Counterfactual Bell-State Analysis

The Bell-state analysis to distinguish between the four maximally entangled Bell states requires the joint measurement on entangled particles. However, spatially separated parties cannot perform the joint measurement. In this paper, we present a counterfactual Bell-state analysis based on the chained quantum Zeno effect. This counterfactual analysis not only enables us to perform a complete Bell-state analysis, but also enables spatially separated parties to distinguish between the four Bell states without transmitting any physical particle over the channel.

i M for outer cycles and = i N for inner cycles, respectively. The Michelson CQZE setup takes H(V) photons as an input, with the polarizing beam splitter PBS 1 (2) H(V) passing H(V) photons and reflecting V(H) as shown in Fig. 1. There are two possible scenarios, 1. If Bob allows the photon to pass, the inner cycles act as an obstacle for outer cycles. After M outer cycles, the photon will end up in the state H(V) . If the photon is found in the transmission channel it will end up at the detector D. For large values of M and N, the probability of finding the photon in the transmission channel approaches to zero. 2. In the case where Bob blocks the channel, the inner cycles act as non-blocking for outer cycles. Unless the photon is absorbed by the absorptive object, the photon will be in the state V(H) after M outer cycles. Table 1 shows the overall action of the I-CQZE gate under the asymptotic limits of M and N, where I, ∈ ⊥ I {H, V}. The basic idea behind the CCNOT gate is direct counterfactual quantum communication 21 where the control bit acts as a quantum absorptive object that can be in the superposition of two orthogonal states as shown in Figure 1. Michelson H(V)-CQZE setup. PBS stands for a polarizing beam splitter, MR stands for a mirror, OD for an optical delay, and OC for an optical circulator. The switchable mirror SM 1(2) is initially turned off to allow passing the photon, and once the photon is passed it will be turned on. After M(N) cycles, SM 1(2) is turned off again allowing the photon out. The switchable polarizing rotator SPR 1(2) rotates the polarization by small angle H(V) . At Bob's end, Bob either allows the photon to pass or block the channel by introducing an electron as a quantum absorptive object. Full counterfactulity is ensured as any photon found in the transmission channel would either absorbed by the electron or else end up at the detector D. allow the photon to pass or in their superposition. Unless the photon is discarded in counterfactual quantum communication, the photon will be in path c or d, or in the superposition of c and d after M outer and N inner cycles depending on the input state of the photon and the control bit. As M, → ∞ N , the probability that the photon is found in the transmission channel approaches to zero.
Counterfactual Bell-state Analysis. In the conventional Bell-state analysis, Alice uses the CNOT gate followed by the Hadamard gate (Bob's side) on the entangled particles. Following the same procedure, we use the CCNOT gate 32 to make our scheme counterfactual as shown in Fig. 3. The CCNOT gate is based on direct counterfactual quantum communication where quantum absorptive object act as control bit as shown in Fig. 2. In direct counterfactual quantum communication, the probability that the photon is found in the transmission channel for each of the Bob's choices (allow the photon to pass or block the channel) depends on the number of M outer and N inner cycles of the CQZE. In the case photon is found in the transmission channel, the photon would either be absorbed by the absorptive object or ends up at detector D (see Fig. 1). This probability is asymptotically zero as M and N approach to infinity.
To demonstrate our scheme for counterfactual Bell-state analysis, we consider that Alice and Bob have an entangled pair of photon and electron, where the electron acts as the control bit and the photon acts as the target bit, respectively. Then, the Bell states are given by  Here BS stands for 50 : 50 beam splitter, the H stands for the Hadamard gate and the CCNOT gate is the counterfactual CNOT gate. By means of the CCNOT gate, Bell states can be analyzed without transmitting any physical particle over the channel where Bob's entangled particle (electron) acts as a control bit. Initially the electron and photon are in entangled state. Alice starts the protocol by sending her photon towards the CCNOT gate. At t = T 1 , a polarization of the photon determines either the state: Φ ± or Ψ ± . After the Hadamard gate, Bob needs to transmit classical information ( 0 or 1 ) e e counterfactualy. For this, Alice send her photon in the CCNOT gate and the photon will be detected at one of the four detectors (D 1 , D 2 , D 3 , and D 4 ) at t = T 2 . Each detector corresponds to one of the four Bell states which enables us to distinguish between the Bell states with certainty. where |pass〉 e = |0〉 e , |block〉 e = |1〉 e , |H〉 p = |0〉 p , and |V〉 p = |1〉 p ; and the subscripts e and p denote the electron and photon, respectively. Alice starts by sending her photon into the CCNOT gate. At t = T 1 , the photon will be horizontally polarized if the initial state is Φ ± , while it will be vertically polarized if the initial state is Ψ ± . Then, the photon and electron will be in the separable state, and the Bell states in (3) and (4)  To make our scheme fully counterfactual, we use a feedback system to determine the absence or presence of the electron as shown in Fig. 4. At t = T 2 , the photon is either in path a or b. If the photon is in path a, the initial state is Φ ± , while if the photon is in path b, the initial state is Ψ ± . The polarization of the photon determines either the state x + or the state x − where ∈ Φ Ψ x { , }. The photon will be detected by one of the four detectors. Each detector corresponds to one of the four Bell states which enables us to distinguish between the four Bell states with certainty. Table 2 shows the estimated initial state corresponding to each detector.
From the counterfactual Bell-state analysis in the paper, we conclude the following observations and remarks.  Table 2. Bell-state analysis under the asymptotic limits of M and N.

Completeness:
The complete Bell-state analysis is itself a challenge. From Table 2, the counterfactual Bell-state analysis in this paper enables us to distinguish between the four Bell states without using the hyperentanglement. It is verified for finite numbers of M outer and N inner cycles for the CQZE (see Sec. Discussion). 2. Counterfacuality: We challenged the long-lasting assumption that spatially separated parties cannot perform the Bell-state analysis without transmitting any physical particle over the channel. In our scheme, the probability of finding the photon in the transmission channel approaches to zero under the asymptotic limits of M and N. Even for finite values of M and N, if the photon is found in the transmission channel either it will be absorbed by the electron or it will be discarded at detector D (see Fig. 1); and no detector will click. Unless the photon is discarded, one of the four detectors click which enables spatially separated parties to perform the complete Bell-state analysis without transmitting any physical particle over the channel. 3. Resource Efficiency: Quantum superdense coding is a prime application to utilize shared entanglement between two parties. In general, both of the entangled particles get destroyed at the end of the protocol to decode a two-bit classical message. Counterfactual Bell-state analysis enables spatially separated parties to implement quantum superdense coding without transmitting any physical particle over the channel at the cost of only one particle. As shown in Fig. 3, only the photon is absorbed by the detector to estimate the initial state.

Discussion
In the previous section, we discussed the counterfactual Bell-state analysis under the ideal scenario. In this section, we show that counterfactual Bell-state analysis can be true: i) for finite values of M and N, and ii) in the presence of channel noise. Then, we further discuss the experimental feasibility for our counterfactual Bell-state analysis.

Finite Values of M and N.
We first show that all results are also true for finite values of M and N. For the finite values of M and N, the counterfactuality of the SLAZ13 has been analyzed 33,34 . It was shown that the protocol can be counterfactual only for one value of the transmitted bit ( 1 ) e . Recently, experimental realization 35 of SLAZ13 21 has been presented to preserve the counterfactual property for both values of the transmitted bit via the CQZE setup using the single-photon source. In the case of multi-photon source or coherent state light, the counterfactuality is not ensured if Bob allows the photon to pass ( 0 ) e . In the presence of the absorptive object, the amplitude of the coherent state which is found in the transmission channel will be absorbed by the absorptive object and no detector clicks. In the absence of the absorptive object, the amplitude of the coherent state found in the transmission channel is nonzero even when the detector D does not click (see Fig. 1) which violates the counterfactuality of the protocol. To ensure the couterfactuality for both values of the transmitted bit for finite values of M and N, they used the single-photon source. The overall action of the modified SLAZ13 scheme is same as shown in Table 1. The only difference is that there exists a success probability corresponding to each Bob's choice, either block the transmission channel or allow the photon to pass. These probabilities are given by 21 where θ π = j /(2 ) j for ∈ j M N { , }; P pass is the success probability when Bob allows the photon to pass while P block is the success probability where Bob blocks the photon path; and y M { ,0} can be obtained from the recursion relations: . After M outer and N inner cycles, the photon and electron will be in the separable state, which is given by Note that (14) and (15) are not orthonormal because the probability that the photon is discarded in the counterfactual quantum communication is nonzero for the finite values of M and N.
From (12) and (13), the probability that the photon is not discarded, denoted by P s , till t = T 1 for any input Bell state is given by 32 Unless the photon is discarded, the photon will be detected at one of the four detectors at t = T 2 , and we can estimate the initial state with the probability one. The success probabilities P D 1 , P D 2 , P D 3 , and P D 4 for the corresponding initial Bell states Φ − , Φ + , Ψ − , and Ψ + are respectively given by   Figure 5. Success probabilities P D 1 , P D 2 , P D 3 , and P D 4 as a function of (M, N) under the ideal channel conditions. ( 1 1) 2 . In Fig. 6, we plot the probabilities P D 1 (3) , P D 2(4) , P D 3 (1) , and P D 4 (2) as a function of p for the corresponding input Bell states (a) Φ Ψ Experimental Feasibility. For an experimental realization of counterfactual Bell-state analysis, there are two major problems to be concerned. The first one is to guarantee the phase stability of the CQZE gate which is the building block of our scheme as shown in Fig. 4. The practical realization of quantum counterfactual-like communication has been recently demonstrated based on the single-photon source 35 and the weak coherent light 36 .
Another problem in the experimental realization is to generate a superposition state of the absorptive object. In our scheme, we used the electron as the quantum absorptive object which can take the superposition of two paths. To overcome the problem of superposition of presence and absence, we introduce a mirror between two paths as shown in Fig. 7. If the electron is in the path A, it shows the absence of the absorptive object. In case the electron is in the path B, the absorptive object is blocking the transmission channel.