4.2.2. Case *pj* 1, *j* = 1, 2, 3

In some practical cases, the expected values for the entangled resource |*χ* vary slightly from a perfect entangled state: *pj* 1 for *j* = 1, 2, 3. Thus, the outcome described through Formula (15) becomes in this case (developing to first order for *pj*, *j* = 1, 2, 3 the factor ∏<sup>3</sup> *<sup>j</sup>*=<sup>0</sup> *p tj <sup>j</sup>* there):

$$
\Lambda^N[\rho \otimes \rho\_\mathbb{c}] \approx \left[ \left( 1 - N \sum\_{j=1}^3 p\_j \right) \rho + N \sum\_{j=1}^3 p\_j \sigma\_j \rho \sigma\_j \right] \otimes \rho\_\mathbb{c} \equiv \rho\_{\text{out}} \otimes \rho\_\mathbb{c}. \tag{27}
$$

Note that under this approximation, *ρ<sup>c</sup>* becomes unaltered and separated from the system state. Thus, the optimal way to teleport the state implies to measure the control state considering |*ψm* = ∑*<sup>k</sup>* <sup>√</sup>*qk*|*k*. In the following, we assume such an optimal measurement made on the control state.

For the particular case where *pj* = <sup>1</sup> <sup>4</sup>*<sup>N</sup>* with *j* = 1, 2, 3, last formula can be written as:

$$
\Lambda^N[\rho \otimes \rho\_\mathfrak{c}] \approx \frac{1}{2} \sigma \eta \otimes \rho\_\mathfrak{c}.\tag{28}
$$

Obtaining the totally depolarized state <sup>1</sup> <sup>2</sup>*σ*0. Notice that it is only applicable for very large values of *N* (due to the assumption *pj* 1, *j* = 1, 2, 3). This aspect is advised in the Figure 7 where the fidelity drops more rapidly to <sup>1</sup> <sup>2</sup> when *N* grows around of *p* = 0.

In general, the probability and fidelity given in (27) will become respectively (developing to first order in *pj*, *j* = 1, 2, 3):

$$\mathcal{P}\_{\rm m} \approx \text{Tr}[\rho\_{\rm out}] = 1 \tag{29}$$

$$\mathcal{F}\_N \approx \frac{\text{Tr}[\rho \rho\_{\text{out}}]}{\mathcal{P}\_m} = 1 - N \sum\_{j=1}^3 p\_j (1 - n\_j^2) \equiv 1 - Np\_{\text{lb}} \sum\_{j=1}^3 a\_j (1 - n\_j^2) \equiv 1 - Np\_{\text{lb}} \Delta\_{\theta, \phi}^{a\_1, a\_2, a\_3}, \tag{30}$$

where *ρ* was written as in (24). We are introduced the reduced parameters *α<sup>j</sup>* ∈ [0, 1] and the threshold probability *p*ts 1 to limit the validity of the current approximation (*pj* = *p*ts*α<sup>j</sup>* 1, *j* = 1, 2, 3). We note in any case that the increasing of *N* worsens the fidelity. Note each term in the sum in (30) is non-negative, thus the fidelity becomes commonly reduced. Because only one of *n*<sup>2</sup> *<sup>j</sup>* , *j* = 1, 2, 3 could be one at the time, then it is necessary in addition that two *pj* become zero to get F*<sup>N</sup>* = 1. Otherwise, F*<sup>N</sup>* < 1 with a notable decreasing if *N* is large. The outcome in (29) exhibits a combination of the three error-syndromes *<sup>σ</sup>*1*ρσ*1, *<sup>σ</sup>*2*ρσ*2, *<sup>σ</sup>*3*ρσ*<sup>3</sup> reflected through the terms *<sup>α</sup>j*(<sup>1</sup> <sup>−</sup> *<sup>n</sup>*<sup>2</sup> *<sup>j</sup>*) as function of *αj*. Thus, for each syndrome *σjρσ<sup>j</sup>* the best states being teleported are those closer to the eigenstates of *σj*, otherwise while several *α<sup>j</sup>* = 0 the teleportation capacity is widely reduced.

Considering *<sup>ρ</sup>* <sup>=</sup> <sup>|</sup>*ψψ*<sup>|</sup> with <sup>|</sup>*ψ* <sup>=</sup> cos *<sup>θ</sup>* <sup>2</sup> <sup>|</sup>0 <sup>+</sup> sin *<sup>θ</sup>* <sup>2</sup> *<sup>e</sup>i<sup>φ</sup>* <sup>|</sup>1 on the Bloch sphere: *<sup>n</sup>*<sup>1</sup> <sup>=</sup> sin *θ* cos *φ*, *n*<sup>2</sup> = sin *θ* sin *φ*, *n*<sup>3</sup> = cos *θ*. Then, we analyze each syndrome and its impact on the fidelity through the quantity Δ*α*1,*α*2,*α*<sup>3</sup> *<sup>θ</sup>*,*<sup>φ</sup>* . As lower it becomes, higher becomes F*N*. Figure 9a shows the simple behavior of Δ*α*1,*α*2,*α*<sup>3</sup> *<sup>θ</sup>*,*<sup>φ</sup>* for each state on the Bloch sphere under each syndrome: *p*<sup>1</sup> = 1, *p*<sup>2</sup> = *p*<sup>3</sup> = 0; *p*<sup>2</sup> = 1, *p*<sup>1</sup> = *p*<sup>3</sup> = 0; and *p*<sup>3</sup> = 1, *p*<sup>1</sup> = *p*<sup>2</sup> = 0 in such order. We have denoted as |0*j* and |1*j* to the eigenstates of *σj*, *j* = 1, 2, 3 (or *j* = *x*, *y*, *z*). Note the behavior commented in the previous paragraph.

Despite, the most interesting issue is centered in the fact that the entanglement resource |*χ* is normally unknown but with a tiny variation of |*β*0 through the deformation parameters *p*1, *p*2, *p*3. By calculating the average and the standard deviation of Δ*α*1,*α*2,*α*<sup>3</sup> *<sup>θ</sup>*,*<sup>φ</sup>* on the parameters *α*1, *α*2, *α*<sup>3</sup> ∈ [0, 1]:

$$\mu\_{\Delta\_{\theta,\phi}^{a\_1a\_2a\_3}} = \int\_0^1 \int\_0^1 \int\_0^1 \Delta\_{\theta,\phi}^{a\_1,a\_2a\_3} \text{d}a\_1 \text{d}a\_2 \text{d}a\_3} = 1 \to \mu\_{\mathcal{F}\_N} = 1 - Np\_{\text{ts}} \tag{31}$$

$$\begin{array}{rcl} \sigma\_{\Delta\_{\theta,\theta}^{a\_1 a\_2 a\_3}} & = & \sqrt{\mu\_{(\Delta\_{\theta,\theta}^{a\_1 a\_2 a\_3)})^2} - (\mu\_{\Delta\_{\theta,\theta}^{a\_1 a\_2 a\_3}})^2} \\ &=& \frac{1}{\sqrt{2}} \sqrt{53 + \sin^4(\theta)\cos(4\phi) + 4\cos(2\theta) + 7\cos(4\theta)} \in \left[\frac{1}{2}, \frac{1}{\sqrt{7}}\right] \end{array} \tag{32}$$

$$=\left.\frac{1}{8\sqrt{6}}\sqrt{53+\sin^4(\theta)\cos(4\phi)+4\cos(2\theta)+7\cos(4\theta)}\in[\frac{1}{3},\frac{1}{\sqrt{6}}]\tag{32}$$

$$
\sigma\_{\mathcal{F}\_{\mathcal{N}}} = Np\_{\mathfrak{k}\mathfrak{b}} \sigma\_{\Lambda^{a\_1}\_{\vartheta,\mathfrak{b}}e^{\mathfrak{a}\_{\mathfrak{b}}}}.\tag{33}
$$

We note that the average value of fidelity F*<sup>N</sup>* = 1 − *N p*ts becomes independent from the state being teleported. While, the dispersion for Δ*α*1,*α*2,*α*<sup>3</sup> *<sup>θ</sup>*,*<sup>φ</sup>* on the values *p*1, *p*2, *p*<sup>3</sup> depends from the teleported state and it becomes lowest for the eigenstates of *σ*1, *σ*2, *σ*3. In fact, the exact result for the case of *<sup>N</sup>* <sup>=</sup> 1 is precisely (30) with such value in (1): <sup>F</sup><sup>1</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>∑</sup><sup>3</sup> *<sup>j</sup>*=<sup>1</sup> *pj*(<sup>1</sup> <sup>−</sup> *<sup>n</sup>*<sup>2</sup> *<sup>j</sup>*), thus the values in (33) are scaled from it by a factor *N*. The reason is easily noticed, the *ρ*out in (27) obtained by linearization from (3) coincides with the sequential case (3) under linearization, so both cases exactly meet under the current limit. It implies that indefinite causal order procedure in teleportation becomes unpractical in this limit.

**Figure 9.** Bloch sphere showing under the assumption *pj* 1, *<sup>j</sup>* <sup>=</sup> 1, 2, 3 for each state: (**a**) <sup>Δ</sup>*α*1,*α*2,*α*<sup>3</sup> *θ*,*φ* in color obtained for each syndrome in (27), *σ*1*ρσ*1, *σ*2*ρσ*2, *σ*3*ρσ*<sup>3</sup> respectively, and (**b**) the standard deviation *<sup>σ</sup>*Δ*α*1,*α*2,*α*<sup>3</sup> *<sup>θ</sup>*,*<sup>φ</sup>* in (33). Red is the best fidelity in (**a**) and the lower dispersion in (**b**).

### *4.3. Notable Behavior on the Frontal Face of Parametric Region: Case p*<sup>0</sup> = 0

The behavior of F<sup>2</sup> on the frontal face (*p*<sup>0</sup> = 0) in Figure 6 can be now better advised in Figure 10. There, we have calculated numerically (for 10<sup>5</sup> states covering the frontal face), the best fidelity obtained using two teleportation channels under indefinite causal order by taking the optimal measurement on the control state together with the best state able to be teleported. Thus, it represents naively the best possible scenario.

In the last process, for each <sup>|</sup>*χ* on the frontal face, we have additionally taken a sample of 10<sup>2</sup> sets of values for *q*<sup>0</sup> ∈ [0, 1] (the initialization value for the control state for *N* = 2), *θ* ∈ [0, *π*], *φ* ∈ [0, 2*π*] for <sup>|</sup>*ψ*m and *<sup>θ</sup>*<sup>0</sup> <sup>∈</sup> [0, *<sup>π</sup>*], *<sup>φ</sup>*<sup>0</sup> <sup>∈</sup> [0, 2*π*] for the teleported state <sup>|</sup>*ψ* <sup>=</sup> cos *<sup>θ</sup>*<sup>0</sup> <sup>2</sup> <sup>|</sup>0 <sup>+</sup> sin *<sup>θ</sup>*<sup>0</sup> <sup>2</sup> *<sup>e</sup>iφ*<sup>0</sup> <sup>|</sup>1. Each value is used as initial condition to find a local maximum for the fidelity F2. Then, those values are used to predict the global maximum of F<sup>2</sup> for each point on the frontal face. Figure 10a shows the best fidelity on the face together with the statistical distribution of the fidelities on the frontal face in the upper image of Figure 10c, which suggests that F<sup>2</sup> = 1 could be obtained on the face always (the little dispersion with lower values of F<sup>2</sup> ∈ [0.9, 1] are due to the numerical procedure followed). The same follows for P*<sup>m</sup>* (Figure 10b,c lower) but denoting that such probabilities of success are centrally distributed around <sup>1</sup> <sup>2</sup> (note they are not the best probabilities because the process is centred

on maximize F2). As in Figure 6, images in Figure 10 appear blurred due to the limited number of points considered because the time processing.

**Figure 10.** Optimal fidelity using two teleportation channels in indefinite causal order followed by an appropriate measurement |*ϕm*. (**a**) The best fidelity obtained for certain teleported state if optimal control measurement is obtained, (**b**) the probability P*<sup>m</sup>* of success for the last process, and (**c**) the statistical distribution for F<sup>2</sup> and P*m*.

Nevertheless, the last fact is in reality a blind strategy. A more critical view of Formulas (18) and (19) and referring to [22] which numerically suggests that *q*<sup>0</sup> = sin<sup>2</sup> *<sup>θ</sup>* <sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> (<sup>1</sup> <sup>−</sup> cos *<sup>θ</sup>*), *<sup>φ</sup>* <sup>=</sup> 0 is related with the optimal case for the case *<sup>p</sup>* <sup>=</sup> *<sup>p</sup>*<sup>1</sup> <sup>=</sup> *<sup>p</sup>*<sup>2</sup> <sup>=</sup> *<sup>p</sup>*<sup>3</sup> <sup>=</sup> <sup>1</sup> <sup>3</sup> . In fact, in such case, last formulas become reduced to:

$$\mathcal{F}\_2 = \begin{array}{c} \frac{\sum\_{i,j=0}^3 p\_i p\_j \left( \text{Tr} \left( \rho \sigma\_i \sigma\_j \rho \sigma\_j \sigma\_i \right) + \text{Tr} \left( \rho \sigma\_i \sigma\_j \rho \sigma\_i \sigma\_j \right) \right)}{\sum\_{i,j=0}^3 p\_i p\_j \left( \text{Tr} \left( \sigma\_i \sigma\_j \rho \sigma\_j \sigma\_i \right) + \text{Tr} \left( \sigma\_i \sigma\_j \rho \sigma\_i \sigma\_j \right) \right)} \tag{34}$$

$$\mathcal{P}\_m = -\frac{\sin^2 \theta}{2} \sum\_{i,j=0}^3 p\_i p\_j \left( \text{Tr}(\sigma\_i \sigma\_j \rho \sigma\_j \sigma\_i) + \text{Tr}(\sigma\_i \sigma\_j \rho \sigma\_i \sigma\_j) \right). \tag{35}$$

Last formula explains the reason because the case *θ* = *<sup>π</sup>* <sup>2</sup> is optimal for P*m*. Moreover, on the frontal face *p*<sup>0</sup> = 0 (then *i*, *j* = 1, 2, 3), then (34) and (35) clearly become (by splitting the cases *i* = *j* from *i* = *j*, noting for the last case *σiσ<sup>j</sup>* = −*σjσ<sup>i</sup>* and the fact that we are dealing with pure states):

$$\begin{array}{rcl} \mathcal{F}\_2 & = & 1 \end{array} \tag{36}$$

$$\mathcal{P}\_{\mathfrak{m}} = \left. \sin^2 \theta \sum\_{i=1}^{\mathfrak{d}} p\_i^2 \right| \quad \text{with} : \sum\_{i=1}^{\mathfrak{d}} p\_i = 1. \tag{37}$$

Thus, the last conditions make the teleportation optimal not only for *p* = *p*<sup>1</sup> = *p*<sup>2</sup> = *p*<sup>3</sup> = <sup>1</sup> <sup>3</sup> but also for the entire cases on the frontal face, being independent from the teleported state. Nevertheless, the probability of success depends entirely from the values of *pi* (considering only the best case *θ* = *<sup>π</sup>* 2 ). Figure 11 shows the distribution of P*<sup>m</sup>* on the frontal face (in some cases we will denote this probability by <sup>P</sup>ff,{*pi*} *<sup>m</sup>*,*N*=<sup>2</sup> to state *<sup>θ</sup>* <sup>=</sup> *<sup>π</sup>* <sup>2</sup> , *p*<sup>0</sup> = 0 and *pi* arbitrary but fulfilling *p*<sup>1</sup> + *p*<sup>2</sup> + *p*<sup>3</sup> = 1), which ranges on [ 1 <sup>3</sup> , 1]. In fact, the case *<sup>p</sup>* <sup>=</sup> *<sup>p</sup>*<sup>1</sup> <sup>=</sup> *<sup>p</sup>*<sup>2</sup> <sup>=</sup> *<sup>p</sup>*<sup>3</sup> <sup>=</sup> <sup>1</sup> <sup>3</sup> corresponds to the worst case for P*<sup>m</sup>* in the center of the face. We have constructed the norm on the frontal face to report such distribution. The mean *<sup>μ</sup>*P*<sup>m</sup>* <sup>=</sup> <sup>1</sup> 2 and the standard deviation *σ*P*<sup>m</sup>* ≈ 0.13 were calculated using such distribution.

In order to solve the cases for *N* > 2 by including further teleportation channels under indefinite causal order, last analysis suggests for arbitrary *N* that the procurement of an analytical formula for (15) is in order at least for the case *p*<sup>0</sup> = 0, implying *t*<sup>0</sup> = 0:

$$\Lambda^{N}\left[\rho\otimes\rho\_{\mathfrak{c}}\right] \quad = \sum\_{k}\sum\_{k'}\sqrt{q\_{k}q\_{k'}}|k\rangle\langle k'|\sum\_{t\_{1}=0}^{N}\prod\_{t\_{2}=0}^{N-t\_{1}}\prod\_{j=1}^{3}p\_{j}^{t\_{j}}\otimes\tag{38}$$

$$\sum\_{p=1}^{N'}\pi\_{k}\left(\pi\_{k\_{p}^{t\_{1}t\_{2}t\_{3}}}\left(\sigma\_{1}^{t\_{1}}\sigma\_{2}^{t\_{2}t\_{3}}\right)\right)\rho\left(\pi\_{k'}\left(\pi\_{k\_{p}^{t\_{1}t\_{2}t\_{3}}}\left(\sigma\_{1}^{t\_{1}}\sigma\_{2}^{t\_{2}t\_{3}}\right)\right)\right)^{\dagger}$$

$$=\sum\_{k}\sum\_{k'}\sqrt{q\_{k}q\_{k'}}|k\rangle\langle k'|\sum\_{t\_{1}=0}^{N}\sum\_{t\_{2}=0}^{N-t\_{1}}\prod\_{j=1}^{3}p\_{j}^{t\_{j}}\otimes\sum\_{p=1}^{N'}\Sigma\_{k\_{p}}^{k}\Sigma\_{k\_{p}}^{k'}\left(\sigma\_{1}^{t\_{1}}\sigma\_{2}^{t\_{2}}\sigma\_{3}^{t\_{3}}\right)\rho\left(\sigma\_{1}^{t\_{1}}\sigma\_{2}^{t\_{2}}\sigma\_{3}^{t\_{3}}\right)^{\dagger}\tag{39}$$

and *t*<sup>3</sup> = *N* − *t*<sup>1</sup> − *t*2. As it was previously mentioned, factors generated by *π<sup>k</sup>* and *π<sup>k</sup>* are equal until a sign. In addition, they always evolve to *σ*0, *σ*1, *σ*<sup>2</sup> or *σ*<sup>3</sup> (easily depending on the parity of *t*1, *t*2, *t*3). Thus, those factors and their signs state the introduction of syndromes on *ρ* together with interference among them and the different orders. Such interference could be manipulated through the parameters *qk*, *pj*.

**Figure 11.** (**a**) Values of P*<sup>m</sup>* on *<sup>p</sup>*<sup>0</sup> = 0 face , and (**b**) its corresponding statistical distribution *<sup>ρ</sup>*P*<sup>m</sup>* for two teleportation channels in indefinite causal order.

Even so, this formula is not easy to address in order to get a simpler closed result because the sign Σ*<sup>k</sup> kp* , Σ*<sup>k</sup> kp* introduced in the permutation with respect *<sup>σ</sup>t*<sup>1</sup> <sup>1</sup> *<sup>σ</sup>t*<sup>2</sup> <sup>2</sup> *<sup>σ</sup>t*<sup>3</sup> <sup>3</sup> cannot be advised easily (see a parallel analysis in [8]). Nevertheless, we can still to analyse computationally the cases for the lowest values of *N* (analytical cases addressed by computer aided methods due to the factorial increasing number of terms). Thus, formulas for <sup>P</sup>ff,{*<sup>p</sup> i* } *<sup>m</sup>*,*<sup>N</sup>* and F for *N* larger than two have been obtained using a computational treatment. The formulas obtained in the analysis are reported in Appendix B. As in our previous discussion for the case *p*<sup>1</sup> = *p*<sup>2</sup> = *p*<sup>3</sup> = *p* in the Section 4.2.1, F = 1 is obtained for all cases on the frontal face if the measurement in the indefinite causal order becomes <sup>|</sup>*ϕ*<sup>+</sup> *<sup>m</sup>* for *N* = 2, 4 and |*ϕ*<sup>−</sup> *<sup>m</sup>* for *N* = 3 independently of the teleported state. Again, it is a consequence of the order interference due to the indefinite causal order together the post-selection induced by the measurement. For complementary cases using other measurement outcomes, we get F ≡ 1 depending from *p*1, *p*2, *p*<sup>3</sup> or still undefined, and additionally depending from the teleported state (see Appendix B).

#### **5. An Alternative Procedure Introducing Weak Measurement**

In spite of the previous outcomes, we guess the indefinite causal order could not work properly at any point inside of region depicted in the Figure 6. Nevertheless, due to the outcomes in [7] for the case *p* = *p*<sup>1</sup> = *p*<sup>2</sup> = *p*<sup>3</sup> and those exhibited in the Figure 6, the teleportation process assisted by indefinite causal order (at least for two channels) becomes optimal on *p*<sup>0</sup> = 0 and *p*<sup>0</sup> = 1 (the origin and the frontal face in Figure 6a). Then, we propose an alternative strategy beginning with a weak measurement on the entangled resource.

#### *5.1. General Case for N* = 2 *Assisted by a Weak Measurement*

By considering first the following weak measurements on |*χ*, we get the post-measurement states and their probabilities of occurrence:

$$\begin{array}{rcl} \text{Po} &=& |\not\!\beta \mathbf{o}\rangle \langle \not\!\mathbf{o} \mathbf{o}| \to |\chi \mathbf{o}\rangle = (\not\!\mathbf{o} | \chi\rangle)\_{\text{norm}} = |\not\!\mathbf{o} \mathbf{o}\rangle, & \mathbf{\tilde{p}o} = \not\!\mathbf{p} \mathbf{o} \end{array} \tag{40}$$

$$P\_1 \quad = \quad \mathbb{I} - P\_0 \to |\chi\_1\rangle = (P\_1|\chi\rangle)\_{\text{norm}} = \sum\_{i=1}^3 \sqrt{\frac{p\_i}{\tilde{p}\_1}} |\beta\_i\rangle \equiv \sum\_{i=1}^3 \sqrt{p\_i'} |\beta\_i\rangle, \quad \tilde{p}\_1 = \sum\_{i=1}^3 p\_i \tag{41}$$

which projects the entangled state on one of the two states |*χ*0 or |*χ*1 with probabilities *p*˜0 or *p*˜1 respectively. Each state is located on the origin or otherwise on the frontal face of the region in Figure 6. Then, if |*χ*0 is obtained, the teleportation process can go as in the Figure 1, otherwise, if |*χ*1 is obtained, we can try the teleportation assisted by indefinite causal order (at this point the reader could note that clearly, we need two entangled resources). We will come back to discuss the weak measurement strategy widely in the last section).

The entire process is depicted in the Figure 12, a schematic diagram of the process as it was originally proposed by [7]. Given certain state to teleport, we use certain entangled resource |*χa*. It goes through the weak measurement in (40) to get |*χa*0 = |*β*0 with probability *p*0. Then we perform a single teleportation. Instead, by obtaining |*χa*1 with probability 1 − *p*0, then we prepare a second entangled resource |*χb* repeating with it the same procedure, if after of the weak measurement |*χb*0 = |*β*0 is obtained with probability *p*0, we disregard |*χa*1 proceeding with a single teleportation using such state. Otherwise, if |*χb*1 is obtained, we perform a two-channel teleportation assisted by indefinite causal order using the states previously obtained. There, the teleportation will become successful with probability *p* <sup>2</sup> <sup>1</sup> <sup>+</sup> *<sup>p</sup>* <sup>2</sup> <sup>2</sup> <sup>+</sup> *<sup>p</sup>* <sup>2</sup> <sup>3</sup>, otherwise it becomes unsuccessful and we need disregard the process. Thus, the global probability of success is (there, <sup>P</sup>ff *<sup>m</sup>*,*N*=<sup>2</sup> corresponds to (37) with *θ* = *<sup>π</sup>* <sup>2</sup> , *φ* = 0, renaming *pi* as *p i* , with *p* <sup>1</sup> + *p* <sup>2</sup> + *p* <sup>3</sup> = 1):

$$\begin{aligned} \mathcal{P}\_{\text{Tot}} &= \quad p\_0 + (1 - p\_0)p\_0 + (1 - p\_0)^2 \mathcal{P}\_{\text{m,N=2}}^{\text{fit,\{p\_i'\}}} \\ &= \quad p\_0 + (1 - p\_0)p\_0 + (1 - p\_0)^2 \sum\_{i=1}^3 p\_i'^2 = 1 - 2(p\_1 p\_2 + p\_2 p\_3 + p\_3 p\_1) \end{aligned} \tag{42}$$

**Figure 12.** Schematic teleportation process assisted by weak measurement.

The last function has been represented in the plots of Figure 13. For each initial set (*p*1, *p*2, *p*3) of the entangled resources (assumed identical), PTot is plotted in color in agreement with the bar besides in the Figure 13a. One-third of the plot has shown, due to its symmetry, to exhibit its inner structure. The corresponding statistical distribution is obtained numerically in the Figure 13b by uniformly

sampling the space in the figure on the left. The mean value of PTot becomes 0.70 and their standard deviation 0.16.

**Figure 13.** Distribution of PTot: (**a**) as function of (*p*1, *p*2, *p*3), and (**b**) as statistical distribution by itself obtained numerically from the data of (**a**).
