## 2 TF-DPSQSS Protocol with Asymmetric Source Intensities  
The schematic diagram of our asymmetric protocol is shown in Figure 1, where [PERSON] and [PERSON] have partial keys for deciphering, and [PERSON] has a full key for ciphering. The two senders, [PERSON] and [PERSON], independently prepare two trains of weak coherent pulses whose intensities are different and phases are randomly modulated to be 0 or \(\pi\). The coherent pulses are sent through the quantum channels and received by the trusted third party, [PERSON], who measures them using an unbalanced interferometer. The details of the protocol are shown as follows. Preparation: [PERSON] ([PERSON]) prepares a weak coherent pulse train and phase-modulates each pulse randomly by 0 or \(\pi\). The coherent states can be denoted as \(\left|\psi_{a}\right\rangle=\underset{n=1}{\overset{N}{\underset{\otimes}{ }}}\left|\sqrt{\mu_{a}}e^{i\phi_{n}^{n}}\right\rangle\)  
\(\left(\left|\psi_{b}\right\rangle=\underset{n=1}{\overset{N}{\underset{\otimes} {}}}\left|\sqrt{\mu_{b}}e^{i\phi_{n}^{n}}\right\rangle\right)\). Then, she (he) sends out the coherent state pulse whose period is 2T to [PERSON] with an average photon number less than one per pulse. [PERSON] ([PERSON]) records her (his) logic bits of each time slot as \"0\" (\"1\") when her (his) modulated phase is 0 (\(\pi\)). We denote the sequence of detection events time slots as \(n\in\{1,2,3,\ldots,2N\}\), where \(N\) is the total number of pulses sent by [PERSON] ([PERSON]). The phase shift \(\phi_{n}^{n}\left(\phi_{n}^{b}\right)\in\{0,\pi\}\) is the  
Figure 1: Configuration of our quantum secret sharing protocol.Weak coherent pulse sources (Laser); phase modulator (PM); signal attenuator (Att); transmittance of channels between [PERSON] ([PERSON]) and [PERSON] (\(\eta_{a},\eta_{b}\)); polarization control (PC); beam splitter (BS); single photon detector (D1, D2). In [PERSON]’s measurement area, two pulse trains are first polarization-modulated by polarization controls to correct their polarization for interference. Then, [PERSON] divides each incoming pulse into two paths and recombines them by a 50:50 beam splitter, where the path-length difference is set equal to the time T. Detectors are placed at the two outputs of the recombining beam splitter. At the detectors, the partial wave functions of two senders’ pulses that are in the same time slots interfere with each other. phase induced by the phase modulator on pulse \(n\) ignoring the global phase, and intensities \(\mu_{a}\), \(\mu_{b}\) are corresponding to [PERSON] and [PERSON], respectively. Measurement: As illustrated in Figure 1, while measuring the signal, [PERSON] records the photon detection time and which detector clicks. When the detection event time slots are corresponding to the time \(2 kT\), detector 1 will click for 0 phase difference between the two senders' pulses and detector 2 will click for \(\pi\) phase difference. When the detection event time slots are corresponding to the time \((2k+1)T\), detector 1 will click for \(\pi\) phase difference and detector 2 will click for 0 phase difference. Note that if both detectors click, [PERSON] randomly chooses one detector click to record. Here, a photon is detected occasionally and randomly because the received signal power is smaller than one photon per pulse. Using the above setup, [PERSON] creates their key shown in Table 1. \"0\" means that the modulated phases in one time slot imposed by [PERSON] and [PERSON] are \(\{0,0\}\) or \(\{\pi,\pi\}\), and \"1\" means that the modulated phases in one time slot imposed by [PERSON] and [PERSON] are \(\{0,\pi\}\) or \(\{\pi,0\}\). Thus, [PERSON]'s bits are exclusive OR of [PERSON]'s and [PERSON]'s bits. That is, [PERSON] and [PERSON] know [PERSON]'s key bits only when they cooperate, and the QSS operation is accomplished. Parameter estimation: [PERSON] randomly chooses recorded detection times and [PERSON] and [PERSON] alternatively disclose her or their test bit first in the chosen time slots through a public channel.