Abstract
We propose an enhanced discrimination measurement for tripartite 3dimensional entangled states in order to improve the discernible number of orthogonal entangled states. The scheme suggests 3dimensional Bell state measurement by exploiting composite two 3dimensional state measurement setups. The setup relies on stateoftheart techniques, a multiport interferometer and nondestructive photon number measurements that are used for the postselection of suitable ensembles. With this scheme, the sifted signal rate of measurementdeviceindependent quantum key distribution using 3dimensional quantum states is improved by up to a factor of three compared with that of the best existing setup.
Introduction
Quantum cryptography is a mature research field that exploits the principles of quantum mechanics to ensure its information theoretical security. The core protocol of quantum cryptography is quantum key distribution (QKD), which is the process of generating a secret key that is shared between two distant parties, called Alice and Bob. These parties are assumed to be exposed to a potential malicious eavesdropper, conventionally called Eve. Since the proposal of the first QKD protocols^{1,2}, many efforts have been made to improve the security of QKD based on quantum principles^{3,4,5,6}. Various types of QKD have also been experimentally demonstrated to date^{7,8,9,10}.
The earliest proposal for QKD used 2dimensional quantum states, called qubits^{11}. After this proposal, significant efforts were made to increase the key rate of QKD protocols. For example, protocols involving highdimensional quantum states, called qudits, were introduced. It is well known that higherdimensional quantum states can carry more information per quantum. In fact, there have been many theoretical proposals of the exploitation of qudits in various types of quantum information processing, such as nonlocality testing^{12,13,14,15,16} and quantum teleportation^{17,18}. Highdimensional quantum states have been experimentally demonstrated in various quantum systems, energytime eigenstates^{19,20}, multipathentangled states^{21,22,23,24}, and quantized orbital angular momentum (OAM) modes of photons^{25,26}.
Furthermore, applying highdimensional states in QKD is known to increase the efficiency of key distribution under a potential attack by Eve in the ideal case^{27,28,29,30}. The results show that QKD based on qudits can achieve a higher key rate and a higher upper bound on the allowed error rate than the original QKD protocol can. Such protocols have been demonstrated using various photon degrees of freedom, such as energytime states^{31,32,33,34} and OAM modes^{35,36,37}.
Along another branch of investigation, the security of the original QKD system has been scrutinized in more detail. Deviceindependent QKD (DIQKD) has been proposed to extend the notion of ultimate security in the device attack scenario^{38,39,40,41,42,43,44}. In this protocol, Alice and Bob can generate a secret key without any a priori assumptions regarding device performance. This scheme is designed to protect the key from sidechannel attack when the measurement device is not very reliable. In this case, the security of the protocol is guaranteed only by nonlocal correlations as identified by the ClauserHorneShimonyHolt (CHSH) inequality^{45}. However, the DIQKD protocol is not easy to be implemented in practice since it requires a loopholefree Bell test experiment, which poses high technological demands^{46,47,48,49,50}.
To compensate for this practical difficulty, a measurementdeviceindependent QKD (MDIQKD) protocol was proposed in 2012^{51}. This protocol can be more easily implemented than DIQKD can because the MDIQKD procedure does not rely on entanglement. In MDIQKD, an untrusted third party, called Charlie, maintains quantum state detectors separately from Alice and Bob. After Alice and Bob send quantum states, e.g. single photon states, to Charlie, he performs a Bell state measurement (BSM) on the incoming photons to generate the correlation between Alice and Bob. In this protocol, Charlie acts as a referee to build up the necessary correlation. Due to the selective construction of the correlation, an eavesdropper who attacks the detector cannot obtain exact information about the secret key. For this reason, MDIQKD possesses unbounded security against any detector attacks. Using MDIQKD, most sidechannel attacks made possible by detector imperfections can be resisted^{52}.
However, MDIQKD suffers from a low secret key rate. When the BSM in MDIQKD is performed using linear optical elements, the success probability of the setup is only 50%^{53}. In the original QKD protocol, the secret key can be extracted when Alice and Bob use the same encoding basis, which is true for half of the generated ensembles on average. However, in MDIQKD, Alice and Bob can share information only when they use the same encoding basis and the BSM is successful, meaning that 75% of their trials must be discarded. This shortcoming makes this key distribution scheme quite inefficient.
To increase the key rate, aversion of MDIQKD using highdimensional quantum states was recently studied^{54,55}, and a performance increase was demonstrated for MDIQKD using 3dimensional quantum states, called qutrits, instead of qubits^{54}. However, the practicality of this scheme is still questionable. This is because a generalized BSM of a bipartite highdimensional maximally entangled state cannot be implemented using only linear optical elements^{56,57}. For the implementation of highdimensional entangled state discrimination, multimode quantum scissors^{58} and a linear optical setup for multipartite highdimensional entangled state discrimination^{59} have been proposed, although their success probabilities are still not sufficient for the implementation of efficient highdimensional MDIQKD.
In the present work, we propose an efficient discrimination setup for tripartite entangled qutrit states to enhance the discernible number of orthogonal entangled states. The setup relies on stateoftheart techniques, a multiport interferometer called a tritter^{60}, and nondestructive photon number measurements^{61,62}. The generation of multipartite highdimensional entangled photonic states has not been well studied even now, and such states have been experimentally demonstrated only very recently^{26,59,63,64}. With the technologies listed above, we construct a setup that can discriminate subsets of tripartite entangled qutrit states and show that qutrit MDIQKD can be implemented using this setup. Moreover, we generalize the measurement to ddimensional dphoton entangled state discrimination measurement and analyse the secret key rate of highdimensional MDIQKD using the generalized setup.
This article is organized as follows. We present a schematic description of the tripartite entangled qutrit state discrimination setup and the set of entangled states that the setup can discriminate. We also present schematic descriptions of the teleportation process and MDIQKD using qutrit states with the setup we propose. We analyse the security of the MDIQKD protocol using qutrit states as well. We generalize the proposed setup to a setup for ddimensional dphoton entangled state discrimination and analyse the secret key rate of highdimensional MDIQKD. The efficiency of the proposed MDIQKD setup when experimental factors are considered is described as well.
Results
Tripartite entangled qutrit state discrimination setup
Before we introduce the discrimination setup, we describe the physical system we are considering and introduce the related notation. We consider 3dimensional photonic states, where a〉, b〉, and c〉 denote 3dimensional orthonormal states. Such 3dimensional quantum states can be realized by exploiting various degrees of freedom, such as, highdimensional timebin states^{20,31} and OAM modes of single photons^{25,26}. As an example, Fig. 1 shows a schematic setup for generating 3dimensional timebin states^{31}. The source generates a single photon and the photon is injected into a delay line. The delay lines have different lengths, and the user can choose into which of the delay lines the photon is injected. a〉, b〉, and c〉 denote the photonic timebin states when the photon passes though the shortest, intermediate, and longest delay lines, respectively.
With regard to the qutrit states, we will focus on tripartite entangled states. To realize interference among three photons, we use a multiport interferometer called a tritter^{60}. The tritter we consider has three input ports and three output ports. We consider the threepath operation described in Eq. (1):
where \({\hat{a}}_{y}^{\dagger }\) is a photon creation operator on path y and \({\hat{U}}_{3}\) is a threedimensional discrete Fourier transformation defined in terms of ω, which, in turn, is defined as ω = exp(2πi/3). The input and output ports are distinguished by the subscripted numbers 0, 1, and 2 and 3, 4, and 5, respectively.
In our notation, a single photon that is timebin mode x in the port labelled y is represented by \({\hat{a}}_{xy}^{\dagger }0\rangle ={x}_{y}\rangle \). We consider nine states, as given in Eq. (2):
where i ∈ {0, 1, 2}, ω is defined in Eq. (1), and we omit (mod 3) in the subscripts on the righthand side. In the states described in Eq. (2), the three photons are in different timebin modes, a〉, b〉, and c〉. The tripartite entangled qutrit states Ψ_{0}〉, Ψ_{1}〉, and Ψ_{2}〉 are the quantum states in which each photon exists in a separate input port, and the other states are those in which two photons of different timebin modes exist in one port and the other photon is in another port. The orthogonality of the nine tripartite entangled qutrit states is easily provable.
A schematic diagram of the entire tripartite entangled qutrit state discrimination setup is shown in Fig. 2. Before photons that are in one of the states described in Eq. (2) are injected into the tritter, nondestructive photon number measurements are performed on each input port for postselection. The details of the postselection process will be described later. Subsequently, the photons enter the tritter, and the tritter performs the \({\hat{U}}_{3}\) operation on the photons. After interference, we perform a qutrit state discrimination measurement on each output port. Since we are using the timebin modes of single photons, the states can be discriminated by measuring the arrival times of the photons at onoff detectors^{65,66} or by using ultrafast optical switches^{67}. We obtain a certain combination of clicked detectors when one of our input states described in Eq. (2) is injected into the entire setup, yielding an output state as shown in Eq. (14); the combination of clicked detectors for each state is given in Eq. (3):
where i ∈ {0, 1, 2} and D_{xy} denotes a click of the detector corresponding to the singlephoton state y_{x}〉. Each possible detection listed in Eq.(3) has an equal probability of 1/6. Since we are using onoff detectors, we cannot discriminate phase differences among the three states. This means that we can infer groups of entangled states from the measurement results, but we cannot discriminate exact entangled states. To discriminate the exact state, we postselect only those trials in which all three photons exist in different input ports by means of nondestructive photon number measurements. In a nondestructive measurement, the absorption of the photons during the measurement is ideally avoided, and the other degrees of freedom of the photons also remain unaffected. Nondestructive measurements of the photon number state can be successfully realized using nonlinear effects^{68,69} or an atomcavity system^{70,71}. The remaining states after postselection are Ψ_{0}〉, Ψ_{1}〉, and Ψ_{2}〉, which can be exactly discriminated from the combinations of clicked detectors given in Eq. (3). The nondestructive photon number measurements do not affect other degrees of freedom of the photons in the case that all photons are distributed in different ports, so the interference pattern among the three photons in the tritter is the same when there is no nondestructive photon number measurement. We note that it is sufficient to use a nondestructive photon number parity measurement instead of a full nondestructive photon number measurement. If nondestructive photon number parity measurements on the input ports indicate an odd photon number, then it is guarantee that all of the photons are in different input ports.
Pathencoded qutrit teleportation
In this section, we show that a qutrit state teleportation protocol can be implemented with the proposed tripartite entangled qutrit state discrimination setup. The protocol is very similar to the previously studied teleportation protocol using linear optical elements^{59}. Figure 3 shows a schematic diagram of the qutrit teleportation protocol. The target state that we want to teleport is a pathencoded qutrit state, and its timebin mode is a〉. The target state can be written as shown in Eq. (4):
where α_{i} is an arbitrary complex number satisfying the normalization condition, \({\sum }_{i=0}^{2}{{\alpha }_{i}}^{2}=1\). One user, called Alice, possesses the target state. She wants to send that state to the other user, called Bob. To teleport the state, Alice and Bob share a photonic entangled state Ψ_{0}〉. Previous studies have investigated the generation of multipartite photonic entangled states by means of an array of nonlinear crystals^{64} and by means of tritter and nondestructive photon number measurements^{59}. Alice has the target state and two photons generated from Ψ_{0}〉 in timebin states b〉 and c〉, and Bob has a photon generated from Ψ_{0}〉 in timebin state a〉. Then, the entire system can be written as described in Eq. (5):
where we omit (mod 3) in the subscripts. Alice applies the tripartite entangled qutrit state discrimination operation to her three photons. The entire system can be rewritten in the basis of Alice’s measurement, as shown in Eq. (6):
where ω = exp(2πi/3) and we omit (mod 3) in all subscripts except those of Ψ. After Alice’s measurement, Alice sends the result of the measurement to Bob. Bob can recover the initial target state by performing a unitary operation on his state corresponding to Alice’s result. The operation that Bob should perform for each possible result is specified in Table 1. The operations that Bob needs to perform are two path rotating operations, which are described in Eq. (7):
The unitary operators \({\hat{P}}_{1}\) and \({\hat{P}}_{2}\) can be implemented with phase shifters on Bob’s photonic paths B0, B1, and B2. For instance, to realize \({\hat{P}}_{1}\), the phase shifters should simultaneously apply no phase shift on B0, a phase shift of ω^{2} on B1, and a phase shift of ω on B2. After the unitary operation, Bob’s photonic path state is as described in Eq. (8):
This state is the same as the initial target state that is given in Eq. (4).
Measurementdeviceindependent quantum key distribution using qutrits
In this section, we study the conceptual implementation of MDIQKD using qutrit states. MDIQKD is proposed to prevent sidechannel attacks against imperfect measurement devices^{51}. In MDIQKD, Alice and Bob, who want to share a secret key, send their encoded photonic states to an untrusted third party, Charlie. Charlie then performs a BSM to measure the correlation of the photons and announces the result. Alice (Bob) can infer the state that Bob (Alice) sent to Charlie from the announced result and her (his) own encoded state. Since the measurement setup detects only the correlation of the photons, an eavesdropper Eve cannot obtain the information that Alice and Bob share by attacking the measurement setup. It has been proven that QKD with highdimensional quantum states has a higher secret key rate than that of traditional qubit QKD^{27,28,29,30}, and there are also studies that have shown that MDIQKD with highdimensional states has some advantages compared with qubit MDIQKD^{54,55}. The procedure for MDIQKD using qutrit states was introduced in our previous work^{54}. Since the proposed setup is designed to discriminate tripartite entangled qutrit states, the quantum states in which Alice and Bob encode their information must be different from those used in the original protocol in order to exploit the setup. Here, we will show that MDIQKD can be implemented with our proposed setup.
In the original MDIQKD protocol, Alice and Bob use two encoding bases at each site. As an extension of that protocol, our protocol also uses two encoding bases, a pathencoding basis and one of its mutually unbiased bases (MUBs). The condition for the two bases to be MUBs in threedimensional Hilbert space, \({\langle \bar{i}j\rangle }^{2}=\mathrm{1/3}\), should be satisfied for all i, j ∈ {0, 1, 2}, where {0, 1, 2} and \(\{\bar{0}\rangle ,\bar{1}\rangle ,\bar{2}\rangle \}\) are orthonormal bases. In our physical system, {x_{0}〉, x_{1}〉, x_{2}〉} is a pathencoding basis, and \(\{{x}_{\bar{0}}\rangle ,\,{x}_{\bar{1}}\rangle ,\,{x}_{\bar{2}}\rangle \}\) is one of its MUBs, as defined in Eq. (9):
The states in the MUB can be generated with the tritter operation \({\hat{U}}_{3}\). For example, state \({x}_{\bar{0}}\rangle \) can be generated by inputting the photonic state x〉 into input port 0 of the tritter depicted in the methods section.
A description of MDIQKD using qutrit states is given as follows. Since the measurement setup projects an incoming state onto tripartite entangled states, the total number of encoded photonic states that Alice and Bob send to Charlie should be three. We assume that Alice sends two photonic states and that Bob sends one photonic state as shown in Fig. 4. First, Alice randomly chooses encoding information from among the ordinary numbers 1, 2 and 3 and the bar numbers, \(\bar{0}\), \(\bar{1}\) and \(\bar{2}\). After that, Alice generates the corresponding bipartite path entangled state. If Alice chooses a number x, Alice generates the state \(\mathrm{(1/}\sqrt{2})({b}_{x}\mathrm{,\ }{c}_{x+1}\rangle {b}_{x+1}\mathrm{,\ }{c}_{x}\rangle )\). Bob chooses a number y from among the numbers Alice used and generates the state a_{y}. Alice and Bob send their states to Charlie; then, Charlie performs tripartite entangled qutrit state discrimination on the incoming photons and announces the result through a public channel. Subsequently, Alice and Bob discard trials in which their encoding bases were different after a basis comparison through a public channel. The remaining data become the sifted key, and Alice and Bob can synchronize their information by performing the appropriate postprocessing as described in our previous work^{54}.
To evaluate the usefulness of the proposed protocol, a security analysis of this protocol is necessary. Such an analysis can be performed through an inspection of the equivalent protocol using the entanglement distillation process (EDP)^{3,5,6}. If the two parties, Alice and Bob, share the maximally entangled state generated via the EDP, then an eavesdropper cannot establish correlations between her state and the states of Alice and Bob^{72}. In this sense, if Alice and Bob share a maximally entangled state, then they are assured that their protocol is secure against eavesdroppers. Thus, the security of the proposed protocol can be analysed with respect to the number of maximally entangled states generated via the EDP.
A schematic diagram of the equivalent protocol that exploits the maximally entangled state is shown in Fig. 5. In this protocol, we assume that Alice has a tripartite qutrit pathentangled state generator and that Bob has a bipartite qutrit path entangled state generator. Implementations of multipartite highdimensional pathentangled states using linear optical elements and nondestructive photon number parity measurements^{59} and using overlapping paths of photon pairs created in different crystals^{64} have recently been proposed. The generation of bipartite highdimensional pathentangled states using nonlinear crystals has previously been studied and demonstrated^{22,23}. With regard to the states, Alice keeps the state that is in timebin mode a and sends the other photons to Charlie by using state discrimination elements, and Bob also keep one photon and sends the other to Charlie. Then the whole system can be described as written in Eq. (10):
The entire system can be rewritten as given in Eq. (11):
In Eq. (11), the state Ψ〉 represents the tripartite state that Alice and Bob sent to Charlie for the entangled state discrimination measurement. The bipartite states in the square brackets denote the pathentangled qutrit state that Alice and Bob share. With the proposed setup, the states Ψ_{0}〉, Ψ_{1}〉, and Ψ_{2}〉 can be exactly discriminated. The other states cannot be exactly discriminated, so Alice and Bob should discard any trials in which Charlie’s announced result is not one of Ψ_{0}〉, Ψ_{1}〉, and Ψ_{2}〉. After sifting, Bob performs a unitary operation on his state based on Charlie’s result; then, the final state Alice and Bob share is the maximally entangled state \(\mathrm{1/}\sqrt{3}{\sum }_{j\mathrm{=0}}^{2}{a}_{Aj},{a}_{Bj}\) when there is no error. Alice and Bob privately choose their measurement bases from between the photon path measurement and the corresponding MUB measurement and perform their measurements. The measurement results of Alice and Bob are strongly correlated only when Alice and Bob choose the same basis, so they discard any trial in which different bases are used after a basis comparison step conducted through public communication. The remaining data are correlated, so these data can be used as a secret key. Then, the security analysis of the MDIQKD protocol using entangled qutrit states becomes equivalent to that of the QKD protocol using 3dimensional maximally entangled states, which has already been studied^{28,29,54}. According to the results, the secret key rate per sifted signal of the protocol, r_{3},can be evaluated as shown in Eq. (12):
where H(x) is the Shannon entropy, defined as \(H(x)=x{\mathrm{log}}_{2}x\mathrm{(1}x){\mathrm{log}}_{2}(1x)\), and Q denotes the state error rate in the pathencoding basis. Each sifted signal is the result of a trial in which Alice and Bob generated the Ψ_{0}〉 state and chose the same measurement basis. The error rate can be obtained as follows (number of signals that contain errors)/(number of sifted signals). An error corresponds to a case in which Alice and Bob share Ψ_{x}〉 such that x ≠ 0 at the end of the protocol.
ddimensional dphoton state discrimination setup and its efficiency
We investigate applications of a ddimensional generalized setup and their efficiency. With the generalized setup and states, the qutrit teleportation scheme and qutrit MDIQKD scheme described in previous sections can be extended to the teleportation of arbitrary ddimensional pathencoded states and ddimensional MDIQKD, respectively. For instance, the steps of the ddimensional MDIQKD protocol are very similar to those of the qutrit MDIQKD protocol, except that Alice and Bob should prepare ddimensional information instead of 3dimensional information and Alice and Bob use the states A_{i} and 0_{j} to encode their information, where A_{i} is defined in the Methods section and i, j ∈ {0, 1, 2, …, d − 1}. The security proof for ddimensional QKD using maximally entangled states has already been studied^{28,29}. Using the results, the secret key rate per sifted signal of ddimensional QKD can be written as given in Eq. (13):
where Q is the state error rate. As previously mentioned, MDIQKD has the disadvantage of a lower secret key rate than that of the original BB84 protocol since Alice and Bob must discard more trials in MDIQKD than in BB84 because of the success probability of the BSM. Here, we investigate the secret key rate per total signal of ddimensional MDIQKD. The secret key rate per total signal R is obtained from (sifted signal rate) × (secret key rate per sifted signal), where the sifted signal rate contains the probability that the ddimensional dphoton entangled state discrimination succeeds and that Alice and Bob chose the same basis. In ddimensional MDIQKD, the number of possible combinations that Charlie can receive is d^{2} since Alice and Bob each use d different orthonormal states. Among the possible combinations, only the states in which all photons exist in different ports are postselected by means of nondestructive photon number measurements. They are projected onto {Φ_{i}〉 Φ_{i}i = 0, 1, 2, …, d − 1}, and all other states are discarded. Thus, the success probability of ddimensional dphoton entangled state discrimination is 1/d in the ideal situation in which we ignore all experimental factors and there is no eavesdropper. With this probability, the secret key rate per total signal of ddimensional MDIQKD can be calculated as r_{d}/(2d), where the 2 in the denominator comes from the probability that Alice and Bob choose the same basis. The secret key rates per total signal of ddimensional MDIQKD with various values of d are shown in Fig. 6. It is already known that highdimensional QKD is more robust against state error than qubit QKD is^{27,28,29,30}, this phenomenon is also reflected in this plot. In the case of no error, when Q = 0, only MDIQKD using qutrit states has a higher key rate than that of the original qubit MDIQKD protocol. The expression for the secret key rate per total signal when Q = 0 is \(({\mathrm{log}}_{2}d)/d\). In this expression, the denominator increases linearly with the number of dimensions while the numerator increases logarithmically, so the secret key rate decreases in the highdimensional case. From the plot, we can identify that qutrit MDIQKD has the highest secret key rate per total signal when 0 ≤ Q ≤ 0.0294. This means that qutrits are best highdimensional quantum states for MDIQKD in the lowerror range.
In real experiments, our proposed setup can fail even if the input state is one of {Φ_{i}〉i = 1, 2, …, d − 1} since the setup involves nondestructive photon number measurements and the success probability of such measurements is not 100%. The generalized ddimensional dphoton entangled state discrimination setup involves d nondestructive photon number measurements, so the efficiency of this setup is exponentially affected by the success probability of these measurements. To investigate its usefulness, we compare the generalized setup with a “Bell filter” that consists only of linear optical elements^{59}. The Bell filter can discriminate Ψ_{0}〉 regardless of dimensionality, so the sifted signal rate of ddimensional MDIQKD using the Bell filter is always 1/d^{2}, where we ignore the probability that Alice and Bob choose the same basis since it is the same for all considered protocols. For the generalized setup we propose, the sifted signal rate is 1/d × (η)^{d}, where η is the success probability of each nondestructive photon number measurement. For the sifted signal rates of the two setups to be the same, η should be η = (1/d)^{(1/d)}; examples of the value of η are 0.693, 0.707, and 0.725 for 3, 4, and 5 dimensions, respectively. Since the efficiency of nondestructive photon number measurements using an atomcavity system was reported to be 0.66 in 2013^{70}, the success probability of the proposed setup is lower than that of a Bell filter with current technology, even though the proposed setup can discriminate more states than the Bell filter can.
Discussion
We investigated a tripartite entangled qutrit state discrimination setup and its applications, especially teleportation and MDIQKD. We showed that the proposed setup can discriminate three tripartite entangled qutrit states. We showed that the setup can be generalized to ddimensional dphoton entangled state discrimination and that the secret key rate per total signal of ddimensional MDIQKD using the generalized setup is highest when qutrit states are used. We considered the efficiency of the nondestructive photon number measurements performed in the setup and calculated the efficiency bound of the proposed setup for which the sifted signal rate becomes higher than that of a Bell filter that consists of linear optical elements^{59}. Since the success probability of the nondestructive photon number measurements using an atomcavity system is 66%^{70}, the proposed setup is not as efficient as a Bell filter with current technology. However, we expect that the proposed setup will be more efficient with future technologies.
Methods
Tritter operation in the proposed setup
A schematic diagram of the tritter that corresponds to the unitary operator \({\hat{U}}_{3}\) is shown in Fig. 7. The tritter contains only linear optical elements: two 50:50 beam splitters, one beam splitter whose reflectivity is 1/3 and several phase shifters. When the input states is one of the state described in Eq. (2), the output state after the tritter operation can be easily obtained from the inputoutput relation of the tritter given in Eq. (1); the possible states are given in Eq. (14):
where i ∈ {0, 1, 2}, the value of a subscript j in Eq. (14) is equal to as 3 + [j (mod 3)] if a number in the subscript is larger than 5, and an unimportant global phase is ignored. The results for the discrimination setup (Eq. (3)) are obtained directly from Eq. (14).
ddimensional dphoton state discrimination setup
Our proposed setup can be easily generalized to a setup for ddimensional dphoton entangled state discrimination. For ddimensional dphoton entangled state discrimination, we need to generalize the threepath operation in Eq. (1) to a dpath operation, which is described by a d × d discrete Fourier transform operation \({\hat{U}}_{d}\), as shown in Eq. (15):
where χ = exp(2πi/d). The unitary operation \({\hat{U}}_{d}\) can be realized by means of a tritter with d inputs and d outputs. As an extension of the tripartite entangled qutrit state discrimination setup, the ddimensional dphoton entangled state discrimination setup consists of a dinput doutput tritter, d nondestructive photon number measurements, and d^{2} onoff detectors. One of the states that the generalized setup can discriminate is very similar to the state that can be discriminated by means of the existing setup^{59}, this state is given in Eq. (16):
where
and vac〉 means the vacuum state. Let us recall that in our notation, \({\hat{a}}_{xy}^{\dagger }\) is the photon creation operator whose orthonormal mode is x and whose path label is y; thus, it can be rewritten as \({\hat{a}}_{xy}^{\dagger }\mathrm{vac}\rangle ={x}_{y}\rangle \). If we extend the state with respect to the first row, then the state is rewritten as shown in Eq. (18):
where
and Λ_{0i} is the (d − 1) × (d − 1) submatrix obtained by omitting the (i + 1)th column and first row of Λ. Similar to the case of tripartite entangled qutrit state discrimination, all states that the generalized setup can discriminate are orthogonal to Φ_{0}〉 with a phase factor χ; these states can be described as given in Eq. (20):
where i ∈ {0, 1, 2, 3, …, d − 1}. These d states can be discriminated from the combinations of clicked detectors in the generalized setup after postselection using nondestructive photon number measurements.
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Acknowledgements
We acknowledge the financial support from the National Security Research Institute (NSRI). W. Son acknowledges the support provided by KIAS through the Open KIAS Program.
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Y.J. designed and analysed the protocols. K.B. contributed to the analysis. W.S. supervised the whole project. All authors reviewed the manuscript.
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Y. Jo was funded by the Agency for Defense Development (ADD). K. Bae and W. Son were funded by The National Security Research Institute (NSRI). W. Son received a salary from Sogang University. W. Son also acknowledges the University of Oxford and KIAS for their visitorship programme.
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Jo, Y., Bae, K. & Son, W. Enhanced Bell state measurement for efficient measurementdeviceindependent quantum key distribution using 3dimensional quantum states. Sci Rep 9, 687 (2019). https://doi.org/10.1038/s4159801836513x
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DOI: https://doi.org/10.1038/s4159801836513x
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