Constructing method of 2-EPP with different quantum error correcting codes Daichi SASAKI1 ∗ Tsuyoshi Sasaki USUDA1 †

1 School of Information Science and Technology, Aichi Prefectural University, 1522-3 Ibaragabasama, Nagakute city, Aichi 480-1198, Japan. Abstract. For quantum information systems, ‘entanglement’ is an important resource. However, entan- gled states are affected by noisy quantum channels if a sender transmits a portion of an entangled state to a receiver creating an entanglement between sender and receiver. To handle noise, entanglement purification protocols (EPPs) were proposed. In a previous study, the superiority of 2-EPPs to 1-EPPs with finite entangled states was shown for a phase-damping channel. We propose here a method of constructing an EPP, which uses two (or more) quantum error correcting codes. Also for phase-damping channels, we assess the performance of the EPP constructed by our method. Keywords: entanglement, entanglement purification, quantum error correction

1 Introduction We consider two stabilizer codes, C1 :[n, k1]code and C :[n, k ]code. Let G(1) and G(2) be sets of generators Entanglement purification protocols(EPPs) are im- 2 2 for C and C , respectively: portant to share entangled states over a noisy quan- 1 2 (1) (1) (1) (1) tum channel[1]. EPPs consist of fundamental proce- G = {g , g , ··· , g − }, 1 2 n k1 (3) dures called “local operations and classical communica- (2) (2) (2) (2) G = {g , g , ··· , g − }. tion(LOCC)”. In Ref.[2], a method was given that con- 1 2 n k2 verted an arbitrary [n, k]stabilizer code to a 2-EPP. In Defining the present paper, we consider EPPs constructed from (1) (1) (1) (1) S = ⟨g , g , ··· , g − ⟩, two (or more) quantum error correcting codes and show 1 2 n k1 (4) S(2) = ⟨g(2), g(2), ··· , g(2) ⟩, that our method has higher performance in comparison 1 2 n−k2 with those using individual codes. as the stabilizers of C1 and C2, we introduce the set C obtained from these two stabilizers as 2 Preliminaries C = S(1) ∩ S(2). (5) In this study, we consider the following problem. Alice ′ { ′ ′ ··· ′} prepares n Bell states Let C = c1, c2, , cl be a set of generators of C. Because C′ ⊂ C ⊂ S(1),S(2), we have the following gen- 1 |Φ+⟩ = √ (|00⟩ + |11⟩), (1) erators of C1 and C2: 2 ′ ′ ′ ′ G(1) = {g(1) , g(1) , ··· , g(1) }, 1 2 n−k1 ′ ′ ′ ′ (6) and sends half of each to Bob over a noisy quantum chan- (2) (2) (2) (2) G = {g , g , ··· , g − }, nel. Then, each one applies an EPP to these. If they keep 1 2 n k2 their pairs, they share k entangled states. where In Ref.[3], the superiority of 2-EPPs to 1-EPPs with finite entangled states was shown for the phase-damping ′ ′ (1) (2) ′ ··· channel, which is represented as gi = gi = ci (i = 1, , l). (7) (1)′ (2)′ † The proposed protocol using G and G is per- E (ρ) = (1 − p)ρ + pZρZ , (2) PD formed as follows: where 0 ≤ p ≤ 1. However, better 2-EPPs are desired • ′ ··· ′ Alice measures c1, , cl on her own quantum for phase-damping channels. For this reason and also for states and obtains a measurement outcome aC′ = simplicity, we shall assume communications are over a (ac′ , ··· , ac′ ). phase-damping channel to examine the performance of 1 l • ′ ··· ′ our construction method. Bob measures c1, , cl on his own quantum states and obtains a measurement outcome bC′ = (bc′ , ··· , bc′ ). 3 Method of constructing EPP 1 l • 3.1 Construction method Alice and Bob send their measurement outcomes to the other and each performs the following two In this section, we show a method to construct an EPP processes according to error syndromes that uses two (or more) quantum error correcting codes. sC′ = aC′ ⊕ bC′ ∗[email protected] † = (a ′ ⊕ b ′ , ··· , a ′ ⊕ b ′ ). (8) [email protected] c1 c1 cl cl 1.0 1.0 Proposed scheme Proposed scheme 0.8 [31,21]2-EPP(ECM) 0.8 [31,21]2-EPP(ECM) Proposed scheme is better than [31,21] 0.6 0.6 Proposed scheme is better than [31,21] 0.4 0.4 Fidelity 0.2 0.2 Purification Rate 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 p p

Figure 1: proposed method

• If sC′ ∈ R, Alice and Bob measure the remaining 4 Conclusion ′ ′ operators g(1) , ··· , g(1) , then calculate syndromes l+1 k1 In this paper, we proposed a method to construct a 2- and perform further processing depending on all EPP which consists of different quantum error correcting the error syndromes. codes and by simulations investigated the performance of the 2-EPPs for a phase-damping channel. The pro- • If sC′ ̸∈ R, Alice and Bob measure the remaining (2)′ (2)′ posed protocol showed improved fidelity and purification operators g , ··· , g , then calculate syndromes l+1 k2 rate compared with an EPP from a single code when the and perform further processing depending on all number of initial shared entanglement is 31. the error syndromes. Although we have shown that the EPP by our method R ⊂ Fl Here, 2 is the subset of all syndromes which are achieves a higher performance, we need to evaluate our obtained by measuring C′. Therefore, a procedure in the method in general quantum channels(e.g. i.i.d. depolar- protocol is changed according to whether sC′ is or is not izing channel). in R. Acknowledgment: This work has been supported in 3.2 Performance part by JSPS KAKENHI Grant Number 24360151. In this section, we evaluate by simulations the per- References formance of 2-EPPs consisting of different quantum er- [1] C.H. Bennett, D.P. DiVincenzo, J.A. Smolin and ror correcting codes. Here, we use [31, 21]code and W.K. Wootters, “Mixed state entanglement and [31, 16]code. Because we are considering phase-damping quantum error correction,” Phys. Rev. A54, pp.3824- channels, each generator consists of I or X. The param- 3851, (1996). eters of the protocol are n = 31, k1 = 21, k2 = 16, and l = 10. [2] R. Matsumoto, “Conversion of general quantum sta- We use the fidelity between k Bell state |Φ+⟩⊗k and bilizer code to an entanglement distillation protocol,” shared states ρout after purification and a purification J. Phys. A36, no.29, pp.8113-8127, (2003). rate which is defined as [3] Y. Yoshiaki, S. Usami, T.S. Usuda, “Superiority of 2-EPPs to 1-EPPs with finite entangled resources,” k Proc. of ISITA2012, pp.206-210, (2012). R = P , (9) P n S [4] A. Ambainis and D. Gottesman, “The minimum dis- where PS is the success probability of the EPP. tance problem for two-way entanglement purifica- In the following, we consider the 2-EPP from the tion,” IEEE Trans. Inform. Theory 52, pp.748-753, [31, 21]code as the ‘standard’ protocol and compare it (2006). with the other protocols. We compared the proposed protocol with the standard [5] H. Aschauer, “Quantum communication in noisy en- protocol (Fig.1), from which we find that the proposed vironments,” Ph.D.Thesis, LMU Munich, (2004). protocol is superior to [31, 21]2-EPP both in fidelity and [6] D. Gottesman, “Stabilizer codes and quantum er- in purification rate. ror correction,” Ph.D.Thesis, California Institute of Technology, Pasadena, CA, (1997).