Optimal nonlocal conversion of photonic four-partite entanglement from two Bell pairs in quantum networks
Yuki Takeuchi Nobuyuki Imoto Toshiyuki Tashima
quant-ph



takeuchi@qi.mp.es.osaka-u.ac.jp
Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan

imoto@mp.es.osaka-u.ac.jp
Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan

tashima.toshiyuki.5e@kyoto-u.ac.jp
Department of Electronic Science and Engineering, Kyoto University, Kyoto Daigaku-Katsura, Nishikyo-ku, Kyoto 615-8510, Japan


We analyze optimal schemes and also propose some practical schemes for the nonlocal conversion from two shared Bell pairs to four-qubit entangled states in optical quantum networks. In the analysis, we consider two-qubit operations as nonlocal operations and minimize the number of access to ancillary qubits as possible.
First, we consider two-qubit unitary operations without using ancillary qubits and derive a necessary and sufficient condition for convertible states. Second, we consider nonlocal optical systems composed of passive linear optics and postselection. For the passive linear optical systems, we derive achievable upper bounds of success probabilities of the conversion in the case without ancillary qubits. We also compare the optimal success probabilities with those of previously proposed schemes. Finally, we discuss success probabilities of the conversion in the case with ancillary qubits.


Optimal nonlocal conversion of photonic four-partite entanglement from two Bell pairs in quantum networks
    Toshiyuki Tashima
    December 30, 2023
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§ INTRODUCTION

Multipartite entanglement is an important resource of a quantum network that enables several multiparty quantum information processing tasks such as quantum key distribution (QKD) <cit.>, quantum teleportation <cit.>, and distributed quantum computing <cit.>. For each of the multiparty quantum information processing tasks, different entangled resource states are required. Particularly, Bell pairs, graph states <cit.>, hypergraph states <cit.>, W states <cit.>, and Greenberger-Horne-Zeilinger (GHZ) states <cit.> are used in a variety of protocols as follows: Bell pairs can be used for QKD <cit.>; quantum teleportation of a general one-qubit state <cit.>; and blind quantum computing <cit.>. Graph states and hypergraph states can be used for universal quantum computing <cit.>. W states can be used for leader election in anonymous quantum networks <cit.> and asymmetric telecloning <cit.>. GHZ states can be used for achieving consensus in distributed networks without classical postprocessing <cit.> and secure delegated classical computing <cit.>. Recently, in addition to these, Dicke states <cit.> and χ states <cit.> have also been studied. For example, Dicke states can be used for Grover's quantum search algorithm <cit.> and certain quantum versions of classical games <cit.>. χ states can be used for achieving optimal violation of a Bell inequality <cit.> and quantum teleportation of a general two-qubit state <cit.>.

So far, several generation methods of multipartite entangled states have been proposed <cit.>. In quantum networks, we normally consider the state generation of various multipartite entangled states by local operations and classical communication (LOCC) because of the characteristics of quantum entanglement. This is fundamentally interesting. On the other hand, it is well known that the conversion between any multipartite entangled state cannot be achieved using LOCC, because GHZ states, W states, graph states, Dicke states, and χ states come from inequivalent entanglement classes under LOCC in the four-qubit case <cit.>. Recently, in order to circumvent such no-go result, the nonlocal conversion required as the realistic networks has been actively studied. For example, it has been shown that a four-qubit linear cluster state can be probabilistically converted to a four-qubit GHZ state, a four-qubit Dicke state, and two Bell pairs <cit.> using a tunable polarization-dependent beam splitter (PDBS) as a nonlocal operation. In another work, a universal optimal gate for transforming Dicke states has been proposed in the case where some qubits can be accessed from one node <cit.>.
Furthermore, a scheme to fuse three W states, which requires an ancillary qubit and access to one qubit of each of the W states, has also been proposed <cit.>.
However, the utility of nonlocal operations for the quantum network has not yet been fully understood. It is still challenging to achieve entanglement generation efficiently in restricted situations, where the number of access to qubits is minimized.

In this paper, we analyze the optimal state conversion from two Bell pairs to four-qubit entangled states and also propose some practical conversion schemes using restricted nonlocal operations. Here, we focus on a situation where one node is close to another one while far from the other two nodes for shared two Bell pairs. We also assume that nonlocal operations can be performed on only two nodes being close to each other. First, in Sec. <ref>, we consider general two-qubit unitary operations as nonlocal operations. We then derive a necessary and sufficient condition for convertible states in this case. Second, in Sec. <ref>, we consider optical systems composed of passive linear optics and postselection as nonlocal operations.
In Sec. <ref>, we define nonlocal operations using only passive linear optics and postselection for our conversion schemes.
Then in Sec. <ref>, we consider the nonlocal conversion from two Bell pairs to well-known four-qubit entangled states such as linear cluster states, GHZ states, W states, Dicke states, different two Bell pairs, and χ states using the definition in Sec. <ref>. In particular, we derive optimal success probabilities of the nonlocal conversion and show how to achieve the optimal success probabilities in passive linear optical systems. In Sec. <ref>, we show improvements of success probabilities of the conversion when we use ancillary qubits. In Sec. <ref>, we compare our scheme with some previous protocols and show advantages of our scheme. Section <ref> is devoted to the conclusion. In Appendix A, we provide details of derivation of optimal success probabilities for our nonlocal conversion methods. In Appendix B, we derive the success probability and the fidelity of an output state of a nonlocal converter given in Sec. <ref> when the transmittance of PDBSs is deviated from an ideal value.





§ STATE CONVERSION BY NONLOCAL TWO-QUBIT UNITARY OPERATIONS


We focus on a shared four-qubit state in quantum networks, as shown in Fig. <ref>. This four-qubit state is represented as two qubits close to each other (nodes 2 and 3) and other two qubits far from the others (nodes 1 and 4). In this situation, we consider the state conversion from two Bell pairs to four-qubit entangled states using two-qubit unitary operations performed on nodes 2 and 3. Two Bell pairs are given by |Φ^+⟩_1,2|Φ^+⟩_3,4, where |Φ^+⟩_i,j≡(|HH⟩_i,j+|VV⟩_i,j)/√(2). Here, |H⟩_i (|V⟩_i) represents a horizontally (vertically) polarized photon in node i of the network. We define V_2,3 as a two-qubit unitary operator performed on nodes 2 and 3. We also define U_1 and W_4 as a single-qubit unitary operator performed on each node 1 and 4, respectively. Using these definitions and the initial state |Φ^+⟩_1,2|Φ^+⟩_3,4 with a relation given by 
    U_j|Φ^+⟩_i,j=U_i^T|Φ^+⟩_i,j,
 
a converted four-qubit state |f⟩ can be written as

    |f⟩   ≡    U_1V_2,3W_4|Φ^+⟩_1,2|Φ^+⟩_3,4
       =   V'_2,3|Φ^+⟩_1,2|Φ^+⟩_3,4
       =   12∑_i,j∈{H,V}|ij⟩_1,4V'_2,3|ij⟩_2,3,

where V'_2,3≡ V_2,3U_2^TW_3^T. From Eq. (<ref>), it is known that when |f⟩ is divided into system A (the nodes 1 and 4) and system B (the nodes 2 and 3), the Schmidt rank of |f⟩ is four, and all of Schmidt coefficients are equal to each other, i.e., 1/2. It is a necessary condition for |f⟩. Next, we show that this condition is also a sufficient condition for |f⟩. Any four-qubit state that satisfy this condition can be written as

    12∑_i,j=0^1|ϕ_i,j⟩_1,4|ψ_i,j⟩_2,3=12∑_i,j∈{H,V}Ṽ_1,4|ij⟩_1,4Ṽ'̃_2,3|ij⟩_2,3,

where {|ϕ_i,j⟩|0≤ i,j≤1} and {|ψ_i,j⟩|0≤ i,j≤1} are orthonormal bases of systems A and B, respectively. Ṽ_1,4 and Ṽ'̃_2,3 are two-qubit unitary operators on systems A and B, respectively. Moreover, any two-qubit unitary operator can be decomposed as

    (Ũ⊗W̃) exp[i(θ_1X⊗ X+θ_2Y⊗ Y+θ_3Z⊗ Z)](Ũ'̃⊗W̃'̃),

where θ_i∈ℝ (i=1,2,3) <cit.>. Ũ, W̃, Ũ'̃, and W̃'̃ are single-qubit unitary operators. X, Y, and Z are Pauli X, Y, and Z operators, respectively. For simplicity, we define R≡ exp[i(θ_1X⊗ X+θ_2Y⊗ Y+θ_3Z⊗ Z)]. From Eqs. (<ref>) and (<ref>),

    12∑_i,j∈{H,V}Ṽ_1,4|ij⟩_1,4Ṽ'̃_2,3|ij⟩_2,3
       =   Ũ_1W̃_4R_1,4Ũ'̃_1W̃'̃_4Ṽ'̃_2,3|Φ^+⟩_1,2|Φ^+⟩_3,4
       =   Ṽ'̃_2,3Ũ'̃_2^TW̃'̃_3^TR_2,3Ũ_2^TW̃_3^T|Φ^+⟩_1,2|Φ^+⟩_3,4
       =   |f⟩.

This implies that the condition is also a sufficient condition for |f⟩. As a result, the following theorem holds:


A quantum state can be converted from two Bell pairs |Φ^+⟩_1,2|Φ^+⟩_3,4 using two-qubit unitary operations performed on nodes 2 and 3 if and only if when the quantum state is divided into system A and system B, the Schmidt rank of the quantum state is four, and all of Schmidt coefficients are equal to 1/2.

Note that in general, any two-qubit unitary operation is required for the conversion. Accordingly, this conversion scheme requires some non-linearities. For example, cross-Kerr non-linearity, an ancillary coherent state, and linear optics are sufficient to perform any two-qubit unitary operation on polarization-encoded qubits <cit.>.

Using Theorem <ref> and this situation, we examine the state conversion from two Bell pairs to well-known four-qubit entangled states. As well-known four-qubit entangled states, we focus on the linear cluster state |C_4⟩, the GHZ state |GHZ_4⟩, the W state |W_4⟩, the symmetric Dicke state |D_4^(2)⟩, |Φ^+⟩_1,3|Φ^+⟩_2,4, |Φ^+⟩_1,4|Φ^+⟩_2,3, and the χ state |χ⟩ defined as follows:


    |C_4⟩   ≡   |HHHH⟩_1,2,3,4+|HHVV⟩_1,2,3,4+|VVHH⟩_1,2,3,4-|VVVV⟩_1,2,3,42
       =   |HH⟩_1,4|HH⟩_2,3+|HV⟩_1,4|HV⟩_2,3+|VH⟩_1,4|VH⟩_2,3-|VV⟩_1,4|VV⟩_2,32
    
    |GHZ_4⟩   ≡   |HHHH⟩_1,2,3,4+|VVVV⟩_1,2,3,4√(2)
       =   |HH⟩_1,4|HH⟩_2,3+|VV⟩_1,4|VV⟩_2,3√(2)
    
    |W_4⟩   ≡   |HHHV⟩_1,2,3,4+|HHVH⟩_1,2,3,4+|HVHH⟩_1,2,3,4+|VHHH⟩_1,2,3,42
       =   1√(2)(|HH⟩_1,4|HV⟩_2,3+|VH⟩_2,3√(2)+|HV⟩_1,4+|VH⟩_1,4√(2)|HH⟩_2,3)
    
    |D_4^(2)⟩   ≡   1√(6)(|HHVV⟩_1,2,3,4+|HVHV⟩_1,2,3,4+|VHHV⟩_1,2,3,4+|HVVH⟩_1,2,3,4+|VHVH⟩_1,2,3,4
          +|VVHH⟩_1,2,3,4)
       =   √(23)|HV⟩_1,4+|VH⟩_1,4√(2)|HV⟩_2,3+|VH⟩_2,3√(2)+1√(6)(|HH⟩_1,4|VV⟩_2,3+|VV⟩_1,4|HH⟩_2,3)
    
    |Φ^+⟩_1,3|Φ^+⟩_2,4   =   |HH⟩_1,4|HH⟩_2,3+|HV⟩_1,4|VH⟩_2,3+|VH⟩_1,4|HV⟩_2,3+|VV⟩_1,4|VV⟩_2,32
    
    |Φ^+⟩_1,4|Φ^+⟩_2,3   =   |HH⟩_1,4+|VV⟩_1,4√(2)|HH⟩_2,3+|VV⟩_2,3√(2)
    
    |χ⟩   ≡   12√(2)(|HHHH⟩_1,2,3,4-|HHVV⟩_1,2,3,4-|HVHV⟩_1,2,3,4+|VHHV⟩_1,2,3,4+|HVVH⟩_1,2,3,4
          +|VHVH⟩_1,2,3,4+|VVHH⟩_1,2,3,4+|VVVV⟩_1,2,3,4)
       =   1√(2)(|HH⟩_1,4+|VV⟩_1,4√(2)|HH⟩_2,3+|VV⟩_2,3√(2)+|VH⟩_1,4-|HV⟩_1,4√(2)|VH⟩_2,3+|HV⟩_2,3√(2))


From Eqs. (<ref>)-(<ref>) and Theorem <ref>, the following corollary holds:


Two Bell pairs |Φ^+⟩_1,2|Φ^+⟩_3,4 cannot be converted to |GHZ_4⟩, |W_4⟩, |D_4^(2)⟩, |Φ^+⟩_1,4|Φ^+⟩_2,3, and |χ⟩ using two-qubit unitary operations performed on nodes 2 and 3. On the other hand, |Φ^+⟩_1,2|Φ^+⟩_3,4 can be converted to |C_4⟩ and |Φ^+⟩_1,3|Φ^+⟩_2,4 using the same operations.

Note that even if we perform above nonlocal operations and two-qubit unitary operations on nodes 1 and 4, Theorem <ref> and Corollary <ref> hold. It is clear from Eqs. (<ref>) and (<ref>).



§ NONLOCAL CONVERSION USING PASSIVE LINEAR OPTICS AND POSTSELECTION


Corollary <ref> means that |Φ^+⟩_1,2|Φ^+⟩_3,4 cannot be converted to any four-qubit state using only two-qubit unitary operations performed on nodes 2 and 3. In order to convert to more various classes of the entanglement composed of four-qubit states in optical quantum networks, we consider optical systems based on a postselection as nonlocal operations. The optical systems considered in this section are composed of passive linear optics, i.e., PDBSs, polarization-independent beam splitters, phase shifters (PSs), and wave plates. We also assume the same situation as considered in Sec. <ref>. In this situation, we show that the passive linear optical systems enable us to convert from two Bell pairs to entangled states that cannot be converted in Sec. <ref>. To this end, in Sec. <ref>, we define nonlocal operations using only passive linear optics and postselection. In Sec. <ref>, using the definition of the nonlocal operation given in Sec. <ref>, we derive achievable upper bounds of success probabilities of the conversion from shared two Bell pairs to well-known four-qubit entangled states. We then show that existing optical systems can achieve the upper bounds. In Sec. <ref>, we give improvements of the conversion when we use ancillary qubits.





 §.§ Nonlocal operation for our conversion method


A general passive linear optical system can be represented as shown in the left-hand side of Fig. <ref>, because we consider the situation where nonlocal operations can be performed on only nodes 2 and 3, which are close to each other. Note that positions of detectors and input photons can be fixed because swap operations between different spatial modes can be realized using only passive linear optics. In order to simplify the form of the optical system without loss of generality and define our nonlocal operations, we prove the equivalence shown in Fig. <ref>. In other words, we show that an output state is the same in both optical systems in Fig. <ref> when a postselection succeeds. Here, we assume that the postselection succeeds when all of threshold detectors detect a photon. 

Let c_is^† and d_is^† be the s(∈{H,V})-polarized photon creation operators for input spacial mode i in node 1 and for output spacial mode i in node 1, respectively. When a photon in node 1 is input into the first spacial mode of the optical system shown in the left-hand side of Fig. <ref>, c_1s^† is transformed as follows:

    Uc_1H^† U^†   =   ∑_j=1^L'(u_jHd_jH^†+u_jVd_jV^†)
    
    Uc_1V^† U^†   =   ∑_j=1^L'(ũ_jHd_jH^†+ũ_jVd_jV^†),

where L' is the number of input (output) special modes. Here, complex numbers u_js and ũ_js satisfy ∑_j=1^L'(|u_jH|^2+|u_jV|^2)=∑_j=1^L'(|ũ_jH|^2+|ũ_jV|^2)=1 due to the unitarity of U. Accordingly, when the postselection in node 1 succeeds, i.e., a photon is detected by a threshold detector in node 1, the Bell pair |Φ^+⟩_1,2 is converted to

    (u_1H|H⟩_1+u_1V|V⟩_1)|H⟩_2
          +(ũ_1H|H⟩_1+ũ_1V|V⟩_1)|V⟩_2
       =   |H⟩_1(u_1H|H⟩_2+ũ_1H|V⟩_2)
          +|V⟩_1(u_1V|H⟩_2+ũ_1V|V⟩_2)

up to normalization. Equation (<ref>) implies that a state converted from |Φ^+⟩_1,2 in the optical system corresponding to U_1 is the same as a state converted from |Φ^+⟩_1,2 in the optical system corresponding to U_2^T. For |Φ^+⟩_3,4 and the optical system corresponding to W, we can explain in the same way. Furthermore, in order to detect a photon by each of two threshold detectors shown in the right-hand side of Fig. <ref>, two photons have to go through the optical systems corresponding to U_2^T and W_3^T, respectively. From these facts, we conclude that the equivalence shown in Fig. <ref> is satisfied. For our nonlocal conversion methods, it is enough to consider only the optical system for two photons in nodes 2 and 3, which corresponds to V'≡ V(U^T⊗ W^T) (see also the right-hand side of Fig. <ref>).



 §.§ Nonlocal four-qubit state conversion


In this section, we derive success probabilities of optimal nonlocal four-qubit state conversion from |Φ^+⟩_1,2|Φ^+⟩_3,4 using the optical system given in Sec. <ref>. Without loss of generality, we assume that two photons in nodes 2 and 3 are input from spacial modes 1 and 2, respectively. After the unitary operator V' is performed on two photons in nodes 2 and 3, each of creation operators is transformed as follows:

    V'a_1H^† V'^†   =   ∑_j=1^L(β_jHb_jH^†+β_jVb_jV^†)
    
    V'a_1V^† V'^†   =   ∑_j=1^L(γ_jHb_jH^†+γ_jVb_jV^†)
    
    V'a_2H^† V'^†   =   ∑_j=1^L(α_jHb_jH^†+α_jVb_jV^†)
    
    V'a_2V^† V'^†   =   ∑_j=1^L(η_jHb_jH^†+η_jVb_jV^†),

where L(≥ 2) is the number of input (output) spacial modes. Here, complex numbers α_js, β_js, γ_js, and η_js satisfy ∑_j=1^L∑_s=H,V|α_js|^2=∑_j=1^L∑_s=H,V|β_js|^2=∑_j=1^L∑_s=H,V|γ_js|^2=∑_j=1^L∑_s=H,V|η_js|^2=1. When the postselection succeeds, i.e., two photons are detected by two threshold detectors in output spacial modes 1 and 2, an output state |F⟩ is given by

    |F⟩   ≡   Π_ postV'|Φ^+⟩_1,2|Φ^+⟩_3,4√(p_ suc),

where p_ suc(>0) is the success probability of the postselection, Π_ post is defined as

    Π_ post   ≡   ∑_s_1=H,V∑_s_2=H,V(∏_l=1^2b_ls_l^†)| vac⟩⟨ vac|(∏_l'=1^2b_l's_l'^†),

and | vac⟩ is a vacuum state. In order to convert to a target state |t⟩, it is necessary to satisfy that |F⟩=|t⟩. By taking inner product with |s_1s_2⟩_1,4, we obtain

    Π_ postV'|s_1s_2⟩_2,3   =   2√(p_ suc)⟨ s_1s_2|_1,4|t⟩.

By substituting (s_1,s_2)=(H,H),  (H,V), (V,H), and (V,V) into Eq. (<ref>), we obtain following four equations:

    Π_ postV'|HH⟩_2,3   =   2√(p_ suc)⟨ HH|_1,4|t⟩_1,2,3,4
    Π_ postV'|HV⟩_2,3   =   2√(p_ suc)⟨ HV|_1,4|t⟩_1,2,3,4
    Π_ postV'|VH⟩_2,3   =   2√(p_ suc)⟨ VH|_1,4|t⟩_1,2,3,4
    Π_ postV'|VV⟩_2,3   =   2√(p_ suc)⟨ VV|_1,4|t⟩_1,2,3,4.

Furthermore, by taking inner product with |s_1s_2⟩_2,3 in both sides of each of Eqs. (<ref>), (<ref>), (<ref>), and (<ref>), we obtain 16 constraints in total. Considering these constraints, we derive the optimal value of the success probability p_ suc for well-known four-qubit entangled states (See Appendix A for details). The optimal success probabilities are summarized in Table <ref>.




Next, we consider optical systems that achieve the optimal values of p_ suc (See Fig. <ref>). In fact, the optical systems given in Fig. <ref> are equivalent to existing optical systems proposed for well-known four-qubit entangled states, which are given in Table <ref>. The optical system shown in Fig. <ref> (a) is essentially equivalent to the controlled-Z gate and the controlled-X gate proposed in Refs. <cit.>. The optical systems shown in Fig. <ref> (b) and (c) are also essentially equivalent to a nonlocal gate proposed in Ref. <cit.>. With respect to an optical system for conversion to |Φ^+⟩_1,3|Φ^+⟩_2,4, which is not shown here, it can obviously be constructed using only mirrors. Note that these optical systems are not unique ones that achieve optimal success probabilities. In fact, |GHZ_4⟩ can also be converted from |Φ^+⟩_1,2|Φ^+⟩_3,4 with probability 1/2 by performing the type-II fusion gate, which is not shown here <cit.>. While it was not known whether these existing optical systems are optimal or not, our results show the optimality of these existing optical systems from a network perspective. Our results also show that |W_4⟩, |D_4^(2)⟩, and |χ⟩ cannot be converted from |Φ^+⟩_1,2|Φ^+⟩_3,4 using only passive linear optics and postselection (see Table <ref>).

In the last of this section, we analyze practicality of our schemes. First, we focus on the optical system shown in Fig. <ref> (a). We consider how change in transmittance of PDBSs affects the success probability p_ suc and the fidelity F between an output quantum state and |C_4⟩. For simplicity, we assume that all of PDBSs have the same transmittance T_H and T_V for H- and V-polarized photons, respectively. In the ideal case, T_H and T_V is set to 1 and 1/3, respectively. Furthermore, we assume that HWPs and the PS ideally work because we can adjust their function by rotating angles of them. The efficiency of two threshold detectors are denoted by η and η'. In this case, p_ suc and F are written as

    p_ suc
       =   ηη'(T_H^2+2T_HT_V+T_V^2-6T_H^2T_V-6T_HT_V^2+12T_H^2T_V^2)4        
          F
       =   |T_H+2T_HT_V-T_V|2√(T_H^2+2T_HT_V+T_V^2-6T_H^2T_V-6T_HT_V^2+12T_H^2T_V^2)

and are plotted in Fig. <ref> in the case of η=η'=1 (See Appendix B for the derivation of p_ suc and F). Accordingly, the efficiency of detectors does not affect the fidelity. Furthermore, from Fig. <ref> (b), it is known that F≥ 0.9 is satisfied even when the deviation from the ideal values of T_H and T_V is 0.14, i.e., T_H=0.86 and T_V=(1±0.14)/3. This implies that our scheme is robust against experimental imperfections and detector inefficiencies. With respect to photon distinguishability, we can adopt the same analysis used in Ref. <cit.>. As a result, considerable deviation from the optimal performance occurs due to photon distinguishability as with Ref. <cit.>.
Second, we focus on the optical systems shown in Fig. <ref> (b) and (c). Since these optical systems are based on the Mach-Zehnder interferometer, stabilization is important to realize them. By using a Sagnac interferometer, we can improve stability of them. In fact, essentially equivalent optical systems have already been experimentally characterized by using the Sagnac interferometer <cit.>. Accordingly, by using the same technique, it would be possible to realize our method.





 §.§ Improvement of success probabilities using ancillary qubits


We show that |W_4⟩, |D_4^(2)⟩, and |χ⟩ can be probabilistically converted from |Φ^+⟩_1,2|Φ^+⟩_3,4 if ancillary qubits are available in addition to passive linear optics and postselection. Note that we minimize the access to ancillary qubits as possible. To this end, we also consider the conversion from |C_4⟩ to |D_4^(2)⟩ and |χ⟩. Such conversion can be done with probabilities 3/10 and 1/2 using optical systems given in Ref. <cit.> and Fig. <ref>, respectively.
Based on this fact, we construct conversion methods for |D_4^(2)⟩ and |χ⟩ as follows: First, polarization encoded qubits are converted into spatial dual-rail qubits using a PDBS and a wave plate. This can be done with probability 1. Second, the controlled-Z gate is performed on photons in nodes 2 and 3 using Knill-Laflamme-Milburn (KLM) scheme <cit.>. This step requires two ancillary qubits and succeeds with probability 1/16. Third, spatial dual-rail qubits are converted into polarization-encoded qubits using a PDBS and a wave plate. As a result, |Φ^+⟩_1,2|Φ^+⟩_3,4 is converted to |C_4⟩. Finally, |C_4⟩ is converted to |D_4^(2)⟩ or |χ⟩ using the optical systems mentioned above. Accordingly, |Φ^+⟩_1,2|Φ^+⟩_3,4 can be converted to |D_4^(2)⟩ and |χ⟩ using two ancillary qubits with probabilities 3/160 and 1/32, respectively. The optical systems for |D_4^(2)⟩ and |χ⟩ are given in Fig. <ref> (a) and (b), respectively.
For |W_4⟩, we can also construct  a conversion method based on existing methods as follows:
First, we transform |Φ^+⟩_1,2|Φ^+⟩_3,4 to |W_3⟩ using method proposed in Ref. <cit.>. This transformation requires no ancillary qubit and succeeds with probability 3/20. Next we use the expansion method proposed in Refs. <cit.> to generate |W_4⟩ from |W_3⟩ and one ancillary qubit. This expansion succeeds with probability 4/15. Accordingly, this conversion method succeeds with 1/25 and requires only one ancillary qubit. The optical system for |W_4⟩ is given in Fig. <ref> (c).






§ COMPARISON WITH PREVIOUS SCHEMES


In previous conversion schemes using only LOCC, |Φ^+⟩_1,2|Φ^+⟩_3,4 cannot be converted to |C_4⟩, |GHZ_4⟩, |Φ^+⟩_1,3|Φ^+⟩_2,4, and |Φ^+⟩_1,4|Φ^+⟩_2,3. In our optimal scheme, we realize such conversion using nonlocal two-qubit operations without ancillary qubits. Furthermore, in the situation considered in Secs. <ref> and <ref>, our scheme is optimal.

We also compare our optimal scheme given in Sec. <ref> with a conversion scheme based on the KLM scheme <cit.>, which is a scheme for universal quantum computing, in terms of the number of required ancillary qubits and the success probability. Note that we consider the KLM scheme that succeeds with probability 1/16 and requires two ancillary qubits to perform the controlled-Z gate. In order to convert to |C_4⟩ from |Φ^+⟩_1,2|Φ^+⟩_3,4, one controlled-Z gate is required. Accordingly, the KLM-based scheme requires two ancillary qubits and succeeds with probability 1/16. In order to convert to |GHZ_4⟩, |Φ^+⟩_1,3|Φ^+⟩_2,4, or |Φ^+⟩_1,4|Φ^+⟩_2,3 from |Φ^+⟩_1,2|Φ^+⟩_3,4, three controlled-Z gates are required. As a result, the KLM-based scheme requires six ancillary qubits and succeeds with probability 1/4096. This argument implies that our method is more efficient than the KLM-based scheme in terms of the number of required ancillary qubits and the success probability when |Φ^+⟩_1,2|Φ^+⟩_3,4 is converted to |C_4⟩, |GHZ_4⟩, |Φ^+⟩_1,3|Φ^+⟩_2,4, or |Φ^+⟩_1,4|Φ^+⟩_2,3. On the other hand, when |Φ^+⟩_1,2|Φ^+⟩_3,4 is converted to |W_4⟩, |D_4^(2)⟩, or |χ⟩, the KLM-based scheme is more efficient than our proposed scheme because such conversion cannot be achieved using our scheme.



§ CONCLUSION


In this work, we have considered two kinds of two-qubit operations to analyze optimal nonlocal conversion of two Bell pairs. First, we have derived a necessary and sufficient condition for two-qubit unitary operators. Second, we have derived optimal success probabilities for nonlocal conversion using passive linear optical systems and a postselection. In these arguments, we have assumed that no ancillary qubits are available. Furthermore, we have shown that the optimal success probabilities can be improved using ancillary qubits. Finally, we compare our scheme with previously proposed schemes. Our method is more efficient than the KLM-based scheme when |Φ^+⟩_1,2|Φ^+⟩_3,4 is converted to |C_4⟩, |GHZ_4⟩, |Φ^+⟩_1,3|Φ^+⟩_2,4, or |Φ^+⟩_1,4|Φ^+⟩_2,3. On the other hand, when |Φ^+⟩_1,2|Φ^+⟩_3,4 is converted to |W_4⟩, |D_4^(2)⟩, or |χ⟩, the KLM-based scheme is more efficient than our scheme.

In the quantum networks, we have various situations for sharing entangled resource states. For example, if nonlocal operations performed on three photons are available, |C_4⟩ can be converted from |Φ^+⟩|HH⟩ with probability 1/4 <cit.>. This success probability is greater than one that can be achieved by our method. As another example, an optical system proposed in Ref. <cit.> can also be used for state conversion from superposition of two Bell pairs to several four-qubit graph states. It is an important step to clarify what we can do using nonlocal operations towards the realization of advanced quantum networks. Thus, there still remains an interesting open problem of how to optimally convert to various entangled resource states with a certain initial state and restricted quantum operations in quantum networks.



ACKNOWLEDGMENTS



Y.T. is supported by the Program for Leading Graduate Schools: Interactive Materials Science Cadet Program and JSPS Grant-in-Aid for JSPS Research fellow No.JP17J03503.
N.I. is supported by CREST, JST JPMJCR1671 and JSPS Grant-in-Aid for Scientific Research(A) JP16H02214.



§ APPENDIX A

We derive the achievable maximum values of p_ suc for the conversion from the initial state |Φ^+⟩_1,2|Φ^+⟩_3,4 to the  well-known four-qubit entangled states, such as |C_4⟩, |GHZ_4⟩, |W_4⟩, |D_4^(2)⟩, |Φ^+⟩_1,3|Φ^+⟩_2,4, |Φ^+⟩_1,4|Φ^+⟩_2,3, and |χ⟩.





 §.§ |t⟩=|C_4⟩

The success probability of the state conversion for the state |C_4⟩ is calculated with 

    Π_ postV'|Φ^+⟩_1,2|Φ^+⟩_3,4=√(p_ suc)|C_4⟩.

From inner products of |s_1s_2⟩_1,4 (s_1,s_2∈{H,V}), we obtain

    Π_ postV'|HH⟩_2,3   =   √(p_ suc)|HH⟩_2,3,
    Π_ postV'|HV⟩_2,3   =   √(p_ suc)|HV⟩_2,3,
    Π_ postV'|VH⟩_2,3   =   √(p_ suc)|VH⟩_2,3,

and

    Π_ postV'|VV⟩_2,3   =   -√(p_ suc)|VV⟩_2,3.

Using inner products of |s_1s_2⟩_2,3 in both sides of each of Eqs. (<ref>), (<ref>), (<ref>), and (<ref>), we also obtain

    √(p_ suc)   =   β_1Hα_2H+β_2Hα_1H
       =   β_1Hη_2V+β_2Vη_1H
       =   γ_1Vα_2H+γ_2Hα_1V
       =   -(γ_1Vη_2V+γ_2Vη_1V),

and

    0   =   β_1Hα_2V+β_2Vα_1H
       =   β_1Vα_2H+β_2Hα_1V
       =   β_1Vα_2V+β_2Vα_1V
       =   β_1Hη_2H+β_2Hη_1H
       =   β_1Vη_2H+β_2Hη_1V
       =   β_1Vη_2V+β_2Vη_1V
       =   γ_1Hα_2H+γ_2Hα_1H
       =   γ_1Hα_2V+γ_2Vα_1H
       =   γ_1Vα_2V+γ_2Vα_1V
       =   γ_1Hη_2H+γ_2Hη_1H
       =   γ_1Hη_2V+γ_2Vη_1H
       =   γ_1Vη_2H+γ_2Hη_1V.

From Eqs. (<ref>), (<ref>), (<ref>), and (<ref>),


    0   =   (β_1Vα_2H+β_2Hα_1V)α_1Hα_2V+(β_1Hα_2V+β_2Vα_1H)α_2Hα_1V
       =   (β_1Vα_2V+β_2Vα_2H)α_1Hα_2V+(β_1Hα_2H+β_2Hα_1H)α_1Vα_2V
       =   √(p_ suc)α_1Vα_2V.


Here, α_1Vα_2V=0 is required to satisfy p_ suc>0. Replacing  α with β in Eq. (<ref>), β_1Vβ_2V=0 is also required. Moreover, from the same way with Eqs. (<ref>), (<ref>), (<ref>), and (<ref>),  η_1Hη_2H=γ_1Hγ_2H=0. If α_2V≠0, α_1V=γ_1V=0 is derived from α_1Vα_2V=0 and Eq. (<ref>), and then p_ suc=0 is derived from Eq. (<ref>). In order to satisfy p_ suc>0, α_2V=0 has to be satisfied. According to the same way used in the above proof, it also requires β_1V=η_2H=γ_1H=0. By substituting α_2V=0 into Eq. (<ref>), we obtain 0=β_2Vα_1H. Here, we consider three cases: (I) β_2V=0 and α_1H≠0, (II) β_2V≠0 and α_1H=0, and (III) β_2V=α_1H=0. 

First, we consider the case (I). From Eqs. (<ref>)-(<ref>), the square root of the success probability is given by

    √(p_ suc)   =   β_1Hα_2H+β_2Hα_1H
       =   β_1Hη_2V
       =   γ_1Vα_2H
       =   -γ_1Vη_2V.

As a result, α_2H=-η_2V and β_1H=-γ_1V are satisfied, and they imply that β_2Hα_1H=-2β_1Hα_2H. Since |β_1H|^2+|β_2H|^2≤1 and |α_1H|^2+|α_2H|^2≤1 are satisfied from ∑_j=1^L∑_s=H,V|α_js|^2=∑_j=1^L∑_s=H,V|β_js|^2=1, the relation is rewritten as follows: 

    4|β_1Hα_2H|^2   =   |β_2Hα_1H|^2
       ≤   (1-|β_1H|^2)(1-|α_2H|^2)
    
    |β_1Hα_2H|^2   ≤   1-(|α_2H|^2+|β_1H|^2)3.

Since p_ suc=|β_1Hα_2H|^2 is a monotone increasing function of |β_1H| and |α_2H|, and [1-(|α_2H|^2+|β_1H|^2)]/3 is a monotone decreasing function of |β_1H| and |α_2H|, p_ suc is maximized by |β_1H| and |α_2H| that satisfy equality in Eq. (<ref>). We define ϵ(>0) such that |β_1H|^2=ϵ|α_2H|^2 holds. From above calculation, an upper bound of the success probability is given by

    p_ suc≤ϵ max{2ϵ^2+16ϵ+2-2(1+ϵ)√(ϵ^2+14ϵ+1)36ϵ}.

Since the right-hand side of Eq. (<ref>) is maximized with ϵ=1, p_ suc≤1/9. For the case (II), according to the process in the same way as well as (I), p_ suc≤1/9. When we consider the case (III), it can be divided into four cases: (i) β_2H=γ_2H=γ_2V=0, (ii) α_1V=η_1H=η_1V=0, (iii) β_2H=η_1H=η_1V=0, and (iv) β_2H=α_1V=η_1H=γ_2H=0. For (i) and (ii), from Eqs. (<ref>)–(<ref>), η_2V=α_2H=-η_2V is satisfied. As a result, p_ suc=0. For (iii) and (iv), with the same way as well as (I), p_ suc≤1/9.



 §.§ |t⟩=|GHZ_4⟩

The conversion from  |Φ^+⟩_1,2|Φ^+⟩_3,4 to the state |GHZ_4⟩ is written by

    Π_ postV'|Φ^+⟩_1,2|Φ^+⟩_3,4=√(p_ suc)|GHZ_4⟩.

We also obtain the following equations from inner products of |s_1s_2⟩_1,4:

    Π_ postV'|HH⟩_2,3   =   √(2p_ suc)|HH⟩_2,3,
    Π_ postV'|HV⟩_2,3   =   0,
    Π_ postV'|VH⟩_2,3   =   0,

and

    Π_ postV'|VV⟩_2,3   =   √(2p_ suc)|VV⟩_2,3.

We give the following relations from inner products of |s_1s_2⟩_2,3 with Eqs. (<ref>), (<ref>), (<ref>), and (<ref>): 

    √(2p_ suc)   =   β_1Hα_2H+β_2Hα_1H
       =   γ_1Vη_2V+γ_2Vη_1V,

and

    0   =   β_1Hα_2V+β_2Vα_1H
       =   β_1Vα_2H+β_2Hα_1V
       =   β_1Vα_2V+β_2Vα_1V
       =   β_1Hη_2H+β_2Hη_1H
       =   β_1Hη_2V+β_2Vη_1H
       =   β_1Vη_2H+β_2Hη_1V
       =   β_1Vη_2V+β_2Vη_1V
       =   γ_1Hα_2H+γ_2Hα_1H
       =   γ_1Hα_2V+γ_2Vα_1H
       =   γ_1Vα_2H+γ_2Hα_1V
       =   γ_1Vα_2V+γ_2Vα_1V
       =   γ_1Hη_2H+γ_2Hη_1H
       =   γ_1Hη_2V+γ_2Vη_1H
       =   γ_1Vη_2H+γ_2Hη_1V.

From Eqs. (<ref>), (<ref>), (<ref>), and (<ref>),


    0   =   (β_1Vα_2H+β_2Hα_1V)α_1Hα_2V+(β_1Hα_2V+β_2Vα_1H)α_2Hα_1V
       =   √(2p_ suc)α_1Vα_2V.


Accordingly,  α_1Vα_2V=0 to satisfy p_ suc>0. By replacing α with β in Eq. (<ref>), we also obtain β_1Vβ_2V=0. 
As results of performing the similar calculation for Eqs. (<ref>), (<ref>), (<ref>), and (<ref>), we get the relation of η_1Hη_2H=γ_1Hγ_2H=0. From above results,  α_1V=0 or α_2V=0 must be satisfied at least. For α_1V and α_2V, here, we redefine α_iV as α_iV=0 and α_iV. For η_1H and η_2H, we also redefine η_kH as η_kH=0 and η_kH. From Eqs. (<ref>) and (<ref>) and above redefinitions,

    0=β_kHη_kH=β_kHη_kV.

Accordingly, we then have to consider two cases: (I) β_kH=0 and (II) β_kH≠0.


  (I) β_kH=0

From Eq. (<ref>),

    p_ suc=|β_kH|^2|α_kH|^22≤12.


  (II) β_kH≠0

From Eq. (<ref>), η_kH=η_kV=0 is derived and then

    p_ suc=|γ_kV|^2|η_kV|^22≤12.


Therefore, the maximum value of the success probability is given by p_ suc≤ 1/2.



 §.§ |t⟩=|W_4⟩

The conversion from  |Φ^+⟩_1,2|Φ^+⟩_3,4 to the state |W_4⟩ is given by

    Π_ postV'|Φ^+⟩_1,2|Φ^+⟩_3,4=√(p_ suc)|W_4⟩.

By taking inner products of |s_1s_2⟩_1,4, we obtain

    Π_ postV'|HH⟩_2,3   =   √(p_ suc)(|HV⟩_2,3+|VH⟩_2,3),
    Π_ postV'|HV⟩_2,3   =   √(p_ suc)|HH⟩_2,3,
    Π_ postV'|VH⟩_2,3   =   √(p_ suc)|HH⟩_2,3,

and

    Π_ postV'|VV⟩_2,3=0.

From inner products of |s_1s_2⟩_2,3 with Eqs. (<ref>), (<ref>), (<ref>), and (<ref>), we obtain

    √(p_ suc)   =   β_1Hα_2V+β_2Vα_1H
       =   β_1Vα_2H+β_2Hα_1V
       =   β_1Hη_2H+β_2Hη_1H
       =   γ_1Hα_2H+γ_2Hα_1H,

and 

    0   =   β_1Hα_2H+β_2Hα_1H
       =   β_1Vα_2V+β_2Vα_1V
       =   β_1Hη_2V+β_2Vη_1H
       =   β_1Vη_2H+β_2Hη_1V
       =   β_1Vη_2V+β_2Vη_1V
       =   γ_1Hα_2V+γ_2Vα_1H
       =   γ_1Vα_2H+γ_2Hα_1V
       =   γ_1Vα_2V+γ_2Vα_1V
       =   γ_1Hη_2H+γ_2Hη_1H
       =   γ_1Hη_2V+γ_2Vη_1H
       =   γ_1Vη_2H+γ_2Hη_1V
       =   γ_1Vη_2V+γ_2Vη_1V.

From Eqs. (<ref>), (<ref>), (<ref>), and (<ref>),


    0   =   (γ_1Vα_2H+γ_2Hα_1V)α_1Hα_2V+(γ_1Hα_2V+γ_2Vα_1H)α_2Hα_1V
       =   (γ_1Hα_2H+γ_2Hα_1H)α_1Vα_2V+(γ_1Vα_2V+γ_2Vα_1V)α_1Hα_2H
       =   √(p_ suc)α_1Vα_2V.


 Accordingly, α_1Vα_2V=0 when p_ suc>0 is satisfied. In a case where α_1V=0, from Eqs. (<ref>) and (<ref>), α_2V=0. On the other hand, in another case where α_2V=0, from Eqs. (<ref>) and (<ref>), α_1V=0. Thus, α_1V=α_2V=0. Then, we derive √(p_ suc)=β_2Vα_1H=β_1Vα_2H from Eqs. (<ref>) and (<ref>). This implies that in order to satisfy that p_ suc>0, β_1Vβ_2V≠0 is required. While, from Eqs. (<ref>), (<ref>), (<ref>), and (<ref>), we obtain the following relation:


    0   =   (β_1Vη_2H+β_2Hη_1V)β_1Hβ_2V+(β_1Hη_2V+β_2Vη_1H)β_2Hβ_1V
       =   (β_1Hη_2H+β_2Hη_1H)β_1Vβ_2V+(β_1Vη_2V+β_2Vη_1V)β_1Hβ_2H
       =   √(p_ suc)β_1Vβ_2V.


Then β_1Vβ_2V=0. As a result, p_ suc=0.



 §.§ |t⟩=|D_4^(2)⟩

The state conversion from |Φ^+⟩_1,2|Φ^+⟩_3,4 to the state |D_4^(2)⟩ is written as

    Π_ postV'|Φ^+⟩_1,2|Φ^+⟩_3,4=√(p_ suc)|D_4^(2)⟩.

We obtain the following relations with inner products of |s_1s_2⟩_1,4,

    Π_ postV'|HH⟩_2,32   =   √(p_ suc)|VV⟩_2,3√(6),
    Π_ postV'|HV⟩_2,32   =   √(p_ suc)(|HV⟩_2,3+|VH⟩_2,3)√(6),  
    Π_ postV'|VH⟩_2,32   =   √(p_ suc)(|HV⟩_2,3+|VH⟩_2,3)√(6),

and

    Π_ postV'|VV⟩_2,32=√(p_ suc)|HH⟩_2,3√(6).

From inner products of |s_1s_2⟩_2,3 using Eqs. (<ref>), (<ref>), (<ref>), and (<ref>), we obtain the following relations:

    √(p_ suc6)   =   β_1Vα_2V+β_2Vα_1V2
       =   β_1Hη_2V+β_2Vη_1H2
       =   β_1Vη_2H+β_2Hη_1V2
       =   γ_1Hα_2V+γ_2Vα_1H2
       =   γ_1Vα_2H+γ_2Hα_1V2
       =   γ_1Hη_2H+γ_2Hη_1H2,

and

    0   =   β_1Hα_2H+β_2Hα_1H
       =   β_1Vα_2H+β_2Hα_1V
       =   β_1Hα_2V+β_2Vα_1H
       =   β_1Hη_2H+β_2Hη_1H
       =   β_1Vη_2V+β_2Vη_1V
       =   γ_1Hα_2H+γ_2Hα_1H
       =   γ_1Vα_2V+γ_2Vα_1V
       =   γ_1Hη_2V+γ_2Vη_1H
       =   γ_1Vη_2H+γ_2Hη_1V
       =   γ_1Vη_2V+γ_2Vη_1V.

From Eqs. (<ref>), (<ref>), (<ref>), and (<ref>),


    0   =   (β_1Vα_2H+β_2Hα_1V)α_1Hα_2V+(β_2Vα_1H+β_1Hα_2V)α_2Hα_1V
       =   (β_1Vα_2V+β_2Vα_1V)α_1Hα_2H+(β_2Hα_1H+β_1Hα_2H)α_1Vα_2V
       =   √(2p_ suc3)α_1Hα_2H.


With p_ suc>0, α_1Hα_2H=0. Here, we consider two cases, i.e., (I) α_1H=0 and (II) α_2H=0.


  (I) α_1H=0

From Eq. (<ref>), α_2Vγ_1H≠0. Then, with Eqs. (<ref>) and (<ref>), β_1H=α_2H=0. From Eqs. (<ref>) and (<ref>), β_2H=0 and α_1V≠0 are also derived.

  (II) α_2H=0

From Eq. (<ref>), α_1Vγ_2H≠0. Then, with Eqs. (<ref>) and (<ref>), β_2H=α_1H=0. From Eqs. (<ref>) and (<ref>), β_1H=0 and α_2V≠0 are also derived.

From above results, α_1H=α_2H=β_1H=β_2H=0 and α_1Vα_2V≠0 are given. Then, from Eqs. (<ref>)–(<ref>),

    √(2p_ suc3)   =   γ_1Hη_2H+γ_2Hη_1H
       =   2p_ suc3(1α_2Vβ_1V+1α_1Vβ_2V) 
       =   2p_ suc3α_1Vβ_2V+α_2Vβ_1Vα_1Vβ_2Vα_2Vβ_1V
    α_1Vβ_2Vα_2Vβ_1V   =   2p_ suc3.

From Eqs. (<ref>) and (<ref>), we obtain

    2p_ suc3   =   α_1Vβ_2Vα_2Vβ_1V
       =   (√(2p_ suc3)-β_2Vα_1V)β_2Vα_1V    
    
    e^± iπ/3√(2p_ suc3)   =   β_2Vα_1V
       =   √(2p_ suc3)-β_1Vα_2V
    β_1Vα_2V   =   e^∓ iπ/3√(2p_ suc3)

and then


    2p_ suc3=|β_2Vα_1V|^2=|β_1Vα_2V|^2=|η_1Hβ_2V|^2=|η_2Hβ_1V|^2=|γ_1Hα_2V|^2=|γ_2Hα_1V|^2.


Accordingly, p_ suc is maximized when |α_1V|^2+|α_2V|^2=1 is satisfied. With [V'a_1H^† V'^†,V'a_2H^† V'^†]=0, and Eqs. (<ref>) and (<ref>), we get the following relation:

    0   =   α_1V^∗β_1V+α_2V^∗β_2V
       =   α_1V^∗e^∓ iπ/3α_2V√(2p_ suc3)+α_2V^∗e^± iπ/3α_1V√(2p_ suc3)
       =   √(p_ suc6)[1∓ i√(3)(|α_1V|^2-|α_2V|^2)].

As a result, p_ suc=0.



 §.§ |t⟩=|Φ^+⟩_1,3|Φ^+⟩_2,4

 p_ suc=1 can be achieved because a swap operation between photons in nodes 2 and 3 can be done using passive linear optics with unit probability.



 §.§ |t⟩=|Φ^+⟩_1,4|Φ^+⟩_2,3

The state conversion from |Φ^+⟩_1,2|Φ^+⟩_3,4 to  |Φ^+⟩_1,4|Φ^+⟩_2,3 is written by

    Π_ postV'|Φ^+⟩_1,2|Φ^+⟩_3,4=√(p_ suc)|Φ^+⟩_1,4|Φ^+⟩_2,3.

By taking inner product of |HH⟩_1,4, we obtain the following relation:

    Π_ postV'|HH⟩_2,3   =   √(p_ suc)(|HH⟩_2,3+|VV⟩_2,3).

By also taking inner products of |s_1s_1⟩_2,3 with Eq. (<ref>), we obtain the following equation:

    √(p_ suc)=β_1Hα_2H+β_2Hα_1H=β_1Hα_2V+β_2Vα_1V

and then

    √(p_ suc)   =   β_1Hα_2H+β_2Hα_1H
       =   β_1Hα_2H+β_2Hα_1H+β_1Hα_2V+β_2Vα_1V2
       =   |β_1Hα_2H+β_2Hα_1H+β_1Hα_2V+β_2Vα_1V2|
       ≤   |β_1Hα_2H|+|β_2Hα_1H|+|β_1Hα_2V|+|β_2Vα_1V|2.

Equation (<ref>) is maximized when ∑_j=1^2(|α_jH|^2+|α_jV|^2)=∑_j=1^2(|β_jH|^2+|β_jV|^2)=1 is satisfied. In order to derive the maximum value of p_ suc, we use the method of Lagrange multiplier. First, using variables λ and μ, we define a function f as


    f≡|β_1Hα_2H|+|β_2Hα_1H|+|β_1Hα_2V|+|β_2Vα_1V|2+λ[∑_j=1^2(|β_jH|^2+|β_jV|^2)-1]+μ[∑_j=1^2(|α_jH|^2+|α_jV|^2)-1].


Second, we calculate

    ∂ f∂ x=0,

where x∈{|α_1H|,|α_1V|,|α_2H|,|α_2V|,|β_1H|,|β_1V|,|β_2H|,
|β_2V|,λ,μ}. From Eq. (<ref>), we obtain the following relations:

    |α_2H|   =   -4λ|β_1H|,


    |α_2V|   =   -4λ|β_1V|,


    |α_1H|   =   -4λ|β_2H|,


    |α_1V|   =   -4λ|β_2V|,


    ∑_j=1^2(|α_jH|^2+|α_jV|^2)   =   1,

and

    ∑_j=1^2(|β_jH|^2+|β_jV|^2)   =   1.

From Eqs. (<ref>)–(<ref>), λ=-1/4. Then, we obtain √(p_ suc)≤1/2. Note that λ=-1/4 gives the (local) maximum of Eq. (<ref>). As the result, p_ suc≤ 1/4.



 §.§ |t⟩=|χ⟩

The state conversion  to |χ⟩ from |Φ^+⟩_1,2|Φ^+⟩_3,4 is given as

    Π_ postV'|Φ^+⟩_1,2|Φ^+⟩_3,4=√(p_ suc)|χ⟩.

We take the following relations from inner products of |s_1s_2⟩_1,4:

    Π_ postV'|HH⟩_2,3   =   √(p_ suc2)(|HH⟩_2,3+|VV⟩_2,3),
    Π_ postV'|HV⟩_2,3   =   -√(p_ suc2)(|HV⟩_2,3+|VH⟩_2,3),       
    Π_ postV'|VH⟩_2,3   =   √(p_ suc2)(|HV⟩_2,3+|VH⟩_2,3),

and

    Π_ postV'|VV⟩_2,3=√(p_ suc2)(|HH⟩_2,3+|VV⟩_2,3).

From inner products of |s_1s_2⟩_2,3 with Eqs. (<ref>), (<ref>), (<ref>), and (<ref>), we obtain the following equations:

    √(p_ suc2)   =   β_1Hα_2H+β_2Hα_1H
       =   β_1Vα_2V+β_2Vα_1V
       =   -(β_1Hη_2V+β_2Vη_1H)
       =   -(β_1Vη_2H+β_2Hη_1V)
       =   γ_1Hα_2V+γ_2Vα_1H
       =   γ_1Vα_2H+γ_2Hα_1V
       =   γ_1Hη_2H+γ_2Hη_1H
       =   γ_1Vη_2V+γ_2Vη_1V,

and

    0   =   β_1Hα_2V+β_2Vα_1H
       =   β_1Vα_2H+β_2Hα_1V
       =   β_1Hη_2H+β_2Hη_1H
       =   β_1Vη_2V+β_2Vη_1V
       =   γ_1Hα_2H+γ_2Hα_1H
       =   γ_1Vα_2V+γ_2Vα_1V
       =   γ_1Hη_2V+γ_2Vη_1H
       =   γ_1Vη_2H+γ_2Hη_1V.

When α_is=0 (i∈{1,2},s∈{H,V}) is satisfied,

    √(p_ suc2)=β_isα_is=γ_isα_is,

and

    0   =   β_isα_is=γ_isα_is,

where 1≡2, 2≡1, H≡ V, and V≡ H. Finally, we get p_ suc=0. With the same way of the above process, we also calculate with β_i's', γ_ĩs̃, and η_ĩ'̃s̃'̃ (i',ĩ,ĩ'̃∈{1,2}, s',s̃,s̃'̃∈{H,V}) and then α_isβ_i's'γ_ĩs̃η_ĩ'̃s̃'̃≠0, when p_ suc>0. From Eqs. (<ref>), (<ref>), (<ref>), and (<ref>),


    0   =   (β_1Hα_2V+β_2Vα_1H)α_1Vα_2H+(β_1Vα_2H+β_2Hα_1V)α_1Hα_2V
       =   (β_1Hα_2H+β_2Hα_1H)α_1Vα_2V+(β_1Vα_2V+β_2Vα_1V)α_1Hα_2H
       =   √(p_ suc2)(α_1Vα_2V+α_1Hα_2H).


Replacing α with β in Eq. (<ref>),

    0=√(p_ suc2)(β_1Vβ_2V+β_1Hβ_2H).

With the same process for other equations and p_ suc>0,

    α_1Vα_2V+α_1Hα_2H   =   0,
    β_1Vβ_2V+β_1Hβ_2H   =   0,
    α_1Vα_2H+α_1Hα_2V   =   0,
    β_1Vβ_2H+β_1Hβ_2V   =   0,
    γ_1Vγ_2V+γ_1Hγ_2H   =   0,
    η_1Vη_2V+η_1Hη_2H   =   0,
    γ_1Vγ_2H+γ_1Hγ_2V   =   0,

and

    η_1Vη_2H+η_1Hη_2V   =   0.

From Eqs. (<ref>) and (<ref>),

    -α_1Hα_2Vα_2H   =   -α_1Hα_2Hα_2V
    α_2Vα_2H   =   α_2Hα_2V
    α_2V   =   (-1)^aα_2H.

Here, a∈{0,1}. From the same equations, we also obtain the following equation,

    -α_1Vα_2Vα_1H   =   -α_1Hα_2Vα_1V
    α_1Vα_1H   =   α_1Hα_1V
    α_1V   =   -(-1)^aα_1H.

By using the same way for other equations, β_2V=(-1)^bβ_2H, β_1V=-(-1)^bβ_1H, γ_2V=(-1)^cγ_2H, γ_1V=-(-1)^cγ_1H, η_2V=(-1)^dη_2H, and η_1V=-(-1)^dη_1H are also derived. Here, b,c,d∈{0,1}. From Eqs. (<ref>) and (<ref>),

    √(p_ suc2)   =   β_1Hα_2H+β_2Hα_1H
       =   (-1)^aβ_1Hα_2V+(-1)^bβ_2Vα_1H
       =   [(-1)^a-(-1)^b]β_1Hα_2V.

Here, a=b⊕ 1 has to be held to satisfy p_ suc>0. From Eqs. (<ref>) and (<ref>),

    √(p_ suc2)   =   -(β_1Hη_2V+β_2Vη_1H)
       =   -[(-1)^dβ_1Hη_2H+(-1)^bβ_2Hη_1H]
       =   -[(-1)^d-(-1)^b]β_1Hη_2H.

Then, b=d⊕ 1 holds with p_ suc>0. From Eqs. (<ref>) and (<ref>),

    √(p_ suc2)   =   γ_1Hα_2V+γ_2Vα_1H
       =   (-1)^aγ_1Hα_2H+(-1)^cγ_2Hα_1H
       =   [(-1)^a-(-1)^c]γ_1Hα_2H
       =   [1-(-1)^a⊕ c]γ_1Hα_2V,

and then a=c⊕ 1, when p_ suc>0. From Eqs. (<ref>) and (<ref>),

    √(p_ suc2)   =   γ_1Hη_2H+γ_2Hη_1H
       =   (-1)^dγ_1Hη_2V+(-1)^cγ_2Vη_1H
       =   [(-1)^d-(-1)^c]γ_1Hη_2V
       =   [1-(-1)^c⊕ d]γ_1Hη_2H,

and then c=d⊕ 1, when p_ suc>0. Thus, 


    √(p_ suc2)=(-1)^a2β_1Hα_2V=-(-1)^a2β_1Hη_2H=2γ_1Hα_2V=2γ_1Hη_2H,


and then α_2V=-η_2H=-α_2V. However, this contradicts α_isβ_i's'γ_ĩs̃η_ĩ'̃s̃'̃≠0. Accordingly, p_ suc=0.



§ APPENDIX B

In this Appendix, we derive Eqs. (<ref>) and (<ref>). We define U_ PDBS as a unitary operator denoting the function of the PDBS with transmittance T_H and T_V. By using this definition and Pauli operators X_i and Z_i acting on spatial mode i, the success probability can be written as


    p_ suc
       =   ηη'||Π_ postZ_2'X_2'X_1'U_ PDBSX_2X_1U_ PDBSU_ PDBS|Φ^+⟩|Φ^+⟩||^2
       =   ηη'||Π_ postZ_2'X_2'X_1'U_ PDBSX_2X_1T_H|HHHH⟩+√(T_HT_V)(|HHVV⟩+|VVHH⟩)+T_V|VVVV⟩2||^2
       =   ηη'||Π_ postZ_2'X_2'X_1'U_ PDBST_H|HVVH⟩+√(T_HT_V)(|HVHV⟩+|VHVH⟩)+T_V|VHHV⟩2||^2
       =   ηη'4||T_H(1-2T_V)|HHHH⟩+T_HT_V(|HHVV⟩+|VVHH⟩)
          +√(T_H(1-T_H)T_V(1-T_V))(|HVHV⟩+|VHVH⟩)-T_V(2T_H-1)|VVVV⟩||^2
       =   ηη'4[T_H^2(1-2T_V)^2+2T_H^2T_V^2+2T_H(1-T_H)T_V(1-T_V)+T_V^2(2T_H-1)^2]
       =   ηη'(T_H^2+2T_HT_V+T_V^2-6T_H^2T_V-6T_HT_V^2+12T_H^2T_V^2)4,


where |||ψ⟩||≡√(⟨ψ|ψ⟩). From above calculation, when the postselection succeeds, the output state |ψ_ out⟩ is

    |ψ_ out⟩   =   √(ηη')2√(p_ suc)[T_H(1-2T_V)|HHHH⟩
          +T_HT_V(|HHVV⟩+|VVHH⟩)
          +√(T_H(1-T_H)T_V(1-T_V))(|HVHV⟩+|VHVH⟩)
          -T_V(2T_H-1)|VVVV⟩].

Accordingly, the fidelity F is

    F   =   √(|⟨ C_4|ψ_ out⟩|^2)
       =   |⟨ C_4|ψ_ out⟩|
       =   |T_H(1-2T_V)+2T_HT_V+T_V(2T_H-1)|2√(T_H^2+2T_HT_V+T_V^2-6T_H^2T_V-6T_HT_V^2+12T_H^2T_V^2)
       =   |T_H+2T_HT_V-T_V|2√(T_H^2+2T_HT_V+T_V^2-6T_H^2T_V-6T_HT_V^2+12T_H^2T_V^2).


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