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

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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 y Hyundong Shin1 z 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 entan- glement [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- j ia j ib ! j ia j ib j ia j ib (1) 0 1 cos θ 0 1 sin θ 1 0 ; eral schemes for controlled NOT (CNOT) gate operation j ia j ib ! j ia j ib − j ia j 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 meansj thei j photoni is in path b and θ = π= (2N).j i j 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 z [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 2 f g 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 princi- ple 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) p2 j ie j ip ± j ie j ip ( 1 H(V) H (V) cos θ H (V) + sin θ V (H) ; ± PR j i ! j i j i (4) Ψ = block e H p pass e V p ; (6) V (H) cos θ V (H) sin θ H (V) : p2 j i j i ± j i j i j i ! j i − j 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 blockj i =j i0 j=i 1 j0i ;j andi the subscriptsj i j i ej andi The switchable mirror (SM) is initially turned off to allow e e A B p denotej thei electronj i j andi j 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.
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