Measurement-device-independent quantification of irreducible high-dimensional entanglement

The certification and quantification of entanglement are of great importance in characterizing entangled systems. Recently it is pointed out that the quantum correlation of higher dimensional bipartite states can be simulated with a sequence of lower dimensional states and sequential measurements. Such scheme may render some entanglement tests unreliable---the observed entanglement may not be a genuine high-dimensional one. Here we show that a recently proposed quantitative measurement-device-independent entanglement witness protocol is naturally robust against such scheme, thus can be used as an irreducible high-dimensional entanglement quantifier and irreducible dimension witness. We then experimentally demonstrate this protocol on a 3-dimensional bipartite state, observing an entanglement that exceeds the bound when the cheating scheme with arbitrary 2-dimensional states is used. The result certifies our system is entangled in dimension (at least) d = 3.

Introduction.-One of the core problems of quantum theory lies in the certification and quantification of quantum entanglement, which is the key resource in various quantum information processes [1][2][3]. Great efforts have been made to detection quantum entanglement, see Ref. [4,5] for a review. Imperfect measurement schemes, however, may lead to incorrect estimation of the entanglement under investigated, thus invalidate these methods in practical scenarios [6].
Such imperfect entanglement detection tasks can be described as a three-party game, in which Charlie, a referee, aims to certify the shared entanglement between two parties, Alice and Bob. However, neither of Alice and Bob is reliable, they may be unable to operate on their share state as Charlie required, or may even deliberately adopt cheating schemes to declare more entanglement than they actually possess. Therefore Charlie must find methods to avoid these to happen. A solution comes from the field of device-independent (DI) quantum tests [7][8][9], where Alice's and Bob's experiment setups are treated as a black box. Charlie tells the black box what measurements to perform by giving them classical labels, and the black box returns the measurement outputs. In a DI entanglement detection game, Charlie does not need to know what measurements are performed, but only need to check the correlations on the inputs and outputs using some Bell-like inequalities.
However, Alice and Bob are still possible to cheat in a DI entanglement detection game. One of their schemes is based on local operation and classical communication (LOCC), which leads to the locality loophole in Bell tests [10]. Another one lies in the fact that, as pointed out in a recent work, quantum correlations of high-dimensional states may be simulated by sequential measurements on a sequence of lowerdimensional states with the assistance of classical feedforward [11]. This makes it possible to attain a quantity of entanglement that can only be obtained in a higher-dimensional system with lower-dimensional states in a Bell-type test. We call such scheme "simulation with sequential lower-dimensional states" (SSLDS), and say Charlie can detect irreducible entan- glement if the test he uses is robust against SSLDS. A common reason that these schemes work is Alice and Bob can get exact information of the black box's inputs. For example, the feedforward process in SSLDS scheme requires the knowledge of "what measurement is going to be simulated" to decide the choice of the measurement sequence. A potential way to solve such problems, called measurement device independent (MDI) quantum test [12], was introduced based on semiquantum scenarios [13]. Instead of using classical inputs, Charlie sends Alice and Bob quantum states from a nonorthogonal set. With a compromise of trusting these quantum input states, Charlie prevents Alice and Bob from distinguishing what the inputs are. Such MDI tests have been proved to be robust against LOCC. One may ask intuitively, can we find a MDI test that is also robust to SSLDS scheme? Or alternatively, can we construct a MDI game where Charlie can make sure that the entanglement he observes is irreducible?
Here we show that a recently proposed MDI protocol, called quantitative measurement device independent entanglement witness (QMDIEW) in Ref. [14,15] is intrinsically robust against the SSLDS scheme, thus provides a method to quantify irreducible entanglement in a MDI way. We prove that even if the SSLDS scheme with a sequence of mdimensional bipartite states are used, the witnessed entanglement in the QMDIEW will never exceed the upper bound of a m-dimensional system. This means that the shared system is truly entangled in dimension at least m + 1 when its quantified entanglement exceeds the upper bound of m-dimensional systems. The "dimension" here is exactly the concept "irreducible dimension" for bipartite systems in Ref. [11]. So, our result shows that the QMDIEW can witness irreducible dimension in a MDI scenario. We then experimentally demonstrate the protocol on a two-qutrit system and observe a lower bound of its generalized robustness (GR) [16] that exceeds the value of arbitrary 2-dimensional systems. Our experiment shows the existence of irreducible high-dimensional entanglement for the first time, marking an important step beyond previous experimental demonstration of MDIEW on qubit systems [17][18][19]. Also, we demonstrate the noise-resistant feature of the protocol by certifying the entanglement of 3dimensional Bell states in noisy environment.
Framework of QMDIEW.-We briefly introduce the construction of the QMDIEW, defined as a MDIEW whose expectation on a state provides a lower bound on its entanglement. In a semiquantum scenario, Alice and Bob share a bipartite state ρ AB , with their measurement devices are treated as black boxes. The blackbox-like devices receive pairs of quantum states τ x (for Alice's), τ y (for Bob's) from Charlie who prepares the input states chosen randomly from a given set {τ 1 ...τ n }. To certify the entanglement of the shared state, Alice (Bob) performs a joint measurement (described by the POVM {A a } or {B b }) on her (his) own system and input states. Correlations in this case write where, a (b) is the outcome of A a (B b ). The system above can also be described with an effective POVM {Π ab } acting on input states τ x ⊗ τ y . In this case, correlations in Eq. (1) can be generally expressed as It is proved that the POVM {Π ab } can be used to recover the ensemble which can be extracted from the setup and whose entanglement provides a lower bound on the entanglement of the shared state ρ AB [15]. That is, the entanglement of ρ AB under an arbitrary convex entanglement E [20] is lower bounded by applying the measure E on the set of operators {Π ab }; formulated as where d X (d Y ) is the dimension of the Hilbert space H X (H Y ) of Alice's (Bob's) input state and Q is the lower bound obtained in this method. The entire process from raw correlations in Eq.
(2) to a quantified entanglement Q (a lower bound) of ρ AB can be described as a semidefinit programming (SDP [21]) and a QMDIEW can be recovered from its dual program. Robustness of QMDIEW against SSLDS.-By adopting SSLDS scheme, Alice and Bob may convince Charlie that they have a higher entanglement which they do not actually possess (see SM [22] for more details). In Ref. [11] an additional test to exclude such scheme was given, but only for the two-ququart case. Now we show that QMDIEW is naturally robust against SSLDS scheme. Suppose Alice and Bob want to confirm Charlie that they have a bipartite state entangled in dimension n in a QMDIEW game, however, they only have a source that can produce a sequence of bipartite states entangled in dimension m (m < n) one after one, which allows them to make measurements sequentially. Under such restriction, the operations of Alice and Bob can summed up as two steps: sequential measurements step and decision step. After receiving the quantum input τ x , Alice begins the sequential measurement step (Bob being analogy). She makes a joint measurement on τ x and her part of the first m-dimensional bipartite state, obtaining an outcome a 1 , then she makes joint measurement on the post-measurement input system and her second state, obtaining an outcome a 2 , after a sequence of k measurements, Alice obtains a sequence of outcomes S a = (a 1 , a 2 ...a k ) (Bob obtains a sequence S b , analogously). Then in the decision step, Alice and Bob decide what outcome a, b they would report to Charlie according to some distribution p(a, b|S a , S b ) ( ab p(a, b|S a , S b ) = 1, ∀S a , S b ). Let P Sa,S b |τx,τy be the correlation of their sequential measurement process, then the the correlation P m SSLDS (a, b|τ x , τ y ) Charlie actually get is However, we demonstrate that in such case, the entanglement Charlie finally detected in QMDIEW can not exceed E m , the upper bound of m-dimensional states. The demonstration includes two theorems, whose proves are left in supplementary [22]. Theorem 1. In a SSLDS scheme where a sequence of mdimensional bipartite states and sequential measurements are used to simulate the correlation of some n-dimensional bipartite state, correlation P Sa,S b |τx,τy satisfies . Theorem 1 indicates that in a SSLDS scheme, the effective measurement operators Alice and Bob can simulate is strongly restricted. Let Q m SSLDS be the witnessed entanglement by Charlie, which is reconstructed by P Sa,S b |τx,τy using Eqs. (2), (7) and (4) (|HH u + |HH l + |V V l ) (the subscripts u and l denote upper and lower path respectively)and maximal mixed state I/9 are prepared respectively and the two states are then combined with different proportions at the beam splitters. The photon pair rate was around 2500 counts/s with a detection efficiency of 0.222. In the Trust module, arbitrary three dimensional pure state can be prepared by expanding the distributed photons' path degree of freedom. High quality of the input state is ensured by highly controllable operations HWPS and BDs. In the BSM module, projector onto arbitrary one of the nine 3-dimensional Bell state can be constructed by setting angles of HWPs and blocking proper pathes. BD-beam displacer, PBS-polarizing beam splitter, HWP-half wave plate, QWP-quarter wave plate.
Theorem 2 gives an important result that the entanglement Q m SSLDS Alice and Bob can simulate with an m-dimensional SSLDS scheme can not exceed the bound of E m of mdimensional bipartite states. Therefore, if the entanglement Charlie obtains exceeds the bound of m-dimensional bipartite states, he can be confirmed that Alice and Bob own bipartite states of dimension at least m + 1. Therefore we proved that Charlie can exclude the effect of SSLDS without any additional tests, making it attractive to demonstrate irreducible high-dimensional entanglement in the lab.
Experimental protocol.-We consider the experimental observation of irreducible high-dimensional entanglement with a simplest system composed of a two-qutrit state. To simulate 3-dimensional entanglement states, Alice and Bob should adopt SSLDS scheme of two-qubit states. In the QMDIEW game, however, Charlie can identify the system is truly entangled in dimension 3 as long as he observes its entanglement exceeding the upper bound of two-qubit systems. Here, the entanglement measure is chosen to be GR, whose value is upper bounded by n-1 for bipartite n-dimensional systems [16].
In our experiment, the state is chosen to be one of the maximally 3-dimensional entangled states, i.e. ρ AB = |φ φ|, with |φ = 1 √ 3 (|00 + |11 + |22 ). After the distribu-tion of ρ AB , Charlie prepares input states τ x and τ y randomly selected from a set S ≡ {|0 , |1 , |0 + |1 , |0 + i|1 , |0 + |2 , |1 + |2 } [23] and sends them to Alice and Bob. Then Alice and Bob choose to perform POVM {A a } and {B b } consisting of the projectors onto all nine Bell states on the distributed qutrit and input state. The obtained correlations P (a, b|τ x , τ y ) are reported to Charlie who then performs a regularization process to find out the closest regularized distribution P r (ab|τ x , τ y ) and thus corrects the inconsistencies of raw P (ab|τ x , τ y ) caused by noise and finite statistics [17,22]. Based on P r (ab|τ x , τ y ), Charlie can finally calculate the GR of ρ AB and recover a QMDIEW. With the quantified GR of ρ AB , Charlie can also determine a lower bound on its irreducible dimension of entanglement according to Theorem 2.
As a byproduct, we also apply the QMDIEW to several noisy states ρ noise = pρ AB + (1−p) 9 I (p ∈ (0, 1)) with the same input set S, but only collect 3 outcomes of BSM each side [24]. Although incomplete data collection decreases the quantify of entanglement Charlie can observed, this will not invalidate its efficiency to determine if the state under tested is entangled [17][18][19]. We thus demonstrate the availability of the protocol under noisy environment, informationally incomplete inputs and measurement outcomes.
We realize this procedure using linear optics. The whole experimental setup is illustrated in Fig. 2 and can be divided into three modules: the Source module for state preparation (orange), the Trust module for trusted inputs preparation (blue) and the BSM for three dimensional Bell state measurement (green). For further details, see [22]. Results analysis.-We calculate the quantity of GR of ddimensional Bell states in a noisy environment, i.e. ρ noise , where only white noise is considered for simplicity. Fig. 3 shows the results of entanglement quantification, where GR as an entanglement measure varies with different Bell state fraction p. The results show that with same fraction p, 3-dimensional states (red line) are more robust than 2dimensional ones (blue line). When p = 1, the values of GR reach maximums for both 3-and 2-dimensional cases and are 2 and 1 respectively. The black dot is the experimental result of the state ρ AB and the obtained value of GR is 1.632±0.017 which exceeds the bound of arbitrary 2-dimensional system by more than 37 standard deviations. The deviation of the GR value from its theoretical predictions are due to imperfect state preparation and measurement error. Our experimental result of GR requires a fidelity of our state to ideal ρ AB of approximate f = 0.985. The experimental fidelity f real = 0.986 ± 0.002 matches well with its theoretical expectation [25].
Meanwhile, Fig. 3 indicates that GR of the two-qutrit state ρ noise with p > 5/8 exceeds the upper bound of two-qubit systems, implying our scheme can be used to detect the irreducible entanglement for bipartite states. In our case, the observed GR confirms that our system is genuinely entangled at least in dimension d=3, which means it cannot be simulated with qubit systems through SSLDS scheme. In other words, we demonstrate an irreducible 3-dimensional state with out characterizations on the measurement apparatus.
The results of the QMDIEW of 3-dimensional isotropic states ρ noise are shown in Fig. 4 black dots represent the theory expectation and experimental results respectively and they match very well. When p < 1/4, the witness value is zero, which means the states are separable. Our experiment can certify almost all entangled ρ noise . Theoretically the entanglement of state ρ noise should vary linearly with p, the nonlinearity of our results stems from incomplete information we obtained using tomographically incomplete input set S and incomplete BSM. If the input set is tomographically complete, the result would be linear as the red line in Fig. 4. If complete BSM being implemented, the result would agree with the one in Fig. 3. The error bars of the parameter p are due to the inaccuracy of the ratio of ρ AB to I/9 in the state ρ noise . All of the error bars are estimated by Monto Carlo simulation.
Conclusion.-In this paper, we displayed the power of a recently reported QMDIEW protocol for high-dimensional systems by showing that it can witness irreducible entanglement, which can not be simulated by lower-dimensional systems. Our results shows that QMDIEW not only can be used to quantify how much entanglement there is, but also can help to determine the genuine dimension that the entanglement exists in. As the simplest application, we demonstrated the highdimensional QMDIEW protocol with a 3-level bipartite entangled state and quantified a lower bound of its GR that exceeded the bound of 2-dimensional systems, presenting the existence of irreducible 3-dimensional entanglement.
Our work here may provide inspiration for some future works. From a theoretical point of view, further exploration into the robustness of MDI tests is possible. For instance, we may consider a more general sequential scheme that simulates a higher entanglement by a sequence lower entanglement states, and discuss the robustness of MDI tests against it. From a practical point of view, our experiment leads to important applications based on high dimensional entangled systems, such as randomness generation [26] and high dimen-sional quantum key distribution [27], without assumptions on the measurement apparatus.
We thank D. Rosset

Supplementary Note 1: SDP for entanglement quantification
The semidefinite programming (SDP) [21] we use for entanglement quantification is as follow: subject to tr(Π ab τ x ⊗ τ y ) = P (a, b|τ x , τ y ), ∀abxy, where d X (d Y ) is the dimension of the Hilbert space H X (H Y ) of Alice's (Bob's) input state. The optimized solution Q obtained in SDP (7) is the lower bound of entanglement under measure E.

Supplementary Note 2: data regularization
Due to the existence of experimental imperfections, the correlations P (a, b|τ x , τ y ) we obtained through experiment may not be able to run SDP (7) successfully. For example, there may be no valid operators Π ab that can realize P (a, b|τ x , τ y ) in SDP (7). So the correlations P (a, b|τ x , τ y ) must be re-regularized. We find a set of re-regularized correlations P r (a, b|τ x , τ y ) using another SDP: subject to tr(Π ab τ x ⊗ τ y ) = P r (a, b|τ x , τ y ), ∀abxy, whereP andP r are vectors of dimensions n A × n B × d X × d Y , whose components are P (a, b|τ x , τ y ) and P r (a, b|τ x , τ y ) of different a, b, τ x , τ y . D[P ,P r ] is the Euclidean distance ofP andP r .

Supplementary Note 3: SSLDS scheme
In this note we give a more detailed description to the SSLDS scheme, an show that it can be used to simulate higherdimensional entanglement with bipartite states of lower dimension.
In Ref. [11], it is pointed out that the correlation of some higher dimensional bipartite states in a device-dependent dimensional witness (DIDW) can be simulated with sequential measurements on a sequence of lower-dimensional states. The ququart case is given as an example, that with standard binary encoding on a ququart: a two-ququart maximally entangled state (MES) can be mapped into the product of two two-qubit MESs: where |Ψ = 1 2 (|00 + |11 + |22 + |33 ), |φ = 1 √ 2 (|00 + |11 ). Also, some projective measurement operators (projectors) of local higher dimensional system can be encoded into sequential projective measurements on lower dimensional systems, and the choice of the measurements in the sequence depends on what the higher dimensional projector is and the outcomes of previous measurements in the sequence (classical information and classical feedforward). Under such encoding, the correlation P (a, b|x, y) of some higher dimensional bipartite states can be simulated, therefore the higher dimensional violation of some Bell-type inequality can be reached by lower dimensional states. An example is given under one of the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequalities, CGLMP 4 . It is shown that one can obtain a violation greater than I 4 = 0.315, which lower bounds the dimension of the observed system to two-ququarts, by sequential measurements on two two-qubit MESs. From the result we can deduce that, in a DI entanglement witness game, if Alice and Bob has two two-qubit MESs, they can convince Charlie that they have the entanglement of a two-ququart MES. Therefore, the DI quantum tests are not reliable either as a dimensional witness or an entanglement witness. Although a solution is given in Ref. [11] to exclude the existence of SSLDS scheme, but it requires additional tests, and is only given for special case (two-ququart states).  [30], it is pointed out that that any POVM can be written in a "fine grained" form where measurement operators are shrinked projective measurement operators (projectors) onto a set of non-renormalized, non-orthogonal pure states. Therefore, without lost of generality, the measurement operator of Alice and Bob can be written as M xai = u xai M xai , and M ybi = u ybi M ybi , where 0 ≤ u xai , u ybi ≤ 1, M xai and M ybi are some projectors in M X⊗Ai and M Y ⊗Bi .
Suppose Alice and Bob use a sequence of length k to simulate correlation P (a, b|τ x , τ y ) under each pair of inputs (τ x , τ y ). We denote the quantum state they have by ρ AB = ρ A1B1 ⊗ ρ A2B2 ⊗ ... ⊗ ρ A k B k , the sequences of their outcomes by S a =  (a 1 , a 2 ...a k ), S b = (b 1 , b 2 ...b k ). Then, correlation P Sa,S b |τx,τy can be written as the product of k correlations P aibi|τxτy , that is The correlation for the first-round measurements is where I Aj (I Bj ) are identities in spaces M Aj (M Bj ). We let M xyaibi = M xai ⊗ j =i I Aj ⊗ M yBi ⊗ j =i I Bj , and the postmeasurement system after the first round is And after the second round the correlation becomes P a2b2|τxτy = u xa2 · u yb2 · T r(M xya2b2 ρ a1b1τxτy ).
Comparing Eq. (13) and Eq. (14), we have whereM xya2b2 = M † xya1b1 M xya2b2 M xya1b1 is a effective measurement operator. Analogously, after k rounds of measurements, the correlation they can produce is Eq. (16) indicates that the correlation P Sa,S b |τx,τy can be obtained by some effective measurement operatorM xySaS b acting on the state σ x ⊗ ρ AB ⊗ σ y , and multiplied by some normalizing coefficients f Sa and f S b . Now we come to discuss the form of the effective measurement operators. Since every operatorM xyaibi is composed of tensor product of projectors M xai , M ybi in M X ⊗ M Ai , M Y ⊗ M Bi , and identities in other spaces, there is where π aj and π bj are positive semidefinite operators in M Aj and M Bj , whose form depends on M xyaj bj and M xyaj−1bj−1 . Therefore,M Since every π ai and π bi can be written as mixture of projectors, measurement M xa1 ⊗ k j=2 π aj and M yb1 ⊗ k j=2 π bj can be written as combinations of sequential projectors: where l c l = T r(M xa1 ⊗ k j=2 π aj ) = 1, l c l = 1, and where σ aj ,l , σ bj ,l are projectors on M Aj and M Bj . Since bothÃ Sa,l andB S b ,l are projectors, they can be written as |φ xSa l φ xSa | and |φ yS b l φ yS b |, where |φ xSa l and |φ yS b l are pure states in spaces H X ⊗ k i=1 H Ai and H Y ⊗ k j=1 H Bj . Now we decode them back into n-dimensional representation, we let H A and H B be the n dimensional Hilbert space for Alice and Bob respectively, and let the two pure states |φ xS a l ∈ H X ⊗ H A and |φ yS b l ∈ H Y ⊗ H B be the pure states after decoding. Now we demonstrate that |φ xS a l and |φ yS b l have Schmidt number no more than m, we only discuss |φ xS a l , and the results are analogous for |φ yS b l . Since H A1 has dimension m, an arbitrary pure states |ψ ∈ H X ⊗ H A1 can be written as m i=1 α i |ν i ⊗ |i , where |ν i are some pure states in space H X , {|i } is a set of basis states of H A1 , m i=1 |α i | 2 = 1. Therefore |φ xSa l can be written as |φ xSa l = ( m i=1 α i |ν i ⊗ |i ) ⊗ n i=2 |φ ai l , where |φ ai l φ ai | = σ ai,l . Let |µ i l denote the pure state |i ⊗ k i=2 |φ ai l , then we have We let states |µ i l ∈ H A denote the states |µ i l after decoding, then Since such binary encoding keeps the inner product unchanged, we have where δ ij is the Kronecker delta. Combining Eqs. (23) and (24) we know that {|µ i l } can be seemed as a set of orthogonal basis that generates a subspace of dimension m of H A , thus the Schmidt decomposition of |φ xS a l has at most m nonvanishing terms, the Schmidt number is at most m.
Consequently, we now have E(|φ xS a l φ xS a |) ≤ E m , where E m is the upper entanglement bound of m-dimensional bipartite states. Since the total effective measurement A m Sa of Alice can be written as we have Analogously we have B m Supplementary Note 5: proof of theorem 2 First we propose a lemma before we prove theorem 2. LemmaS1 Let {M i } be a set of k × k positive semidefinite matrices satisfying then any matrix A obtaining by where a i ≥ 0, is of rank at least r.
proof . The proof of Lemma S1 is straight forward so we just state it short. Let M 0 be the matrix of rank r in {M i }, if we represent M 0 in the eigenbasis of A, there must be at least r positive diagonal elements. Since a i ≥ 0, and that the diagonal elements of positive semidefinite matrices are non-negative in any bases, we have A = i =0 a i M i + a 0 M 0 has at least r positive diagonal elements in its own eigenbasis. Therefore rank(A) ≥ r. Now we come to the proof of theorem 2. proof . In theorem 1 we proved that effective measurements A m Sa and B m Sa can be written as the convex combination of some projectors whose entanglement do not exceed E m , multiplied by some normalizing coefficients f Sa and f S b , then there is where l c l = l c l = 1, Sa f Sa = S b f S b = 1, A Sa,l and B S b ,l are some projectors acting on H X Under tomographically complete input sets {τ x } and {τ y }, by comparing Eqs. (1) and (2) in the main text, we have Since in theorem 1 we have demonstrated the quantum states that projectors A m Sa and B m S b project on have Schmidt numbers no more than m, by simple calculation we have As O SaS b are positive semidefinite operators, we can find a set of normalized quantum states {σ SaS b } satisfying where t SaS b = T r(O SaS b ). We know that σ SaS b can always be written as the combination of some bipartite pure states, and we can also affirm that the Schmidt number of these pure states are no more than m, since if any of these pure states has Schmidt number n (n > m), with Lemma S1 and simple calculation we will have T r X (O SaS b ) ≥ n, which contradicts Eq. (32). Then, as bipartite pure states of Schmidt number m has entanglement E m , we have Finally, for the detected entanglement Q is where Sa,S b t SaS b = 1, ab p(a, b|S a , S b ) = 1, therefore we have Q ≤ E m . Since Q is an upper bound of the detected entanglement Q m SSLDS , we have Q m SSLDS ≤ Q ≤ E m .
Supplementary Note 6: experimental details The experimental setup is shown in Fig. 2 of the main text. A cw violet laser at 404nm was separated with a beam displacer (BD40), and then was incident to a Sagnac interferometer to pump a type-II cut ppKTP crystal to generate photon pairs at 808nm. Through adjustments of half-and quarter-plates(HWPs and QWPs), three dimensional pure state |φ = 1 √ 3 (|HH u + |HH l + |V V l ), i.e. ρ AB , was prepared, where the subscripts u and l denote upper path and lower path (refer to our previous works [28,29] for more details). To prepare the mixed state ρ noise , another photon pair was used [17]. Setting the angles of HWPs in the Source module, the state 1 3 (|H u + |H l + |V l ) ⊗ (|H u + |H l + |V l ) was obtained. Inserting four quartz, we destroyed the coherence of this state and obtained the mixed state I/9. Mixing the two states ρ AB and I/9 at beam splitters, we finally prepared the path-polarization hybrid entangled state ρ noise with tunable p by adjusting the ratio of the two parts.
To prepare the input states {τ x , τ y }, Charlie expanded the dimension of photon's path degree of freedom by separating the photons via HWP and BD20 (half the length of BD40) in the Trust modules as shown in Fig. 2. Encoded as h u , h l , and v l , arbitrary trusted input states of the form β 1 |h u + β 2 |h l + β 3 |v l could be prepared by adjusting the angles of the HWPs. The input states τ x (τ y ) are assumed to be trustable and thus should be prepared with high accuracy. This could be met for our scheme restricts in highly controllable operations of polarization elements (HWPs and BDs). Our method to prepare the input states consumes no extra degree of freedom of the photons which is indeed precious resource.
There are 9 Bell states for a 3-dimensional system, which can be divided into 3 categories (each contains 3 of them) as: category 1