Abstract
Measurement can reveal the difference between quantum physics and classical one. Recently, a quantum witness \(W_q\) was proposed to characterize quantumness (Li et al. in Sci Rep 2:885, 2012; Kofler and Brukner in Phys Rev A 87:052115, 2013). It is built upon the no-signaling-in-time condition, and there is only one-time intermediate measurement. As an extension, we consider here multiple intermediate measurements at different moments of time. And we discuss the quantumness of a damped and driven qubit. Uniform, quasiperiodic and random time-interval sequences (TISs) of measurements are considered, respectively. Numerical results show that \(W_q\) depends on the kind of TISs when the number of measurements N is less than 10, while it is almost independent of the kind of TISs when N is larger. Further, \(W_q\le W_q^{max}(N)=(1-\frac{1}{2N}) e^{-\gamma \tau }\) for all cases, where \(\tau \) is the evolution time, \(\gamma \) is the dephasing intensity, and \(W_q^{max}(N)\) is the maximum violation of the quantum witness equality.
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This work was supported by the National Natural Science Foundation of China (11504179, 11705097, 61271238 and 61475075).
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Appendices
Appendix A
For uniform TISs, based on Eq. (17) and at the driving amplitude \(\Delta =0.0\), quantum witness is reduced to
As the number of intermediate measurement times \(N\rightarrow \infty \) and \(\omega _0\tau \) is finite, \(\frac{\omega _0\tau }{N+1}\rightarrow 0\). Using Taylor expansion,
and
Therefore,
At \(\omega _0\tau =\pi \), \(W_q\approx [1-\frac{\pi ^2}{4(N+1)}]e^{-\gamma \tau }<W_q^{max}(N)\).
Appendix B
For uniform, quasiperiodic and random TISs, Fig. 4d shows that quantum witness \(W_q\) as function of time \(\tau \) at intermediate measurement times \(N=10^4\). As given in Eq. (17),
where \(p_+(\tau )\) and \(p''_+(\tau )\) are the probabilities of the system in the \(|+\rangle \) eigenstate at last moment \(\tau \) in the presence and in the absence of multiple intermediate measurements. As given in Eq. (11),
so the probability \(p_+(\tau )\) exhibits underdamped behavior, which is shown in Fig. 6a. Based on Eqs. (15) and (16), we define a factor
then
Figure 6a shows that \(p''_+\) exhibits critically damped or overdamped behavior. The inset in Fig. 6a shows that there exist differences among \(p''^u_+, p''^f_+\) and \(p''^r_+\). On the whole, the differences are very small. In Eq.(B4), two factors, i.e., \(\exp (-\gamma \tau )\) and \(\chi \), make the decay of \(p''_+\). Fig. 6b shows that \(\chi \) smoothly decreases with time \(\tau \). In fact, at relative small \(\tau \), the differences among \(\chi _u, \chi _f\) and \(\chi _r\) are small. On the other hand, at relative large \(\tau \), there exist obvious differences among \(\chi _u, \chi _f\) and \(\chi _r\); at the same time, the factor \(\exp (-\gamma \tau )\) is smaller. Therefore, combined the two aspects, there are no obviously differences in \(p''^u_+, p''^f_+\) and \(p''^r_+\).
Appendix C
The system evolves within the range \([0, \tau ]\). The time interval is defined by \(\mu _n=\tau _n-\tau _{n+1}\), so the measurement moment \(\tau _n=\sum _{k=1}^{n}\mu _{k}\). The time interval \(\mu _n\) is generated from the two-point distribution, exponential distribution, normal distribution, respectively.
The two-point distribution is
We set \(\mu (C):\mu (D)=1:2\) and \(p_c=0.5\).
The exponential distribution function is
We set \(\lambda =1.0\).
The normal distribution function is
where \(\nu \) is the mean value and \(\sigma \) is the standard deviation. In calculation, we set \(\nu =0.0\) and \(\sigma =1\). At the same time, \(\mu >0\) is used.
In calculations, the time interval \(\mu _n\) is generated from the above three different distributions. We also set \(\tau _0=0\) and \(\tau _{N+1}=\tau \). At last, \(\tau _n\) is rescaled by the restriction that \(\tau _{N+1}=\tau \). Quantum witness \(W_q\) as functions of time \(\tau \) is shown in Fig. 7a–d, respectively, where \(N=1,2,10\) and \(10^4\). It shows that for all cases, \(W_q(\tau )\le W_q^{max}(N)=(1-\frac{1}{2N}) e^{-\gamma \tau }\). When intermediate measurement times N are tens [Fig. 7a–c], \(W_q\) depends on the kind of TISs, while when N are larger [Fig. 7d], it is almost independent of the kind of TIS.
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Gong, L., Ma, K., Zhao, X. et al. Quantum witness of a damped and driven qubit by sequential intermediate measurements with uniform and nonuniform time intervals. Quantum Inf Process 19, 260 (2020). https://doi.org/10.1007/s11128-020-02765-8
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DOI: https://doi.org/10.1007/s11128-020-02765-8