Entanglement generation using a controlled-phase gate for time-bin qubits

Quantum logic gates are important for quantum computations and quantum information processing in numerous physical systems. While time-bin qubits are suited for quantum communications over optical fiber, many essential quantum logic gates for them have not yet been realized. Here, we demonstrated a controlled-phase (C-Phase) gate for time-bin qubits that uses a 2x2 optical switch based on an electro-optic modulator. A Hong-Ou-Mandel interference measurement showed that the switch could work as a time-dependent beam splitter with a variable spitting ratio. We confirmed that two independent time-bin qubits were entangled as a result of the C-Phase gate operation with the switch.

been used in most of the fiber-based quantum communication experiments because of their robustness against these fluctuations. [21][22][23] However, a problem remains in that many of the essential quantum logic gates, such as controlled-not (CNOT) or controlled-phase (C-Phase) gates, have not yet been realized for time-bin qubits.
In this study, we demonstrate an implementation of the C-Phase gate for time-bin qubits.
Our scheme is based on a two-input, two-output (2x2) optical switch used as a time-dependent beam splitter. 24) We show that two time-bin qubits can be entangled as a result of C-Phase gate operation with the 2x2 switch.
Here, we describe the C-Phase gate operation for time-bin qubits by using a high-speed 2x2 optical switch, which is a Mach-Zehnder (MZ) interferometer that includes an electrooptic phase modulator (PM) 24) in one of the optical paths as shown in Fig. 1(c). We launch two time-bin qubits as a control and a target state into ports A and B of the switch, whose states are given by |ψ A = c 1A |t 1 A + c 2A e iφ A |t 2 A and |ψ B = c 1B |t 1 B + c 2B e iφ B |t 2 B . The index A and B are the input ports to Alice and Bob. Here, |t x y represents the photon in the time position t x ∈ {t 1 , t 2 } of the input port, y ∈ {A, B}, and c xy is the amplitude of |t x y which is a nonnegative real number that satisfies c 2 1y + c 2 2y = 1, and φ y is the phase difference between temporal states t 1 and t 2 which can be set by adjusting the temperature controller (TC). The ideal C-Phase gate operates on two input time-bin states, as follows: By applying a time-varying signal to the PM, the 2x2 switch can work as a time-dependent beam splitter whose splitting ratio changes in time. The evolution of a time-bin state with the 2x2 switch is described as where θ(t k ) represents the phase difference between the two arms of the MZ interferometer at time t k and the index C and D are the output ports to Charlie and David. For the C-Phase gate operation, we set θ(t 1 ) = 0 and θ(t 2 ) = 2 cos −1 ( 1 √ 3 ), which means that the 2x2 switch passes the first temporal mode and works as a one-third beam splitter for the second mode.
By performing a coincidence measurement between Charlie and David, we obtain a state given by Similarly to the case of previous C-Phase gates realized for path 14) and polarization 12,15) qubits, the amplitude unbalance can be eliminated by applying one-third attenuation only to the t 1 mode. Thus, in the coincidence basis between Charlie and David, we obtain an output state for the C-Phase gate operation, given by The experimental setup is shown in Fig. 1. We generate a 1561-nm pulse train with a 250-MHz repetition rate by modulating continuous-wave laser light from an external-cavity Alice and Bob prepare their time-bin states by launching the signal and idler photons into 1-bit delay interferometers fabricated using planar light-wave circuit (PLC) 25) technologies, as shown in Fig. 1(b). As discussed in the previous section, one-third amplitude attenuation (ATT) should be added to the first temporal mode. Note that additional one-third polarizationdependent beam splitters were integrated in the gate in previous experiments, 14,15,18) while we placed the amplitude attenuation in the stage of state preparation by Alice and Bob. This means that the states of the initial time-bin qubits are given by 3 4 1 3 |t 1 + e iφ |t 2 . To implement this, we fabricated PLC interferometers equipped with additional MZ interferometers in the short arms. With these MZ interferometers of Alice and Bob as shown in the insert of Fig. 1(b), we can apply variable attenuation to the first temporal mode by adjusting the TC individually. Then the time-bin qubits are launched into the 2x2 optical switch, which is based on a lithium niobate waveguide (EO Space). 24) By adjusting the DC bias and RF modulation signal to the PM in the switch as shown in Fig. 1(c), the 2x2 switch works as a one-third beam splitter for the t 2 mode and as a transparent transmission path for the t 1 mode. Because of the post-selection and amplitude compensation, the success probability of the C-Phase gate is 1/9 even when there are no component losses. The detection efficiencies of SSPD for Charlie and David are 57% and 62%, respectively, and the dark count for both detectors is less than 40 cps. In order to erase the polarization Thus, the density matrix of the obtained state after the C-Phase gate operation is given by To confirm generation of an entangled state as a result of C-Phase gate operation, we performed quantum state tomography (QST) 27,28) for time-bin qubits so that we could obtain the density matrix of the output state as shown in Fig. 3. The minus-sign terms in the density matrix of Eq. (6) are clearly visible in Fig. 3(a). The fidelity to the target entangled state was 62±7.8%. We also calculated the von Neumann entropy of 0.817, linear entropy of 0.505, and concurrence of 0.551. 27,28) However, according to Peres,29)  There are several points that may have decreased the fidelity of the entangled state generated by the C-Phase gate. The fluctuation of the splitting ratio of the 2x2 switch, which comes from the DC bias drift of the lithium niobate waveguide modulator, would have been the main source of the errors in the generated state. 11) Moreover, because of the large loss induced by the interferometers and the optical switch, the present experiment required a long measurement time, which increased the fluctuation of the setup further. We believe that we can obtain better fidelity by overcoming these issues. 30) In addition, the use of integrated photonics technologies will enable significant compactification of the gate function, which will lead to better stability. For example, we can integrate the function of amplitude compensation as additional intensity modulators placed in front of the 2x2 switch fabricated in a lithium niobate waveguide.
Although we demonstrated that the C-Phase gate successfully worked for a specific input state, the present experiment does not constitute a full characterization of the quantum gate.
Quantum process tomography (QPT) 31) is now widely used for this purpose. Using QPT to analyze the gate operations requires 16 different input states, which increases the measurement time significantly. Because of the low coincidence rate caused by the relatively large component losses and the limited stability of the 2x2 switch described above, it is difficult to perform QPT with our C-Phase gate with the current setup. Therefore, it is important to reduce the component losses and improve the stability of the setup so that we can undertake QPT for more a comprehensive characterization of the gate operation.
As with the previous C-Phase gates based on post-selection, 12,15) the limited success probability will constrict the application of these gates to systems with a relatively small number of qubits. For example, such probabilistic quantum gates could be useful for demonstrating a quantum communication system based on quantum error correction. 20) We would like to note that a C-NOT gate can be performed by applying a Hadamard transform on the target time-bin qubit before and after the C-Phase gate. 18) In addition, we can tune the amount of phase shift given to the time-bin qubits by changing the amplitude of the modulation signal to the switch. 30) When we use the proposed gate in a quantum network over optical fiber, precise adjustment of the path lengths in front of the 2x2 switch is not a trivial issue. However, such path-length matching in a fiber network has been successfully demonstrated in several longdistance quantum teleportation experiments, 32,33) in which active feedback control based on HOM interference measurement was implemented, together with the sharing of the time ref- Appl. Phys. Express erence between nodes enabled by the use of classical channels. These techniques can be applied to deploy our quantum gate in a real fiber network.
In summary, we demonstrated a C-Phase gate for time-bin qubits by using a 2x2 switch as a beam splitter. By adjusting the DC and RF signal, the optical switch can work as a timedependent beam splitter with different splitting ratios for different temporal modes. Here, the 2x2 switch was operated as a one-third beam splitter of the t 2 mode that passes the t 1 mode.
The HOM experiment showed that the visibility was 0.78±0.02 for the t 2 mode and there was no dip for the t 1 mode. By performing QST, an examination of the density matrix showed that the C-Phase gate successfully entangled the time-bin states prepared by Alice and Bob, and the fidelity was 62±7.8%.
We thank William J. Munro for fruitful discussions.