ϕ B V ϕ A'+ π /2, ϕ B (V ϕ A', ϕ B +V ϕ A'+ π /2, ϕ B ) FIG. 8. (Color online)
Long-term phase stability of ourMM-TQA.
Red squares are expectation values for projec-tion measurements φ A (cid:48) ⊗ φ B ( V φ A (cid:48) ,φ B ) and blue hexagons for φ A (cid:48) + π/ ⊗ φ B ( V φ A (cid:48) + π/ ,φ B ). Each measurement is aver-aged over 3 min. Black circles are average correlation visibil-ity values (cid:113) V φ A (cid:48) ,φ B + V φ A (cid:48) + π/ ,φ B . The phase drifts slowly,on the order of π /2 over an half an hour, showing the stabil-ity of our MM-TQA. The average expectation value is alwayshigher than the required expectation value for entanglementverification. no photon arriving at Bob. Hence, the final state ρ sharedby Alice and Bob is a 2 × ρ is entangled using onlythe measurement results in V Z = 0 . ± .
01 (0 . ± . V φ = 0 . ± .
01 (0 . ± .
01) with Method 1(2) with-out further assumptions on the state. Since the measure-ments of Alice and Bob are block-diagonal with respectto the subspaces of total photon number, as we will showbelow in Eq. (7) and Eq. (8), we can also assume withoutloss of generality that the state ρ shows the same struc-ture. This follows from the fact that the measurementstructure allows us to assume that a quantum nondemoli-tion measurement of the total photon number is executedbefore the actual measurement itself.In order to verify entanglement, we need to know howto accurately describe the measurements on the polariza-tion and the time-bin qubit. For the polarization qubit,we measure it in the horizontal/vertical or diagonal/anti-diagonal basis, i.e. along the Z- or X-axis in the Blochsphere. These measurements are represented as M H = (cid:34) (cid:35) M V = (cid:34) (cid:35) M D = 12 (cid:34) (cid:35) M A = 12 (cid:34) − − (cid:35) , (7) where the subscript indicates measurement outcome, andH, V, D, or A denotes the horizontal, vertical, diago-nal, or anti-diagonal polarization. On the other side, forthe time-bin qubit, the photon loss in the long path orthe short path of the MM-TQA could be different fromeach other. Hence, the operators corresponding to mea-surement of the time-bin qubit in the early/late basis orin the superposition bases could deviate from the idealcase. Without loss of generality, we can choose the rel-ative phase between the early- and late-basis states inthe superposition basis to be zero. Therefore, in the ba-sis in which the basis states are no photon, one photonin the early bin and one photon in the late bin, thesemeasurements can be written as M E = η S
00 0 0 M L = η L (8) M X = η L √ η S η L √ η S η L η S M ∅ = I − M E − M L − M X , where the subscript E, L, X, or ∅ means that the measure-ment outcome is early time, late time, the superpositionof the early and late time, or no detection, respectively.No-detection events are due to detection inefficiency andthe absence of the second output in the MM-TQA. InEq. (8), η S or η L is the respective transmission efficiencyin the short path or the long path of the MM-TQA. Notethat in our experiment η S and η L are very close to eachother.After knowing the description of Alice’s and Bob’s jointstate ρ and that of their measurements, we can verifyentanglement by the negative partial-transpose (NPT)criterion [28]. The NPT criterion is used because thiscriterion is satisfied if and only if a state is entangled,given the state is 2 ×
2- or 2 × ρ subject to ρ ≥ , T r ( ρ ) =1 , ρ Γ ≥ T r [ ρ ( M H ⊗ M E − M V ⊗ M E )]= V +Z , − Z T r [ ρ ( M H ⊗ M E + M V ⊗ M E )] T r [ ρ ( M V ⊗ M L − M H ⊗ M L )]= V +Z , − Z T r [ ρ ( M V ⊗ M L + M H ⊗ M L )] T r [ ρ ( M D ⊗ M X − M A ⊗ M X )]= V φ T r [ ρ ( M D ⊗ M X + M A ⊗ M X )] , (9)where Γ is the partial-transpose operation on a subsys-tem, such as on the polarization-qubit subsystem, and ⊗ denotes the tensor product. Note that, we formulate0the last three constraints according to the measured vis-ibilities. The first two are based on entanglement visi-bilities V +Z , +Z and V − Z , − Z conditioned on measurementoutcomes of the time-bin qubit being early time and latetime, respectively. The last constraint is based on en-tanglement visibility V φ , where the time-bin qubit comesout in the middle bin. Since the MM-TQA has only oneoutput, we cannot differentiate the case when the photoncomes out from the second output if this output existsfrom the case when the photon is lost over the transmis-sion. Hence, we cannot formulate two constraints basedon V φ .In our experiment, we verified that within experimen-tal errors the visibilities V +Z , +Z = V − Z , − Z . So, for solv-ing the SDP in Eq. (9) we set V +Z , +Z = V − Z , − Z = V Z .Using the measured results of Method 1(2) V Z = 0 . ± .
01 (0 . ± .
01) and V φ = 0 . ± .
01 (0 . ± . ρ must be entangled. Furthermore, by numerically check-ing over which values of V Z and V φ the SDP in Eq. (9)is not possible, we are able to upper bound the requiredvisibilities V Z and V φ that certify the presence of entan-glement in the system. The numerical results are shownin Fig. 6. From this figure, one can see that our visibilityresult at any observed incident angle witnesses entangle-ment with high confidence. Finally, we would like to notetwo points. First, the constraints considered in Eq. (9)are independent of the transmission or conversion lossof the photon arriving at the MM-TQA, and even inde-pendent of the common photon loss in the two differentpaths of the MM-TQA. Therefore, the upper bounds onthe visibilities V Z and V φ obtained for verifying entangle-ment are independent of all of these different losses. Sec-ond, our obtained classical boundary [see Fig.6] is evenindependent of the relative loss between the two paths ofthe MM-TQA. ∗ Present address: National Research Council of Canada,1200 Montreal Road, Ottawa, Ontario K1A 0R6,Canada; [email protected] † Present address: Rockefeller University, 1230 York Av-enue, New York, New York 10065, USA ‡ Present address: NTT Basic Research Laboratories,Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198,Japan § [email protected][1] R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H.Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T.Jennewein, J. Perdigues, P. Trojek, B. ¨Omer, M. F¨urst,M. Meyenburg, J. G. Rarity, Z. Sodnik, C. Barbieri, H.Weinfurter, and A. Zeilinger, Nat. Phys. , 481(2007).[2] J. Yin, Y. Cao, Y.-H. Li, S.-K. Liao, J.-G. Ren, W.-Q.Cai, W.-Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H.Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang,M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen,C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W.Pan, Science , 1140 (2017).[3] H. Takenaka, A. Carrasco-Casado, M. Fujiwara, M. Ki-tamura, M. Sasaki, and M. Toyoshima,
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