Entanglement certification and quantification in spatial-bin photonic qutrits
EEntanglement certification and quantification in spatial-bin photonic qutrits
Debadrita Ghosh , Thomas Jennewein , Urbasi Sinha ∗ Light and Matter Physics, Raman Research Institute, Bengaluru-560080, India † and Institute for Quantum Computing, University of Waterloo, 200 University Avenue West, Waterloo, ON Canada
Higher dimensional quantum systems are an important avenue for new explorations in quantumcomputing as well as quantum communications. One of the ubiquitous resources in quantum tech-nologies is entanglement. However, so far, entanglement has been certified in higher dimensionalsystems through suitable bounds on known entanglement measures. In this work, we have, for thefirst time, quantified the amount of entanglement in bi-partite pure qutrit states by analytically relat-ing statistical correlation measures and known measures of entanglement, and have determined theamount of entanglement in our experimentally generated spatially correlated bi-partite qutrit system.We obtain the value of Negativity in our bi-partite qutrit to be 0.85 ± ± ∼
15% deviation while the EOF value demonstrates ∼
24% deviation. This serves as the first experimental evidence of such non-equivalence of entangle-ment measures for higher dimensional systems.
Entanglement [1] is one of the pivotal features ofquantum mechanics that has deep-seated implicationslike nonlocality [2] and is a crucial resource for quan-tum communication and information processing tasks[3–12]. While the two-dimensional (qubit) case contin-ues to be widely studied in the context of various appli-cations of quantum entanglement, it has been graduallyrecognised that higher dimensional entangled states canprovide significant advantages over standard two-qubitentangled states in a variety of cases, like, increasing thequantum communication channel capacity [13, 14], en-hancing the secret key rate and making the quantumkey distribution protocols more robust in the presenceof noise [15–18] as well as enabling more robust tests ofquantum nonlocality by reducing the critical detectionefficiency required for this purpose [19].Against the above backdrop, the enterprise of experi-mentally realising higher-dimensional entangled states,along with studies on the question of optimally cer-tifying and quantifying higher-dimensional entangle-ment is of considerable importance. The usual methodof characterising quantum states i.e. Quantum StateTomography (QST) or estimation of any entanglementmeasure would require determination of an increasinglylarge number of independent parameters as the dimen-sion of the system grows [20]. Therefore, formulating ex-perimentally efficient methods for the characterizationof higher-dimensional entangled states based on limitednumber of measurements has become an active area ofresearch [21–24].While many approaches for characterization of higherdimensional entanglement [25–36] have been studied(refer to [37] for an overview), only few schemes provideboth necessary and sufficient certification together with ∗ Electronic address: [email protected] † now at Laser-Laboratorium, Georg August Universit¨at G¨ottingen,Germany quantification of high-dimensional entanglement. Thequantification schemes that have been suggested so farseem to have focused essentially on providing boundson entanglement measures. This gives rise to a need forexploring quantification of entanglement, by which, oneessentially means determining the actual value of an ap-propriate entanglement measure in terms of a limitednumber of experimentally measurable quantities usingan analytically derived relation, for a given entangledstate. Towards filling this gap, we apply a recentlydeveloped technique of experimentally generating spa-tially correlated bipartite photonic qutrits, whereby thequantumness of the correlated state generated is estab-lished through certification and quantification of entan-glement. This is achieved for pure bipartite qutrits,by formulating and experimentally verifying analyticalrelations between statistical correlation measures andknown measures of entanglement namely Negativityand Entanglement of Formation. A salient feature of ourwork lies in our choice of correlation measures. On theone hand, we have employed commonly used correla-tion measures such as Mutual Predictability and MutualInformation; on the other hand, to the best of our knowl-edge we are the first team to use the Pearson CorrelationCoefficient ( P CC ) for higher dimensional entanglementcharacterization together with quantification.In this context, it is worth noting that common choicesfor photonic higher dimensional systems include thosebased on exploiting the Orbital Angular Momentumdegree of freedom of a single photon [38–42], spatialdegree of freedom by placing apertures or spatial lightmodulators in the path of down-converted photons[43–48] as well as time-bin qudits [49–51]. Recently,our group has demonstrated a novel technique forspatial qutrit generation which is based on modu-lating the pump beam in spontaneous parametricdown-conversion (SPDC) by appropriately placed tripleslit apertures [52]. This leads to direct generation ofbipartite qutrits from the SPDC process which we call a r X i v : . [ qu a n t - ph ] S e p spatial-bin qutrits. This technique has been shown [52]to be more efficient and robust, also leading to a moreeasily scalable architecture than what is achieved by theconventional method [53] of placing slits in the pathof down-converted photons. We used the pump-beammodulation technique for the first time in the qutritdomain yielding spatially correlated qutrits with avery high degree of correlation between measurementsdone in the image plane. However, no complementarybasis (focal-plane) measurement was done to certifyentanglement as has been argued to be necessary forsuch certification [47].In this work, what we have done has a three foldnovelty. One is the certification of entanglement in pumpbeam modulated qutrits by appropriate measurementsin complementary basis. The second is the main sig-nificance wherein we have shown in theory that wecan relate statistical measures of correlations to knownmeasures of entanglement i.e. Negativity( N ) andEntanglement of Formation( E OF ) through analyticallyderived monotonic relations and have applied theserelations to quantify the produced entanglement in ourexperiment. Then the operationally relevant questionarises as to how close is this prepared unknown stateto the maximally entangled state? In this context, ourwork’s final novelty lies in the comparison betweenthe two inferred values of the two measures of en-tanglement i.e. N and E OF vis-a-vis their respectivedeviation from their defined values correspondingto the maximally entangled state. We find that thesetwo measures of entanglement are non-equivalent inthe sense that they yield different estimates of thedeviation from the maximally entangled state. In avery recent study, the feature of non-equivalence hasbeen comprehensively shown and analysed by us fortwo qubit pure states where the measures are seen toremain monotonically related [54]. However, in higherdimensional systems, studies have indicated that forcertain classes of states, the different measures are notmonotonically related to each other [55–57]. Ours is thefirst experimental demonstration of non-equivalence (inthe sense defined above), between different measuresof entanglement in higher dimensional bipartite purestates. An illustration of this feature along with atten-dant non monotonicity is provided in Appendix C interms of theoretical estimates of the amounts of nonequivalence for different choices of Schmidt coefficientsfor two qutrit pure states. This opens up interestingquestions regarding the optimal choice of entanglementmeasure in different higher dimensional quantuminformation protocols.Now, proceeding to the specifics of this paper, we firstestablish the relations between three statistical measuresand two known measures of entanglement, followed bydiscussion of the experimental scheme we have usedand the implications of the results obtained.
Deriving an analytic relation between
PCC and N for purebipartite qutrit states P CC for any two random variables A and B is definedas C AB ≡ (cid:104) AB (cid:105) − (cid:104) A (cid:105) (cid:104) B (cid:105) (cid:113) (cid:104) A (cid:105) − (cid:104) A (cid:105) (cid:113) (cid:104) B (cid:105) − (cid:104) B (cid:105) , (1)whose values can lie between − (cid:104)·(cid:105) is anaverage value.It is noteworthy that P CC has so far been used inphysics only in limited contexts [58, 59] until recentlywhen Maccone et al. [36] suggested its use for entangle-ment characterization.Let us suppose two spatially separated parties, Aliceand Bob, share a bipartite pure or mixed state in an ar-bitrary dimension; Alice performs two dichotomic mea-surements A and A and Bob performs two dichotomicmeasurements B and B on their respective subsystems.Then, Maccone et al. conjectured that the sum of two P CC s being greater than 1 for appropriately chosen mu-tually unbiased bases would certify entanglement of bi-partite systems, i.e., for A = B = ∑ j a j | a j (cid:105)(cid:104) a j | and A = B = ∑ j b j | b j (cid:105)(cid:104) b j | , |C A B | + |C A B | >
1, (2)would imply entanglement. Here, {| a j (cid:105)} is mutuallyunbiased to {| b j (cid:105)} . However, Maccone at al. justifiedthis conjecture only by showing its applicability for bi-partite qubits and the validity of this conjecture has re-mained uninvestigated for dimensions d >
2. In thiswork, we have justified the validity of this conjecture forpure bipartite qutrits by deriving an analytic relation be-tween
P CC and N and tested this relation by applyingit to quantify the amount of entanglement in our exper-imentally generated pure bipartite qutrits. Please referto [37] for a detailed study encompassing a wide rangeof mixed states for qutrits as well as qudits of higher di-mension.Consider a pure bipartite qutrit state written inSchmidt decomposition | ψ (cid:105) = c | (cid:105) | (cid:105) + c | (cid:105) | (cid:105) + c | (cid:105) | (cid:105) (3)where {| (cid:105) , | (cid:105) , | (cid:105)} are the computational bases.Let the basis {| b j (cid:105)} be the generalised ˆ σ x basis [37, 60,61] | b (cid:105) = √ [ | (cid:105) + | (cid:105) + | (cid:105) ] (4) | b (cid:105) = √ [ | (cid:105) + ω | (cid:105) + ω | (cid:105) ] (5) | b (cid:105) = √ [ | (cid:105) + ω | (cid:105) + ω | (cid:105) ] (6)where ω = e i π /3 with i = √− A and ˆ B whose eigen-states are | b (cid:105) , | b (cid:105) and | b (cid:105) given by Eqs.(4),(5) and (6)respectively with the corresponding eigenvalues b , b and b . Then we writeˆ A = ˆ B = b | b (cid:105) (cid:104) b | + b | b (cid:105) (cid:104) b | + b | b (cid:105) (cid:104) b | (7)For the pair of observables given by Eq.(7) and the quan-tum state given by Eq.(3), one can use Eqs.(4),(5) and (6)to evaluate the quantity C A B as given by | C A B | = N (8)where N corresponds to the Negativity measure of en-tanglement corresponding to the state given by Eq.(3)[62]. For A = B = ∑ j a j | a j (cid:105)(cid:104) a j | where the basis {| a j (cid:105)} isthe computational basis, by computing the relevant sin-gle and joint expectation values for the pure two-qutritstates given by Eq.(3), one can obtain the P CC in thiscase to be given by | C A B | = | C A B | + | C A B | = + N (10)Here one may just briefly remark that for deriving thevalue of N from the observed value of P CC , it is empiri-cally advantageous to consider single
P CC because thenthe associated error range is less than when the sum of
P CC s is considered. Hence, from the point of view ofsuch experimental consideration, in the following sec-tions, we consider Mutual Predictability ( MP ) and Mu-tual Information ( MI ) as single quantities for analyti-cally linking them to N and E OF respectively.
Relating Mutual Predictability with Negativity
In showing the relation between MP and N , we in-vestigate the situation in which generalised ˆ σ x observ-able is measured as one observable and its complex con-jugate is measured as the other. We discuss the casewhen both observables are the same (see Appendix A).Consider a pure bipartite qutrit state written inSchmidt decomposition as given in Eq.(3).For the complex conjugate basis of | b j (cid:105) and using ω ∗ = ω and ( ω ) ∗ = ω we obtain | b (cid:105) ∗ = | b (cid:105) , | b (cid:105) ∗ = | b (cid:105) and | b (cid:105) ∗ = | b (cid:105) .The above relations imply that one can obtain theprobability of detecting the quantum state in the | b (cid:105) ∗ or | b (cid:105) ∗ or | b (cid:105) ∗ state pertaining to the complex conjugateof generalised ˆ σ x basis by using the generalised ˆ σ x basisand obtaining the corresponding probability of detect-ing the quantum state in | b (cid:105) or | b (cid:105) or | b (cid:105) state respec-tively. Now, if we consider measuring generalised ˆ σ x op-erator on one system and its complex conjugate on the other, then we can obtain the joint probabilities P ( b i , ¯ b j ) whence P ( b , ¯ b ) = P ( b , ¯ b ) = P ( b , ¯ b ) (see AppendixB for details). The quantities P ( b , ¯ b ) , P ( b , ¯ b ) in thiscase are same as the quantities P ( b , b ) and P ( b , b ) asmeasured by using generalised ˆ σ x operator on both sys-tems. Then, one can obtain [61] the MP as C = ∑ i P ( b i , ¯ b i ) = ( + N ) (11)where N is the negativity of the bipartite qutrit state[62]. Relating Mutual Information with Entanglement ofFormation
Let the common basis of the pair of observables A and B pertaining to Alice and Bob be the computational ba-sis. For this choice of measurements, let the joint prob-abilities be p ( ab | AB ) and the marginal probabilities be p ( a | A ) and p ( b | B ) for the pure two-qudit state | ψ d (cid:105) = d − ∑ i = c i | ii (cid:105) , (12)where 0 ≤ c i ≤ ∑ i c i =
1, are given by p ( ab | AB ) = c i for a = b = i and 0 otherwise while p ( a | A ) = p ( b | A ) = c i for a = b = i . Substitutingthe above joint probabilities and marginal probabilitiesin the expression for MI [36] given by I AB = d − ∑ a , b = p ( ab | AB ) log p ( ab | AB ) p ( a | A ) p ( b | B ) , (13)we obtain I AB = − d − ∑ i = c i log c i (14)Now, note that E OF , for bipartite pure states | ψ (cid:105) AB is equal to the von Neumann entropy of either of the re-duced density matrices, i.e., E ( | ψ (cid:105) AB ) = S ( ρ A ) = S ( ρ B ) ,here S ( ρ ) = − Tr ρ log ρ . For the general pure two-qudit state as given by Eq.(12), E OF is given by the fol-lowing expression: E ( | ψ d (cid:105) ) = − ∑ i c i log c i . (15)since S ( ρ A ) = S ( ρ B ) = − ∑ i c i log c i . From Eqs.(14)and (15), it then follows that MI pertaining to the com-putational basis on both sides equals the E OF for anypure bipartite qudit state.
FIG. 1: Schematic of the experimental set-up. L1, L2, L3: Planoconvex lenses, BBO: Nonlinear crystal for SPDC, LP: Long-pass filter, BS: 50-50 Beamsplitter, BP: Band-pass filter, CL:Cylindrical Lens, D1, D2: Single photon detectors.FIG. 2: Normalized coincidence count (Rc) vs detector position(X) in focal plane. Blue line indicates the theoretical predictionwhereas the red circle indicates the experimental result. Thenormalised coincidence plot exhibiting interference is a certifi-cation of entanglement.
Experimental scheme
A Type-1 BBO source is used to generate spatiallycorrelated qutrit pairs using our previously developedpump beam modulation technique [52]. AppendixD has details on the source. We explore the system(schematic in Fig. 1) at two different positions, i.e. focal(f) and image plane (2f) of the lenses L2 and L3. Whenthe detectors are placed at the focus of lenses L2 and L3,the lenses transfer the triple-slit interference in the cor-relation of the signal and idler photons to the detectorplane. In addition, there are two cylindrical lenses offocal lengths 50 mm and 60 mm in the transmitted (let’scall signal arm) and reflected (let’s call idler arm) armsof BS respectively which focus the single photons alonga line at the detector plane. The individual measuredsingles spatial profiles of signal and idler photons havea flat top Gaussian structure.We fix one detector at the centre of an individual profile and move the other detector to measure the coincidence.The moving detector is scanned over 4 mm in 30 µ m stepsize. At each position, three counts are recorded, withan accumulation time of 90 sec each. Fig.2 shows thecoincidence profile measured in the interference plane.The blue line represents the theoretical prediction whilethe red circles are the measured values. We includeerror bars in terms of position and number uncertainty.The error in the position is limited by the step size ofthe actuator which moves the detector. The chosen stepsizes of the actuator to measure the profiles are 10 µ m and 30 µ m for image plane and focal plane respectively.Experimental and theoretically generated Rc have beenappropriately normalised by their respective maxima.The focal plane measurement shown in Fig.2 has animportant significance. As discussed in [47], whencross correlation measurements (also called coincidencemeasurements) as a function of detector position in thefocal plane for both the signal and idler photons exhibitinterference, this implies certification of entanglement.The image plane corresponds to the generalised σ z likeoperator with eigenstates comprising of the computa-tional basis states (discussed in theory section above).The three slit peak positions represent the three eigenstates with eigen values 0, 1 and -1 respectively. In orderto calculate the P CC for generalised σ z like operatorsapplied to both the signal and the idler photons, weneed to measure the corresponding joint probabilities.We fix one detector at the three peak positions, oneposition at a time, of its singles spatial profile (lets saysignal arm) and measure the coincidence counts whenthe other detector is at the peak positions of its singlesprofile (idler arm). Thus, by measuring the peak to peakcoincidence counts we construct a 3 x 3 matrix with 9components. The maximum coincidence counts are thediagonal elements of the matrix. We measure 5 suchsets of matrices and find the average P CC to be 0.904(2).Table I (left) shows a representative correlation matrix.The focal plane corresponds to the generalised σ x likeoperator with eigenstates given by Eqs.(4), (5) and(6). These correspond to three unique positions inthe measured cross correlation profile (see AppendixE). In order to measure the P CC for generalised σ x like operators applied to both the signal and the idlerphotons, we need to measure the corresponding jointprobabilities like in the case of σ z like operator. Toconstruct such a joint probability matrix, we measurethe singles profile for the signal arm and by fitting itwith a flat-top Gaussian module function, we find outthe centre position naming it y . Similarly, we find thecentre position for the idler singles profile and nameit x . Next, we fix the signal arm at y and scan theidler arm to measure the coincidence profile. From this,we extract the positions x , x and x correspondingto the eigenstates of the generalised σ x like operator.Next, we fix the idler arm at x , x and x respectivelyand for each fixed position we scan the signal arm tomeasure the coincidence profiles. From each of the FIG. 3: Normalized singles count (Rs) vs detector position (X) measured in image plane (left). Normalized coincidence count (Rc)vs detector position (X) measured in focal plane (right). Here, the signal arm detector is fixed at the centre of the singles profile(called y in text) while the idler arm is scanned. x , x and x in the plots represent the positions derived from the generalized σ z and σ x like operators respectively. coincidence profiles, we extract the coincidence countsat the positions y , y and y which are the derivedpositions for generalized σ x like operator for signal arm.Fig. 3 represents the detector positions corresponding togeneralised σ z and σ x eigen states respectively. Thus weconstruct a 3x3 correlation matrix. We measure 5 suchsets of matrices and find the average P CC to be 0.848(2).Table I (right) shows a representative matrix. We workwith operators defined by assigning eigen values 0,1 and -1 respectively to the eigen states of the σ x likeoperator.Then by invoking Eq.(8) derived earlier, we derive N asa measure of entanglement to be 0.848 ± N from measured MP . As shownbefore, when generalised σ x basis is measured on oneside (say the signal arm) and its complex conjugatebasis is measured on the other side (idler arm), thenthe quantities P ( b , ¯ b ) , P ( b , ¯ b ) in this case are sameas the quantities P ( b , b ) and P ( b , b ) respectively, asmeasured by using generalised ˆ σ x basis on both sides.Thus summing over the x − y , x − y and x − y elements of the correlation matrix as represented inTable I, one can derive MP . The MP so derived comesout to be 0.899 ± N derived using Eq.(11)equal to 0.849 ± MI where c i are thenormalised coincidence counts when one detector isfixed at x (then x and x ) and the other detector movesfrom y to y respectively. MI is equal to the E OF in case of computational basis as shown earlier. Ourcalculated
E OF is 1.23 ± N as a measure ofentanglement derived from two independent statisticalcorrelation measures is the same within error bound.An interesting point emerges here. All the derivationsshown in this manuscript as well as experimentally de- rived quantities assume the initial bi-partite qutrit stateto be a pure state. The value of N being the same withinerror bound serves as a consistency check for this as-sumption. Moreover, we know that the concept of co-herence is intimately connected with mixedness of thestate [63] and the measure of coherence is the Visibility ofthe interference [64]. Higher degree of coherence implieshigher state Purity. In our experiment, the interferencein the coincidence plane has a Visibility of ∼
94% whichindicates a high state Purity. The question now arises:what is the percentage deviation from the maximallyentangled state as quantified by two different entangle-ment measures i.e. N and E OF ? Noting that the maxi-mum values of N and E OF are 1 and log = N value quantifiesa ∼
15% deviation from the maximally entangled state,the deviation captured by the
E OF is around ∼ N and E OF as measures of entanglement. Our method of entangle-ment characterisation is applicable to unknown quan-tum states involving limited number of measurementsas opposed to extracting such information from a com-plete QST. This has led to a curious observation thatthe two measures of entanglement differ strikingly inquantifying the deviation of entanglement in the pre-pared unknown state from that in the maximally entan-gled state. This, in conjunction with the feature of nonmonotonicity illustrated by us (see Appendix C) calls fora deeper understanding for the underlying reasons and x x x y y y x x x y y y σ z like operator measured in image plane (left) and σ x like op-erator measured in focal plane (right). N from PCC N from
MP E from MI ± ± ± reexamining the efficacy of these measures for quantify-ing the resource for higher dimensional quantum infor-mation processing protocols. Acknowledgments
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Mutual Predictability when generalized σ x observableis measured on both sides Consider a pure bipartite qutrit state written inSchmidt decomposition | ψ (cid:105) = c | (cid:105) | (cid:105) + c | (cid:105) | (cid:105) + c | (cid:105) | (cid:105) (16)where {| (cid:105) , | (cid:105) , | (cid:105)} are the computational bases.Let the basis {| b j (cid:105)} be the generalized ˆ σ x basis [37, 60,61] | b (cid:105) = √ [ | (cid:105) + | (cid:105) + | (cid:105) ] (17) | b (cid:105) = √ [ | (cid:105) + ω | (cid:105) + ω | (cid:105) ] (18) | b (cid:105) = √ [ | (cid:105) + ω | (cid:105) + ω | (cid:105) ] (19)where ω = e i π /3 with i = √− P ( b i , b j ) = | (cid:104) ψ | | b i (cid:105) (cid:104) b i | ⊗ | b j (cid:105) (cid:104) b j | | ψ (cid:105) | (20)Using Eq.(20) we obtain the following joint probabilities P ( b , b ) = P ( b , b ) = P ( b , b ) (21) = ( + c c + c c + c c ) (22) P ( b , b ) = P ( b , b ) = P ( b , b ) (23) = P ( b , b ) = P ( b , b ) = P ( b , b ) (24) = ( − c c − c c − c c ) (25)It can be checked from Eqs.(21) and (23) that ∑ i , j P ( b i , b j ) = C = ∑ i P ( b i , b i ) (26) =
13 (27)
B. Mutual Predictability when generalized σ x observableis measured on one side and its complex conjugate ismeasured on the other side Consider a pure bipartite qutrit state written inSchmidt decomposition as given in Eq.(16).One can construct the complex conjugate bases of {| b j (cid:105)} as follows | b (cid:105) ∗ = √ [ | (cid:105) + | (cid:105) + | (cid:105) ] (28) | b (cid:105) ∗ = √ [ | (cid:105) + ω ∗ | (cid:105) + ( ω ) ∗ | (cid:105) ] (29) | b (cid:105) ∗ = √ [ | (cid:105) + ( ω ) ∗ | (cid:105) + ω ∗ | (cid:105) ] (30) where ω ∗ is the complex conjugate of ω . Using ω ∗ = ω and ( ω ) ∗ = ω we obtain | b (cid:105) ∗ = | b (cid:105) (31) | b (cid:105) ∗ = | b (cid:105) (32) | b (cid:105) ∗ = | b (cid:105) (33)Eqs.(31)-(33) imply that one can obtain the probabilityof detecting the quantum state in the | b (cid:105) ∗ or | b (cid:105) ∗ or | b (cid:105) ∗ state pertaining to the complex conjugate ofgeneralized ˆ σ x basis by using the generalized ˆ σ x basisand obtain the corresponding probability of detectingthe quantum state in | b (cid:105) or | b (cid:105) or | b (cid:105) state respectively.Now, if we consider measuring generalized ˆ σ x on oneside and its complex conjugate on the other side, thenwe can obtain the joint probabilities P ( b i , ¯ b j ) as P ( b i , ¯ b j ) = | (cid:104) ψ | | b i (cid:105) (cid:104) b i | ⊗ | b ∗ j (cid:105) (cid:104) b ∗ j | | ψ (cid:105) | (34)where ¯ b j denotes the eigenvalue corresponding to thecomplex conjugate of the | b j (cid:105) , whence we obtain P ( b , ¯ b ) = P ( b , ¯ b ) = P ( b , ¯ b ) = ( + c c + c c + c c ) (35)As discussed in the preceding paragraph, the quantities P ( b , ¯ b ) , P ( b , ¯ b ) in this case are same as the quantities P ( b , b ) and P ( b , b ) given by Eq. (21) respectively,as measured by using generalized ˆ σ x basis on both sides.Now, using Eq. (35), one can obtain [61] the MutualPredictability as C = ∑ i P ( b i , ¯ b i ) (36) = ( + c c + c c + c c ) (37) = ( + N ) (38)where N is the negativity of the bipartite qutrit state[62]. C. Analytical study of Relationship between N andEntanglement of Formation( E ) for two qutrit pure states Consider a two qutrit pure state with Schmidt coeffi-cients c , c & c | Φ (cid:105) = c | (cid:105) | (cid:105) + c | (cid:105) | (cid:105) + c | (cid:105) | (cid:105) (39)where 0 < c , c , c < c + c + c = 1 Deviations from the Maximally Entangled State
Two parameters are defined to measure the percent-age deviations of measures from the values correspond-ing to Maximally Entangled State [54] Q E = (( log ( ) − E ) / log ( )) ×
100 (40) Q N = ( − N )) ×
100 (41)To see to what extent these two parameters differ witheach other, the following quantity is an appropriate mea-sure [54] ∆ Q NE = | Q E − Q N | (42) Study of Monotonicity
Rate of change of E w.r.t c d E d c = ( ln ( )) c log ( − ( c + c )) / c ) (43)Similarly, rate of change of E w.r.t c d E d c = ( ln ( )) c log ( − ( c + c )) / c ) (44)Rate of change of N w.r.t c d N d c = c + ( − c c − c − c ) / (cid:113) − ( c + c ) (45)Similarly, rate of change of N w.r.t c d N d c = c + ( − c c − c − c ) / (cid:113) − ( c + c ) (46) Observations : • From Eqs. (43),(44),(45),(46) it can be seen that fora given c ( c ), E and N grow with c ( c ), reacha certain value and then start decreasing w.r.t c ( c ). All the above four Eqs.(43),(44),(45),(46) vanishwhen c = c = 1/ √ • As in the case of two qubit pure state [54], one can-not say that d E d N is always greater than zero exceptwhen c & c = 1/ √
3. This explains the presence ofnon-monotonic nature between these two param-eters. For example, consider a pair of two qutritpure states with Schmidt coefficients c = 0.4 , c =0.9 & c = 0.5 , c = 0.1. Former state has, E = N = E = N = E > E but N < N , showing E and N arenot monotonic w.r.t each other. Study of Deviations from the Maximally Entangled State • Different values of ∆ Q for different values of theSchmidt coefficients have been tabulated in TableIII. TABLE III: Differences in the % deviations from the value cor-responding to the maximally entangled state c c E N Q E Q N ∆ Q NE Observations : • Given a non-maximally entangled two qutrit purestate, one cannot comment as in the case of twoqubit case [54] that one entanglement measure isalways greater than other entanglement measurefor any value of state parameter. • ∆ Q NE takes a maximum value of 12.148% when c = c = • Thus, both as absolute and relative entanglementmeasures, Negativity and Entanglement of Forma-tion do not give equivalent results.
D. Details on the photon source and pump beammodulation technique
A Type-1, non-linear crystal (BBO) with a dimensionof 5 mm x 5 mm x 10 mm generates parametricallydown-converted degenerate photons at 810 nm wave-length with collinear phase matching condition. A diodelaser at 405 nm with 100 mw power pumps the crys-tal. The transverse spatial profile of the pump beamat the crystal is prepared by transferring the laser beamthrough a three-slit aperture with 30 µ m slit width and100 µ m inter-slit distance and imaging it at the crys-tal. A plano-convex lens (L1) of focal length 150 mmis placed such that it forms the image of the three-slitat the crystal. A 50-50 beam-splitter (BS) placed afterthe crystal splits the two down-converted photons in thetransmitted and reflected ports of the BS. A band-passfilter centred at 810 nm with a FWHM of 10 nm passesthe down-converted photons and a long-pass filter withcut-off wavelength 715 nm blocks the residual pump. Ineach arm of the BS, a plano-convex lens (L2 and L3 re-spectively) of focal length 75 mm is placed at 2f distancefrom the crystal to transfer the signal and idler photonprofile to the detectors.0 E. Experimentally defining generalized σ x and generalized σ z basesExperimental realization of σ z The state of the down-converted photon after passingthrough a three-slit can be written as, | ψ (cid:105) = √ ( c | (cid:105) + c | (cid:105) + c | (cid:105) ) (47)The probability to detect a photon prepared in the state | ψ (cid:105) in the position corresponding to the n -th slit imageis proportional to | c n | and the measurement operatorscan be defined as M n f ( n ) = µ n f | n (cid:105) (cid:104) n | (48)where µ n f is the normalization factor[53]. The σ z matrixin 3-dimension is σ z = − (49)which can be written as σ z = | (cid:105) (cid:104) | − | (cid:105) (cid:104) | (50)So the positions corresponding to the eigen bases are thecenter of the first and third slit image profile. Experimental realization of σ x A detection in the position x in the focal plane corre-sponds to the projector onto | k x (cid:105) . k x = xk / f (51) where k x is the transverse wave vector, k is the wavenumber and f is the focal length of the lens. The de-tection probability is proportional to [53] | (cid:114) a π sinc ( k x a /2 ) | φ ( k x d ) (cid:105) (cid:104) ψ | | (52)where | φ ( θ ) (cid:105) = | (cid:105) + exp i θ | (cid:105) + exp 2 i θ | (cid:105) . Hence, themeasurement operators in the far field can be defined as M f f ( θ ) = µ f f | φ ( θ ) (cid:105) (cid:104) φ ( θ ) | and the phase parameter is θ = π xd / λ f (53)the operator σ x can be written as O = | b (cid:105) (cid:104) b | − | b (cid:105) (cid:104) b | (54)where | b (cid:105) , | b (cid:105) and | b (cid:105) are | b (cid:105) = √ ( | (cid:105) + | (cid:105) + | (cid:105) ) (55) | b (cid:105) = √ ( | (cid:105) + e i π /3 | (cid:105) + e − i π /3 | (cid:105) ) (56) | b (cid:105) = √ ( | (cid:105) + e − i π /3 | (cid:105) + e i π /3 | (cid:105) ) (57)which are also given by the Eqs. (4),(5) and (6) in mainpaper.The corresponding angles of the eigenvectors of σ x are0, 2 π and π µ m which are x , x and x as mentioned in main paper; where, λ = µ m , f = cm , d = µ mm