4.2.1. Case *p*<sup>1</sup> = *p*<sup>2</sup> = *p*<sup>3</sup> ≡ *p*

First, we will address with the case *p* = *p*<sup>1</sup> = *p*<sup>2</sup> = *p*<sup>3</sup> widely used in the literature for simplicity. In [7], it has been suggested that for |*ψm* having one of the following forms:

$$|\!\!\!\varphi\_m^{\pm}\rangle \equiv \frac{1}{\sqrt{N!}} \sum\_{i=0}^{N!-1} (\pm 1)^{\sigma(\pi\_i)} |i\rangle. \tag{25}$$

The teleportation fidelity becomes optimal. There, *σ* is the signature of the parity of each order |*i*. By considering (15) together with (25) and the control state with *qk* = <sup>1</sup> *<sup>N</sup>*!∀*k* = 0, 1, ..., *N*! − 1:

$$
\langle \varphi\_m^{\pm} | \Lambda^N \left[ \rho \otimes \rho\_{\vec{c}} \right] | \varphi\_m^{\pm} \rangle \quad \quad = \sum\_k \sum\_{k'} \frac{1}{N!^2} (\pm 1)^{\sigma(\pi\_k) + \sigma(\pi\_{k'})} \sum\_{t\_1}^N \sum\_{t\_2=0}^{N-t\_1} \sum\_{t\_3=0}^{t\_2-t\_1} \prod\_{j=0}^3 p\_j^{t\_j} . \tag{26}
$$

$$
\sum\_{p=1}^{N'} \pi\_k \left( \pi\_{k\_p^{t\_1 t\_2 t\_3}} \left( \sigma\_0^{t\_0} \sigma\_1^{t\_1} \sigma\_2^{t\_2} \sigma\_3^{t\_3} \right) \right) \rho \left( \pi\_{k'} \left( \pi\_{k\_p^{t\_1 t\_2 t\_3}} \left( \sigma\_0^{t\_0} \sigma\_1^{t\_1} \sigma\_2^{t\_2} \sigma\_3^{t\_3} \right) \right) \right)^{\dagger}.
$$

Then, we have developed the Formulas (14) and (16) with |*ψm* = |*ϕ*<sup>±</sup> *<sup>m</sup>* in (25) to get both F*<sup>N</sup>* and P*<sup>N</sup>* for *N* = 2, 3, 4. Those formulas have been plotted (they are not reported here because their complexity, despite they are included in the Appendix A), the outcomes are shown in Figure 7 showing that a perfect fidelity <sup>F</sup>*<sup>N</sup>* <sup>=</sup> 1 for *<sup>p</sup>* <sup>=</sup> <sup>1</sup> <sup>3</sup> is achieved when |*ϕ*<sup>±</sup> *<sup>m</sup>* meets with the same parity to *N* (*p* is indicated in the color-scale besides). Despite, for *p* = <sup>1</sup> <sup>3</sup> the success probabilities P*<sup>m</sup>* decrease while *N* increases. For |*ϕ*− and *N* = 4, we get P*<sup>m</sup>* = 0, thus F<sup>4</sup> becomes undefined in such a case. While *<sup>p</sup>* <sup>∈</sup> [0, <sup>1</sup> <sup>6</sup> ] the best election is the single teleportation channel, for *<sup>p</sup>* <sup>∈</sup> [ <sup>1</sup> 6 , 1 <sup>3</sup> ], the assistance of the causal order becomes an alternative to enhance the fidelity of teleportation, particularly with *N* = 2 channels.

**Figure 7.** Probability P*<sup>m</sup>* to obtain different values of fidelity F*<sup>N</sup>* when the measurement states |*ϕ*+ or |*ϕ*− are applied for cases (**a**) *N* = 2, (**b**) *N* = 3 and (**c**) *N* = 4. Color-scale bar depicts the respective value for *p* for *N* = 2, 3, 4.

Figure 8 again compares the fidelity F*<sup>N</sup>* versus *p* for both measurements with the corresponding sequential case showing the alternated optimization of F*<sup>N</sup>* as function of the parity of *N* and |*ϕ*<sup>±</sup> *<sup>m</sup>*. Despite, the outcomes in Figure 6 suggest analysing the behavior of F*<sup>N</sup>* for independent values of *p*1, *p*2, *p*3.

**Figure 8.** Comparison of fidelity obtained when the channels are applied sequentially (blue) and with indefinite causal order depending on the measurement state <sup>|</sup>*ϕ*<sup>+</sup> *<sup>m</sup>* (red) and |*ϕ*<sup>−</sup> *<sup>m</sup>* (green), for the cases (**a**) *N* = 2, (**b**) *N* = 3, and (**c**) *N* = 4 (in this last case, the fidelity becomes undefined for |*ϕ*<sup>−</sup> *<sup>m</sup>*).
