*5.2. Cases for N* ≥ 2 *Assisted by a Weak Measurement*

For *N* ≥ 2, the procedure follows as in the previous section by introducing *N* imperfect entangled resources, |*χi* (assumed identical for simplicity) but in each step, we decide if after of the weak measurement, the state |*χj*<sup>0</sup> = |*β*0 is used to perform a single teleportation or if we continue the process of weak measurement *N* times on identical entangled resources |*χj* to finally get <sup>|</sup>*χN*1 <sup>=</sup> <sup>∑</sup><sup>3</sup> *<sup>i</sup>*=<sup>1</sup> *p i* |*βi* as in the Figure 12. The corresponding situation is now depicted for the general case in the Figure 14. In this case, the global probability of success becomes:

$$\mathcal{P}\_{\text{TotN}} = \sum\_{j=1}^{N} p\_0 (1 - p\_0)^{j-1} + (1 - p\_0)^N \mathcal{P}\_{m,N}^{\text{ff}, \{p\_i'\}}.\tag{43}$$

Inserting the formulas for <sup>P</sup>ff,{*<sup>p</sup> i* } *<sup>m</sup>*,*<sup>N</sup>* in Appendix B (specialized for the frontal face *p*<sup>0</sup> = 0 and changing *pi* by *p i* ). Then, we can get the outcomes for global probability PTot*<sup>N</sup>* for the last cases with F = 1:

$$\mathcal{P}\_{\text{Tot2}} = 1 - 2(p\_1 p\_2 + p\_1 p\_3 + p\_2 p\_3) \tag{44}$$

$$\begin{array}{rcl}\mathcal{P}\_{\text{Tot}3} &=& 1 - \left(p\_1^3 + p\_2^3 + p\_3^3\right) - 3\left(p\_1^2(p\_2 + p\_3) + p\_2^2(p\_1 + p\_3) + p\_3^2(p\_1 + p\_2)\right) \\\mathcal{P}\_{\text{Tot4}} &=& 1 - 4\left(p\_1^3(p\_2 + p\_3) + p\_2^3(p\_1 + p\_3) + p\_3^3(p\_1 + p\_2)\right) \end{array} \tag{45}$$

$$-12p\_1p\_2p\_3(p\_1+p\_2+p\_3) - \frac{16}{3}(p\_1^2p\_3^2 + p\_2^2p\_3^2 + p\_1^2p\_2^2).\tag{46}$$

Now, we can visualize last outcomes for PTot in Figure 15. Again, all the entangled states used for the teleportation process are assumed to be identical by simplicity. Figures 15a–c depict the probability PTot*<sup>N</sup>* to reach F = 1 in the entire process represented in color. Each color bar shows the entire range of values for such probabilities on the graphs. According to the color, the blue zone represents the region where PTot → 1, observing for the case *N* = 4 a larger blue area, suggesting still the goodness of increase the number of teleportation channels under indefinite causal order combined with post-measurement.

Figure 15d depicts a numerical analysis of statistical distribution for the cases *N* = 2, 3, 4. Note that for *N* = 3, all greater values for the probability occur almost evenly. For the case *N* = 4, it is observed a larger amount of success probabilities than failure probabilities compared with *N* = 3. Despite, *<sup>μ</sup>*PTot2 ≈ 0.702, *<sup>σ</sup>*PTot2 ≈ 0.158 and *<sup>μ</sup>*PTot4 ≈ 0.667, *<sup>σ</sup>*PTot4 ≈ 0.249 (because for *<sup>N</sup>* = 2 there are a

large distribution for medium success probabilities). In any case, the most successful outcomes of teleportation appears for *N* = 4.

**Figure 14.** Schematic teleportation process assisted by indefinite causal order using *N*-teleportation channels and weak measurement.

**Figure 15.** (**a**–**c**) values of PTot as function of (*p*1, *p*2, *p*3), for *N*2, *N*<sup>3</sup> and *N*<sup>4</sup> respectively. (**d**) Statistical distribution numerically obtained for PTot2,PTot3 and PTot4.

#### **6. Experimental Deployment of Teleportation with Indefinite Causal Order**

In this section, we comment on some main experimental developments for a possible deployment of indefinite causal order in teleportation. We begin with the procedure to set the weak measurement used in Section 5.1. Afterwards, we set some elements and experimental developments to propose the implementation of the theoretical proposal before presented.
