Superdense Coding Via Semi-Counterfactual Bell-State Analysis Fakhar Zaman1 ∗ Youngmin Jeong1 † Hyundong Shin1 ‡

Superdense Coding via Semi-Counterfactual Bell-State Analysis Fakhar Zaman1 ∗ Youngmin Jeong1 † Hyundong Shin1 ‡

1 Department of Electronic Engineering, Kyung Hee University, Yongin-si, 17104 Korea

Abstract. Quantum superdense coding is an example of how entanglement can be used to transmit two bits of classical information in one qubit. However, to take the full advantage of quantum superdense coding, one needs to be able to perform the complete Bell-state analysis. In this paper, we present a semi-counterfactual scheme for Bell-state analysis based on dual quantum Zeno effect. We show that our scheme enables one to distinguish between the four Bell states with certainty and plot the data rate for quantum superdense coding versus number of cycles of dual quantum Zeno effect. Keywords: Bell-state analysis, Entanglement, Quantum superdense coding. 1 Introduction b Quantum superdense coding is a prime example of a b quantum information processing task where two bits of BS BS BS BS BS classical information can be transferred between two spa- a a tially separated parties using previously shared entanglement [1]. For a bipartite system, Bell-state analysis is a crucial step in quantum superdense coding [1, 2]. Figure 1: Interaction free measurement. However it has been proven that the complete Bell-state analysis is impossible using only linear operations and single degree of shared entanglement [3, 4, 5]. While uti- data rate Rb for quantum superdense coding as a func- lization of an ancillary photon [6] or an ancillary degree tion of N. of freedom [7, 8]—hyperentanglement—with linear optics The IFM was first proposed by Dicke [19], and Vaid- enable complete Bell-state analysis. Hyperentanglement man et al. [13]. The basic idea was to infer the presence assisted Bell-state analysis has been demonstrated using or absence of an absorptive object without interacting orbital angular momentum [9] or time [10, 11] as an an- with it. The maximum achievable efficiency was limited cillary degree of freedom. by the margin of 50%. This efficiency was further im- In [12], a different approach for Bell-state analysis has proved to 100% by Kwiat et al. [14]. been proposed based on the interaction free measure- We consider an interferometer [14] that consists of N ment (IFM) [13, 14]. First they proposed 2-qubits IFM unbalanced beam splitters (BS) as shown in Fig. 1. We gate that changes one particle’s trajectory according to describe the upper path as b (Bob’s side) and lower path whether or not the other particle exists in the interfer- as a (Alice’s side). The BS in Fig. 1. works as follows: ometer and proposed a scheme for Bell-state analysis by 1 0 cos θ 1 0 + sin θ 0 1 , using combination of four 2-qubits IFM gates. Later, sev- | ia | ib → | ia | ib | ia | ib (1) 0 1 cos θ 0 1 sin θ 1 0 , eral schemes for controlled NOT (CNOT) gate operation | ia | ib → | ia | ib − | ia | ib via IFM have been presented [15, 16]. Although they im- where 1 a 0 b means the photon is in path a, 0 a 1 b proved it in terms of resources used to perform CNOT means| thei | photoni is in path b and θ = π/ (2N).| i | i gate operation but they sacrificed the probability of suc- Alice starts by throwing her photon in path a, Bob has cess which plays an important role to determine the data choice either allow the photon to pass or block the chan- rate of quantum superdense coding. In this article, we nel. If Bob allows the photon to pass, the photon will be present a semi-counterfactual scheme for complete Bell- found in the transmission channel with probability one state analysis based on dual quantum Zeno effect (QZE) and end up in path b. In case Bob blocks the channel with the high probability of success even for the small 1 by introducing absorptive object, if the photon found in number of cycles N of IFM. The rest of the paper is the transmission channel it will be absorbed by the ab- organized as follows: in the later half of this section, we sorptive object. The probability that the photon is not briefly overview IFM and Bell-state analysis via already absorbed by the absorptice object is cos(2N) θ. Unless existing IFM based CNOT gates [12, 16]. In Sec. 2, we the photon is absorbed by the absorptive object, after demonstrate our scheme for quantum superdense coding N cycles it will end up in path a. It shows that IFM is via semi-counterfactual Bell-state analysis and plot the counterfactual only if Bob block the channel. ∗[email protected] In [12], they presented a scheme for Bell-state anal- †[email protected] ysis based on IFM. First they presented the 2-qubits ‡ [email protected] IFM gate. The working principle of 2-qubits IFM gate 1The quantum Zeno effect is inhibition between the quantum states by the frequent measurements of the state, that is, the quan- is exactly the same as IFM gate. The only difference tum state usually collapse back to the initial state if the time be- is they considered absorptive object as quantum rather tween the measurements is short enough [17, 18]. then classical one. For uniform distribution of Bell states, average probability Pavg that the photon is not absorbed MR2 by the absorptive object is given by: ψ = state of quantum absorptive object. ψ ! 1 1 3N 1 N P = 1 sin2 θ + 1 sin2 θ . (2) avg 2 − 2 − 2 channel H(V) H(V) SM PR PBS OD MR1 Later in [16], they presented a CNOT gate using only OC1 OC2 OC3 H (V) one QZE gate and an ancillary photon. They considered both the target bit and the control bit as the quantum absorptive object. In order to distinguish between four Bell states, the ancillary photon needs not to be absorbed Figure 2: H (V)-QZE setup. by the absorptive object. The probability P that the ancillary photon is not absorbed by the absorptive object for any input Bell state is given by: Table 1: Table for I-QZE setup under the asymptotic limits of N, where I and I⊥ H, V . 3 N ∈ { } P = 1 sin2 φ , (3) − 4 Inputs Outputs II⊥ pass block II⊥ pass block where φ = π/N and N > 1. In [16], although they used only one QZE gate but they 0 0 0 0 0 0 0 0 sacrificed the probability of success. In next section, we demonstrate our scheme for quantum superdense coding 1 0 1 0 0 1 1 0 via semi-counterfactual Bell-state analysis and improve 1 0 0 1 1 0 0 1 the probability of success Ps using only two QZE gates instead of using four 2-qubits IFM gates [12]. 0 1 1 0 1 0 1 0 0 1 0 1 0 0 0 1 2 Quantum Superdense Coding We now consider H (V) QZE [20], showing in principle how to perform IFM based− on quantum Zeno effect. unitary operations, identity, pauli X, pauli Y or pauli Here H (V) refers to horizontal (vertical) polarization. Z operator, corresponding to classical bits he wants to The function of BS in the interferometer of Kwiat et al. transmit. is achieved by the combined action of polarizing rotator After performing unitary operation on electron, Alice (PR) and polarizing beam splitter (PBS) as shown in and Bob will perform semi-counterfactual Bell-state anal- Fig. 2. The only difference is, we consider the absorptive ysis based on dual QZE as shown in Fig. 3. where electron object as quantum one instead of classical so that it acts as a quantum absorptive object. The Bell states are can be in superposition of the two orthogonal states respectively given as: (superposition of blocking and not blocking the channel). ± 1 The action of PRH(V), on Alice’s photon is Φ = block V pass H , (5) √2 | ie | ip ± | ie | ip ( 1 H(V) H (V) cos θ H (V) + sin θ V (H) , ± PR | i → | i | i (4) Ψ = block e H p pass e V p , (6) V (H) cos θ V (H) sin θ H (V) . √2 | i | i ± | i | i | i → | i − | i where H = 1 , V = 0 , pass = 1 = 0 1 We set the rotation angle θ = π/ (2N) for PRH(V). p p p p e e A B and block| i =| i0 |=i 1 |0i ;| andi the subscripts| i | i e| andi The switchable mirror (SM) is initially turned off to allow e e A B p denote| thei electron| i | andi | i photon respectively. If the passing the photon, and once the photon is passed it will electron is in path A (B), it shows the presence (absence) be turned on for rest N cycles. It is turned off again of the absorptive object. after N cycles, allowing the photon out. The H (V)- Alice starts the protocol by sending her photon to- QZE setup takes H (V) polarized photons as input, with wards PBS . PBS separates the H and V compo- PBSH(V) passing H (V) photons and reflecting V (H) as 1 1 nents and feed them into the corresponding H (V)-QZE shown in Fig. 2. Table 1 shows the overall action of the setup. In dual-QZE, in the absence of the electron I-QZE gate under the asymptotic limits of N, where I ⊥ pass x where x H, V the photon is found in and I H, V . In the later half of this section we will | ie | ip ∈ { } demonstrate∈ { our} scheme for quantum superdense coding. the transmission channel with probability one which makes our scheme semi-counterfactual. In case the pres- Quantum superdense coding is a method of utilizing shared quantum entanglement to increase the rate at ence of the electron block x , if the photon is found | ie | ip which information may be sent through a noiseless quan- in transmission channel it will be absorbed by the elec- tum channel. We consider that a third party, Charlie gen- tron. After N cycles, the probability Ps that the photon erates an entangled pair ( Φ+ ) of electron and photon | i is not absorbed by the electron for any input Bell state and transmit the electron to Bob (sender) and the pho- is given by: ton to Alice (receiver). Bob will perform one of the four t = T1

B D1 Bob electron AO H A D2 PBS2

PBS1 BS PBS3 H H Alice photon H QZE D − 3 V V V QZE − D4 Dual-QZE

Figure 3: Proposal for semi-counterfactual Bell-state analysis. Initially electron and the photon are in max- imally entangled state where electron act as quantum absorptive object for dual-QZE setup. Presence and absence of the quantum absorptive object (electron) is represented as 1 0 and 0 1 respectively. Alice starts by | iA | iB | iA | iB sending photon towards PBS1 to separate the H and V components and feed them into corresponding H (V)-QZE setup. After Hadamard gate , Bob will measure the path of the electron and send this classical information to Alice using counterfactual quantumH communication [21]. Table 2 shows the estimated initial state corresponding the state of the electron and photon after Hadamard gate.

2.0 Table 2: Bell-state analysis under the asymptotic limits of N. 1.8 After Hadamard Gate 1.6 Photon State Detectors Estimated 1.4 Polar- of the Click Initial 1.2

ization Electron State b 1.0 State R 0.8 − 0 A 1 B D1D4 Φ V | i | i | i 0.6 + 1 0 D2D4 Φ | iA | iB | i 0.4 DQZE − IFM 0 1 D1D3 Ψ H | iA | iB | i 0.2 QZE + 1 A 0 B D2D3 Ψ 0.0 | i | i | i 2 4 6 8 10 12 14 16 N

N Figure 4: Data rate Rb as a function of N where DQZE, 1 2 Ps = 1 sin θ . (7) IFM and QZE shows the Bell-state analysis is performed − 2 based on dual-QZE (our scheme), IFM [12] and QZE [16] respectively. In our scheme, R > 1.8 for N 12 . And the equations 5 and 6 will transform as: b ≥

± 1 N Φ V cos θ block pass , (8) photon and state of the electron (see Table 2). Each Bell → √ | ip | ie ± | ie 2 state carries two bits of classical information. Then the ± 1 N Ψ H cos θ block pass . (9) data rate Rb for quantum superdense coding for a fixed √ p e e → 2 | i | i ± | i value of N is Rb = 2 Ps. In Fig. 4., we plot Rb as a function of N. In our scheme× R > 1.8 for N 12. Unless the photon is absorbed by the electron, the pho- b ≥ ± ton is horizontally polarized if the initial state is Ψ , 3 Acknowledgment in case the initial state is Φ± the photon will be| verti-i | i cally polarized at t = T1. After Hadamard gate, Bob will This work has supported by the National Research measure the path of the electron and send this classical Foundation of Korea (NRF) grant funded by the information to Alice using counterfactal quantum com- Korea government (MSIT) (No. 2016R1A2B2014462), munication [21] to enable Alice to distinguish between and by the Basic Science Research Program through four Bell states. Alice will estimate the initial with cer- the NRF funded by the Ministry of Education tainty corresponding to the polarization of her existing (No. 2018R1D1A1B07050584). References [15] J. Franson, B. Jacobs, and T. Pittman, “Quantum computing using single photons and the Zeno effect,” [1] C. H. Bennett and S. J. Wiesner, “Communica- Phys. Rev. A, vol. 70, no. 6, p. 062302, Dec. 2004. tion via one-and two-particle operators on Einstein- Podolsky-Rosen states,” Phys. Rev. Lett., vol. 69, [16] Y. Huang and M. Moore, “Interaction-and no. 20, p. 2881, Nov. 1992. measurement-free quantum Zeno gates for universal computation with single-atom and single-photon [2] K. Mattle, H. Weinfurter, P. G. Kwiat, and qubits,” Phys. Rev. A, vol. 77, no. 6, p. 062332, A. Zeilinger, “Dense coding in experimental quan- Jun. 2008. tum communication,” Phys. Rev. Lett., vol. 76, no. 25, p. 4656, Jun. 1996. [17] W. M. Itano, D. J. Heinzen, J. Bollinger, and D. Wineland, “Quantum Zeno effect,” Phys. Rev. [3] L. Vaidman and N. Yoran, “Methods for reliable A, vol. 41, no. 5, p. 2295, Mar. 1990. teleportation,” Phys. Rev. A, vol. 59, no. 1, p. 116, Jan. 1999. [18] T. Petrosky, S. Tasaki, and I. Prigogine, “Quantum Zeno effect,” Phys. Rev. A, vol. 151, no. 3-4, pp. [4] J. Calsamiglia and N. Lütkenhaus, “Maximum effi- 109–113, Dec. 1990. ciency of a linear-optical Bell-state analyzer,” Appl. Phys. B: Lasers Opt., vol. 72, no. 1, pp. 67–71, Apr. [19] R. H. Dicke, “Interaction-free quantum measure- 2001. ments: A paradox,” Am. J. Phys., vol. 49, no. 10, pp. 925–930, 1981. [5] S. Ghosh, G. Kar, A. Roy, A. Sen, U. Sen et al., “Distinguishability of Bell states,” Phys. Rev. Lett., [20] P. G. Kwiat, A. White, J. Mitchell, O. Nairz, vol. 87, no. 27, p. 277902, Dec. 2001. G. Weihs, H. Weinfurter, and A. Zeilinger, “High- efficiency quantum interrogation measurements via [6] W. P. Grice, “Arbitrarily complete Bell-state mea- the quantum Zeno effect,” Phys. Rev. Lett., vol. 83, surement using only linear optical elements,” Phys. no. 23, p. 4725, Dec. 1999. Rev. A, vol. 84, no. 4, p. 042331, Oct. 2011. [21] H. Salih, Z.-H. Li, M. Al-Amri, and M. S. Zubairy, [7] P. G. Kwiat, “Hyper-entangled states,” J. Mod. “Protocol for direct counterfactual quantum com- Opt., vol. 44, no. 11-12, pp. 2173–2184, 1997. munication,” Phys. Rev. Lett., vol. 110, no. 17, p. [8] J. T. Barreiro, N. K. Langford, N. A. Peters, and 170502, Apr. 2013. P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett., vol. 95, no. 26, p. 260501, Dec. 2005.

[9] J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys., vol. 4, no. 4, p. 282, Mar. 2008. [10] C. Schuck, G. Huber, C. Kurtsiefer, and H. We- infurter, “Complete deterministic linear optics Bell state analysis,” Phys. Rev. Lett., vol. 96, no. 19, p. 190501, May 2006. [11] B. P. Williams, R. J. Sadlier, and T. S. Humble, “Su- perdense coding over optical ﬁber links with complete Bell-state measurements,” Phys. Rev. Lett., vol. 118, no. 5, p. 050501, Feb. 2017. [12] H. Azuma, “Interaction-free quantum computation,” Phys. Rev. A, vol. 70, no. 1, p. 012318, Jul. 2004.

[13] A. Elitzur and L. Vaidman, “Quantum mechan- ical interaction-free measurement,” Found. Phys., vol. 23, no. 76, pp. 987–997, Jul. 1993. [14] P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M. A. Kasevich, “Interaction-free measurement,” Phys. Rev. Lett., vol. 74, no. 24, p. 4763, Nov. 1995.