A Hermitian operator W ^ is said to be EW if it has non-negative expectation values for all separable states, ρ s e p , and negative expectation values for some entangled states, ρ e n t [10, 12, 13, 33, 34]: T r W ^ σ s e p ≥ 0 ∀ σ s e p , T r W ^   ρ e n t < 0 ∃ ρ e n t . (1)

Extensive efforts have been made to develop suitable witness operators. In Refs [35, 36], the authors have developed EW operators using two orthogonal local observables. To develop EW operators, let us consider an experimental setup that has been constructed to prepare the state Ψ . In practice, the generated state becomes the mixed state ρ, which is in proximity to the state Ψ . To test whether the state ρ is entangled or not, the authors in Refs [35, 36] proposed a witness operator in the form W ^ = c 0 I − ∑ k c k S k , (2) where “ I ” represents the identity operator, the operator S k stabilizes the state Ψ , that is, S k Ψ = Ψ , and c k is the constant. In this work, two time-bin entangled photons are prepared in one of four two-qubit Bell states, e.g., ϕ ± 12 = 00 12 ± 11 12 2 , (3) ψ ± 12 = 01 12 ± 10 12 2 , (4) where the subscript i corresponds to the ith photon. For the aforementioned Bell states, the stabilizing operators are S 1 = X 1 X 2 and S 2 = Z 1 Z 2 , where X i , Z i are Pauli operators acting on the ith photon. The corresponding witness operators certifying the Bell states are as follows [35, 36]: W ^ ϕ + = 1 4 I − 1 4 X 1 X 2 + Z 1 Z 2 , (5) W ^ ϕ − = 1 4 I + 1 4 X 1 X 2 − Z 1 Z 2 , (6) W ^ ψ + = 1 4 I − 1 4 X 1 X 2 − Z 1 Z 2 , (7) W ^ ψ − = 1 4 I + 1 4 X 1 X 2 + Z 1 Z 2 . (8)

The experimentally generated state ρ exp is said to be entangled if min Ψ ∈ { ϕ ± , ψ ± } Tr W ^ Ψ   ρ exp <   0 . (9)

In the ideal scenario, Tr W ^ Ψ Ψ Ψ = − 1 / 4 .

The experimental setup is shown in Figure 1A. The pump laser is a picosecond pulsed laser operating at a wavelength of 1552.52 nm. The repetition rate, spectrum width, and pulse width are 20 MHz, 0.26 nm, and 13.23 ps, respectively. First, this pump pulse is amplified using an erbium-doped fiber amplifier (EDFA), following which a dense wavelength division multiplexing (DWDM) is utilized to remove the amplified spontaneous emission from EDFA [37]. Then, the amplified pump pulse is temporally divided into two peaks with a 3-ns interval by passing through the pump interferometer. We construct each interferometer with an optical fiber splice and optical delay lines, each with an optical path length difference of 3 ns. Faraday mirrors and the Michelson interferometer are utilized to match the polarization of two peaks [38]. Furthermore, we use a variable optical attenuator (VOA) to correct the optical loss difference due to the path difference between the short and long paths in the interferometer. Afterward, a time-bin entangled photon pair state is generated in the dispersion-shifted fiber (DSF) by spontaneous four-wave mixing (SFWM). Here, to reduce spontaneous Raman scattering, DSF was immersed in liquid nitrogen to lower the temperature [37], and an in-line polarizer (ILP) was used. These are divided into the signal at 1549.32 nm and idler channels at 1555.75 nm through the DEMUX composed of DWDM with a 100 GHz bandwidth.

In fiber optics, the phase of the fiber interferometer changes with temperature. Therefore, thermal stabilization is required. The scheme for thermal stabilization is shown in Figure 1B. This stabilization setup is included in the red dotted line in Figure 1A. For feedback, a continuous wave (CW) laser in a range different (1529.7 nm) from the wavelength of the pump was used [39]. Feedback is more stable when the tracking point is minimum. The minimum detector power of the bias controller is −30 dBm, and the minimum optical power of interference used in the CW laser is less than −30 dBm. Therefore, another CW laser for offset is used. Output power before and after stabilization is shown in Figure 2. In addition, we placed the interferometer inside the styrofoam for thermal stabilization.

Phase modulators (PMs), equipped with a 3-dB bandwidth of 10 GHz, are utilized to alter the phase of the signal and idler photons’ second peak (late peak). We apply rectangular voltage pulses to the PMs using an arbitrary pulse generator. Each voltage pulse possesses a rise time less than the 3-ns interval between the first and second pulses and is applied at the second peak time. Notably, the pulse heights are controlled from one pulse to another, leading to alterations in the phase of the second peak of the signal and idler photons, denoted by φ s and φ i , respectively. When these time-bin qubits pass through the interferometer, they are temporally separated into three peaks with an interval of 3 ns. Then, the state of the center peak becomes the superposed state as follows: 1 2 ( 0 + e i ( φ p − φ ) 1 ) , (10) where φ is the phase of the signal or the idler changed by PM. In the interference between the center peak of the signal and the center peak of the idler, the coincidence count rate is given by R c   ∼   1 + cos φ s + φ i − 2 φ p , (11)

where φ p is the phase of the pump changed by the pump’s PM, and in our experiment, φ p = 0 π . In order to know the phase change for voltage applied to PM, we measured the coincidence between the center peaks of the signal and idler by changing the voltage applied to the PM of the signal (idler), while the voltage of the PM of the idler (signal) is fixed. The results are shown in Figure 3. From the fitting curves, we obtain V − 1 π V 0 π and V − 1 2 π V 1 2 π of the signal and idler, respectively. The difference in the voltage values of the signal and idler is due to an error in the PM itself. Finally, we performed projection measurement by using PMs and time-correlated single-photon counting (TCSPC) and adjusting the gate delay of the single-photon detector based on the InGaAs avalanche photodiode (APD). The gate width and quantum efficiency of the two APDs are 1 ns and 20%, respectively. The gate frequency of APDs is synchronized with the repetition rate of the pump pulse laser through the pulse generator.

First, we checked whether the Bell state ϕ + was properly generated by performing QST [40, 41] before the measurement of EW. In order to proceed with QST, we used 0 , 1 , R , − for the signal and 0 , 1 , L , + for the idler, and the coincidence counts for these combinations are shown in Table 1. In addition, since only one detector is connected at the output of each interferometer in our setup, a 50% intrinsic loss occurs in time basis measurement. Therefore, for the project measurement including one-time basis and the project measurement including two-time basis, the coincidence measurement time was doubled and quadrupled, respectively [37]. The coincidence counts were corrected by subtracting the accidental coincidence count, as shown in Table 1. As a result of QST performed using these coincidence counts, the obtained density matrix is given by ρ exp = 0.498 − 0.0064 + i 0.0098 − 0.0064 − i 0.0098 0.0139 − 0.0158 + i 0.0048 0.48 − i 0.0041 0.0029 + i 0.0015 − 0.0093 − i 0.0241 − 0.0158 − i 0.0048 0.0029 − i 0.0015 0.48 + i 0.0041 − 0.0093 + i 0.0241 0.0013 − 0.0181 − i 0.0073 − 0.0181 + 0.0073 0.486 , (12) and is shown in Figure 4. The state ρ exp has a fidelity of 96.31   % ± 1.61 % with the ideal state ϕ + . The concurrence of the state ρ exp is 0.94   ±   0.03 . The aforementioned parameters confirm that the prepared state ρ exp is an entangled and Bell state, ϕ + .