**3. Quantum Teleportation Assisted by Indefinite Causal Order with** *N* **Channels**

In this section, we will consider a generalization of some variants of the process under indefinite causal order as they are presented in [7,22] by considering *N* channels in a superposition of causal orders. By applying *N* channels in a superposition of causal orders, we could have *N*! combinations with different orders. Thus, we will need a control state with such number of dimensions (|0 sets for the normal sequential order of gates *T*1, *T*2, ..., *TN*) to rule the application of each causal order:

$$\rho\_{\mathcal{E}} = \left(\sum\_{i=0}^{N!-1} \sqrt{q\_i}|i\rangle\_{\mathcal{E}}\right) \left(\sum\_{j=0}^{N!-1} \sqrt{q\_j}|j|\_{\mathcal{E}}\right) = \sum\_{i,j=0}^{N!-1} \sqrt{q\_i q\_j}|i\rangle\_{\mathcal{E}} \langle j|.\tag{9}$$

For a definite causal order of teleportation channels *Ti*<sup>1</sup> , *Ti*<sup>2</sup> , ..., *TiN* given by the element *π<sup>k</sup>* ∈ Σ*<sup>N</sup>* in the symmetric group of permutations Σ*<sup>N</sup>* from the ordered case, it has the effect:

$$
\pi\_k = \begin{pmatrix} T\_{\dot{i}\_1} & T\_{\dot{i}\_2} & \cdots & T\_{\dot{i}\_N} \\ T\_{\dot{i}\_{\dot{j}\_1}} & T\_{\dot{i}\_{\dot{j}\_2}} & \cdots & T\_{\dot{i}\_{\dot{j}\_N}} \end{pmatrix} \to \pi\_k(K\_{\dot{i}\_1}K\_{\dot{i}\_2}\cdots K\_{\dot{i}\_N}) = K\_{\dot{i}\_{\dot{j}\_1}}K\_{\dot{i}\_{\dot{j}\_2}}\cdots K\_{\dot{i}\_{\dot{j}\_N}},\tag{10}
$$

and symbolically corresponding to the control state |*kc*. Then, the corresponding Kraus operators *Wi*1,*i*2,...,*iN* are:

$$\mathcal{W}\_{i\_1, i\_2, \dots, i\_N} = \sum\_{k=0}^{N!-1} \pi\_k(K\_{i\_1} K\_{i\_2} \dots K\_{i\_N}) \otimes |k\rangle\_c \langle k|\_r \tag{11}$$

where in the following, we will drop the tensor product symbol ⊗ in the sake of simplicity. Thus, the output for *N*-channels in superposition is given by:

<sup>Λ</sup>*<sup>N</sup>* [*<sup>ρ</sup>* <sup>⊗</sup> *<sup>ρ</sup>c*] <sup>=</sup> <sup>∑</sup>*i*1,*i*2,...,*iN Wi*1,*i*2,...,*iN ρ* ⊗ *ρ<sup>c</sup> Wi*1,*i*2,...,*iN* † (12) = ∑*i*1,*i*2,...,*iN* ∑ *k πk Ki*1*Ki*<sup>2</sup> ... *KiN* |*kk*| *ρ* ⊗ *ρ<sup>c</sup>* ∑ *k π<sup>k</sup> Ki*1*Ki*<sup>2</sup> ... *KiN* |*k k* | † = ∑*i*1,*i*2,...,*iN pi*<sup>1</sup> ··· *piN* ∑ *k πk σi*<sup>1</sup> ··· *σiN* |*kk*| *ρ* ⊗ *ρ<sup>c</sup>* ∑ *k π*† *k σi*<sup>1</sup> ··· *σiN* |*k k* | (13) = ∑*i*1,*i*2,...,*iN k*,*k pi*<sup>1</sup> ··· *piN* <sup>√</sup>*qkqk*|*k<sup>k</sup>* | ⊗ *π<sup>k</sup> σi*<sup>1</sup> ··· *σiN ρπ*† *k σi*<sup>1</sup> ··· *σiN* .

Still, we can use the last formula to reach a simpler expression using combinatorics and then the properties of Pauli operators. In fact, noting that the sum in (14) includes all different values given to each *i*1, *i*2, ... , *iN*, after they are permuted as distinguishable objects by *π<sup>k</sup>* and *π<sup>k</sup>* , it can be transformed into:

$$\sum\_{i\_1=0}^{3} \sum\_{i\_2=0}^{3} \dots \sum\_{i\_N=0}^{3} \longrightarrow \sum\_{t\_1=0}^{N} \sum\_{t\_2=0}^{N-t\_1N-t\_1-t\_2} \sum\_{p=1}^{N'} \dots \tag{14}$$

where *tj* is the number of scripts in *i*1, *i*2, ..., *iN* equal to *j* = 0, 1, 2, 3 (*t*<sup>0</sup> = *N* − *t*<sup>1</sup> − *t*<sup>2</sup> − *t*3). Sum over *p* runs on the distinguishable arrangements obtained with a fix number *tj* of operators *σ<sup>j</sup>* departing from *σt*<sup>0</sup> <sup>0</sup> *<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> by means of a certain permutation *<sup>π</sup><sup>k</sup> t* <sup>1</sup>,*t*2,*t*<sup>3</sup> *<sup>p</sup>* . Then, the permutations among identical operators in each one of the four types *<sup>σ</sup>*0, *<sup>σ</sup>*1, *<sup>σ</sup>*2, *<sup>σ</sup>*<sup>3</sup> are indistinguishable. There, *<sup>N</sup>* <sup>=</sup> *<sup>N</sup>*! *<sup>t</sup>*0!*t*1!*t*2!*t*3! . In such case, Formula (14) can be written as:

$$\begin{split} \Lambda^{N}\left[\rho\otimes\rho\_{\varepsilon}\right] &= \sum\_{k}\sum\_{k'}\sqrt{q\_{k}q\_{k'}}|k\rangle\langle k'|\sum\_{t\_{1}=0}^{N}\sum\_{t\_{2}=0}^{t\_{1}N-t\_{1}}\sum\_{t\_{3}=0}^{t\_{1}-t\_{2}}\prod\_{j=0}^{3}p\_{j}^{t\_{j}}\otimes \\ &\sum\_{p=1}^{N'}\pi\_{k}\left(\pi\_{k\_{p}^{t\_{1}t\_{2}t\_{3}}}\left(\sigma\_{0}^{t\_{0}}\sigma\_{1}^{t\_{1}}\sigma\_{2}^{t\_{2}}\sigma\_{3}^{t\_{3}}\right)\right)\rho\left(\pi\_{k'}\left(\pi\_{k\_{p}^{t\_{1}t\_{2}t\_{3}}}\left(\sigma\_{0}^{t\_{0}}\sigma\_{1}^{t\_{1}}\sigma\_{2}^{t\_{2}}\sigma\_{3}^{t\_{3}}\right)\right)\right)^{\dagger}, \end{split} \tag{15}$$

providing an easier formula for <sup>Λ</sup>*<sup>N</sup>* [*<sup>ρ</sup>* <sup>⊗</sup> *<sup>ρ</sup>c*] in terms of a definite number of sums and with the teleported state separated from the control state. From the properties of Pauli operators algebra, it is clear that both permutation terms besides *ρ* in (15) becomes equal until a sign. In addition, each one becomes in the set {*σj*|*j* = 0, 1, 2, 3}. Thus, (15) becomes a mixed state obtained as a linear combination of syndromes *σjρσj*, *j* = 0, 1, 2, 3 and normally entangled with the control state.

Following to [7], then we select an adequate basis to perform a measurement on the control state: B = {|*ψMi* |*i* = 1, 2, ..., *N*!}. Such a measurement post-selects the original symmetry of the teleported state mixed with the control and the imperfect entangled state. In such a basis, we hope to find a privileged state |*ψm*∈B to stochastically maximize the fidelity with probability P*<sup>m</sup>* (assuming *ρ* is a pure state). P*<sup>m</sup>* sets the probability of success of the process. If the measurement of control does not

conduct to |*ψm*, then other undesired teleportation outcome will be obtained. Then, if the desired outcome is not obtained, we disregard the output state. The fidelity and the success probability are:

$$\mathcal{F}\_N = \frac{\text{Tr}(\rho \langle \psi\_m | \Lambda^N \left[ \rho \otimes \rho\_c \right] | \psi\_m \rangle)}{\mathcal{P}\_m} \tag{16}$$

$$\mathcal{P}\_m = \text{Tr}(\langle \psi\_m | \Lambda^N \left[ \rho \otimes \rho\_c \right] | \psi\_m \rangle). \tag{17}$$

The process is depicted by Figure 3, where *N*! causal orders are considered to arrive to the pictorial representation of a complete superposition of causal orders on the right. Each causal order corresponds to one definite order in the application of channels *Ti* ruled by the control state *ρ<sup>c</sup>* above it.

**Figure 3.** *N*! causal order combinations for *N* identical teleportation channels *Ti*, *i* = 1, 2, ..., *N* finally conforming a superposition of it. Each one is ruled by the control state above.

#### **4. Analysis of Quantum Teleportation Assisted by the First Indefinite Causal Orders**

In the following section, we deal with the analysis for the increasing number of teleportation channels after to remark some outcomes for the case *N* = 2 guiding the further analysis.

#### *4.1. Teleportation with N* = 2 *Teleportation Channels in an Indefinite Causal Order Superposition*

For the case *N* = 2, it has been obtained in [22] that (16) reduces to:

$$\mathcal{F}\_2 = \frac{\sum\_{i,j=0}^3 p\_i p\_j \left( \left( \frac{1}{2} + (q\_0 - \frac{1}{2}) \cos \theta \right) \text{Tr} (\rho \tau\_i \sigma\_j \rho \sigma\_i \tau\_l) + \sqrt{q\_0 q\_1} \sin \theta \cos \phi \text{Tr} (\rho \tau\_i \sigma\_j \rho \sigma\_i \tau\_l) \right)}{\sum\_{i,j=0}^3 p\_i p\_j \left( \left( \frac{1}{2} + (q\_0 - \frac{1}{2}) \cos \theta \right) \text{Tr} (\sigma\_i \sigma\_j \rho \sigma\_j \tau\_l) + \sqrt{q\_0 q\_1} \sin \theta \cos \phi \text{Tr} (\sigma\_i \sigma\_j \rho \sigma\_i \tau\_j) \right)} , \tag{18}$$

then, a measurement on the control is made on the basis <sup>B</sup> <sup>=</sup> {|*ψm* <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, *ψ*⊥ *m* = sin *<sup>θ</sup>* <sup>2</sup> <sup>|</sup>0 <sup>−</sup> cos *<sup>θ</sup>* <sup>2</sup> *<sup>e</sup>*−*i<sup>φ</sup>* <sup>|</sup>1}, being <sup>|</sup>*ψm* the supposed state maximizing <sup>F</sup>2. The corresponding probability to get that outcome becomes:

$$\mathcal{P}\_m = \sum\_{i,j=0}^3 p\_i p\_j \left( (\frac{1}{2} + (q\_0 - \frac{1}{2}) \cos \theta) \text{Tr}(\sigma\_i \sigma\_j \rho \sigma\_j \sigma\_i) + \sqrt{q\_0 q\_1} \sin \theta \cos \phi \text{Tr}(\sigma\_i \sigma\_j \rho \sigma\_i \sigma\_j) \right). \tag{19}$$

Last formulas, (18) and (19) become reduced for pure states *ρ* = |*ψ ψ*| , |*ψ* = *α*|0 + *β*|1 and *p*<sup>0</sup> = 1 − 3*p*, *p*<sup>1</sup> = *p*<sup>2</sup> = *p*<sup>3</sup> = *p* by considering the identities:

$$\sum\_{i,j=0}^{3} p\_i p\_j \text{Tr}(\rho \sigma\_i v\_j \rho \sigma\_j v\_i) \quad = \quad 1 - 4p + 8p^2 \tag{20}$$

$$\sum\_{i,j=0}^{3} p\_i p\_j \text{Tr}(\sigma\_i \sigma\_j \rho \sigma\_j \sigma\_i) \quad = \quad 1 \tag{21}$$

$$\sum\_{i,j=0}^{3} p\_i p\_j \text{Tr}(\rho \sigma\_i \sigma\_j \rho \sigma\_i \sigma\_j) \quad = \quad (1-2p)^2 \tag{22}$$

$$\sum\_{i,j=0}^{3} p\_i p\_j \text{Tr}(\sigma\_i \sigma\_j \rho \sigma\_i \sigma\_j) \quad = \quad 1-12p^2. \tag{23}$$

Note that the combination of the two first formulas gives the sequential case in (5). The other two terms correspond to the interference terms. First and third formulas can be demonstrated noting that:

$$\begin{aligned} \rho &=& \frac{1}{2} (\sigma\_0 + \hbar \cdot \vec{\sigma}) \\ \text{with } : \quad \hbar &= (|a|^2 - |\beta|^2, a\beta^\* + a^\*\beta, i(a\beta^\* - a^\*\beta)), \\ \vec{\sigma} &= (\sigma\_1, \sigma\_2, \sigma\_3). \end{aligned} \tag{24}$$

This fact is not exclusive of the case *N* = 2. Due to the Pauli operators algebra and the regarding they are traceless (while, Tr(*σ*0) = 2), introducing (24) in (16) and (17), we note for P*<sup>m</sup>* that only the terms containing *σ*<sup>0</sup> become different from zero. For F*N*, only the quadratic terms in *σ*<sup>0</sup> and *n*ˆ ·*σ* become different from zero. For the terms quadratic in *n*ˆ ·*σ*, the additional condition *pi* = *pj*∀*i* = *j*(*i*, *j* = 0) is required in order to reduce the terms containing *σασβ* to the magnitude of *n*ˆ, thus removing all reference of the teleported state.

In [7], it has been demonstrated that for <sup>|</sup>*ψm* <sup>=</sup> <sup>|</sup>+ the worst deformed state <sup>|</sup>*χ* with *<sup>p</sup>* <sup>=</sup> <sup>1</sup> <sup>3</sup> still lets a perfect teleportation with probability <sup>P</sup>*<sup>m</sup>* <sup>=</sup> <sup>1</sup> <sup>3</sup> . In fact, Figure 4 summarizes the findings for the fidelity considering the two families of measurements with |− (dashed orange lines) and |+ (dashed blue lines). The sequential case with *N* = 2 is reported as a continuous line black together with the single teleportation channel *N* = 1 (continuous red line). Dashed blue and orange lines go folded from *q*<sup>0</sup> = 0, 1 (two channels in definite causal order) nearest to the two sequential channels case in black to the outermost lines for *q*<sup>0</sup> = <sup>1</sup> <sup>2</sup> (the evenly distributed control state) reaching <sup>F</sup> <sup>=</sup> 1 in *<sup>p</sup>* <sup>=</sup> 0, <sup>1</sup> <sup>3</sup> (blue for <sup>|</sup>*ψm* <sup>=</sup> <sup>|</sup>+) and <sup>F</sup> <sup>=</sup> <sup>1</sup> <sup>3</sup> , ∀*p* (orange for |*ψm* = |−).

For the case *N* = 2, [22] has shown that for different values of *q*<sup>0</sup> = <sup>1</sup> <sup>2</sup> , other measurements <sup>|</sup>*ψm* <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 are possible in order to achieve <sup>F</sup> <sup>=</sup> 1 when *<sup>p</sup>* <sup>=</sup> <sup>1</sup> <sup>3</sup> giving *φ* = 0 and *θ* distributed as in the Figure 5 as function of *q*0. Thus, the best fidelities F<sup>2</sup> depend entirely from *p* (see the color-scale besides in Figure 5) but the corresponding values of <sup>P</sup>*<sup>m</sup>* go down far from *<sup>q</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup> 2 (*θ* = *<sup>π</sup>* <sup>2</sup> ). The red dotted line is the threshold setting the minimum fidelity reached in the optimal case for *p* = <sup>3</sup><sup>−</sup> √3 <sup>6</sup> , <sup>F</sup><sup>2</sup> <sup>=</sup> <sup>√</sup><sup>1</sup> <sup>3</sup> [22]. Thus, we conclude that for *<sup>p</sup>* <sup>=</sup> *<sup>p</sup>*<sup>1</sup> <sup>=</sup> *<sup>p</sup>*<sup>2</sup> <sup>=</sup> *<sup>p</sup>*3, the best state for the control is *q*<sup>0</sup> = <sup>1</sup> <sup>2</sup> in order to maximize <sup>P</sup>*m*, despite only for *<sup>p</sup>* <sup>=</sup> <sup>1</sup> <sup>3</sup> and *p* → 0 it is possible to approach F<sup>2</sup> → 1. The last outstanding outcome for *p* = <sup>1</sup> <sup>3</sup> is a consequence of the two-folded interference introduced by the indefinite causal order together with the post-selection induced by the measurement which filters only constructive interference among the terms belonging to the original state.

**Figure 4.** Fidelity for the case of two channels in indefinite causal order as function of *p*. The blue dashed upper line corresponds to <sup>|</sup>*ψm* <sup>=</sup> <sup>|</sup>+ and *<sup>q</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> reaching <sup>F</sup> <sup>=</sup> 1 in *<sup>p</sup>* <sup>=</sup> <sup>1</sup> 3 .

**Figure 5.** Condensed outcomes for the case *N* = 2. The respective probability P*<sup>m</sup>* of measurements are included as function of *<sup>q</sup>*<sup>0</sup> and *<sup>θ</sup>* in <sup>|</sup>*ψm* <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 (*<sup>φ</sup>* <sup>=</sup> 0 in the optimal measurement). Fidelity depends entirely from *<sup>p</sup>*, and <sup>P</sup>*<sup>m</sup>* goes down while *<sup>p</sup>* <sup>→</sup> <sup>1</sup> 3 .

Fidelity (18) can be still analysed for independent values of *p*1, *p*2, *p*3. Figure 6 shows a numerical analysis to search the best possible fidelity (achieved for certain teleported state) max|*ψm*,*q*<sup>0</sup> (F2) for all possible |*ψm* and 0 ≤ *q*<sup>0</sup> ≤ 1. The value of fidelity F<sup>2</sup> is represented in color in agreement with the color-scale bar besides. Figure 6a shows a cut from the entire plot showing the inner core where fidelity goes down (three parts are symmetric). The higher values of fidelity on the faces of polyhedron suggest that better solutions can be reached for other cases with unequal values of *pi*, *i* = 1, 2, 3, particularly for the frontal face *p*<sup>0</sup> = 0 completely colored in blue in Figure 6. The case *p*<sup>1</sup> = *p*<sup>2</sup> = *p*<sup>3</sup> ≡ *p* falls in the central red dashed division crossing the clearer core reflecting the outcome in Figure 4, where not good values of <sup>F</sup><sup>2</sup> are inevitably obtained far from *<sup>p</sup>* <sup>=</sup> 0 and *<sup>p</sup>* <sup>=</sup> <sup>1</sup> <sup>3</sup> . In addition, complementary information for such cases is given by P*<sup>m</sup>* in Figure 6b, the probability to reach the corresponding higher fidelity in each process assisted by an intermediate optimal measurement on the control qubit. The plot depicts disperse outcomes barely around of P*<sup>m</sup>* ≈ 0.5. Note that the computer process to obtain Figure 6a,b requires optimization on lots of parameters, thus requiring a considerable time of processing. The region (*p*1, *p*2, *p*3) was divided in 10<sup>7</sup> points to perform such optimization. After, each point is reported as a colored sphere to fill the space in order to give a representation in

color about the continuity of F<sup>2</sup> and P*m*. Such an approach gives a certain impression of blurring in the figures, but they are reported with the best precision available under numerical processing. Particularly, Figure 6b is a collage of colored dots due to P*<sup>m</sup>* is reported on an average basis, due the optimization was made on F<sup>2</sup> on the left. By performing a numerical statistics of our outcomes for each P*m*, we get an approximation to its statistical distribution *ρ*P*<sup>m</sup>* included in the upper inset in Figure 6b. This distribution shows symmetric behavior around of P*<sup>m</sup>* = 0.5 as it could be expected for the numerical optimization.

**Figure 6.** (**a**) Best fidelity F<sup>2</sup> for the two-channels case as function of *p*1, *p*2, *p*3. Each point inside the polyhedron corresponds to their acceptable values and it is coloured in agreement with its fidelity value (see the color-scale besides); the cut of polyhedron region exhibits the inner structure; (**b**) The corresponding values for measurement probabilities P*<sup>m</sup>* denoting disperse values around 0.5. The upper inset confirms the statistical distribution *<sup>ρ</sup>*P*<sup>m</sup>* exhibiting symmetry around P*<sup>m</sup>* = 0.5.

#### *4.2. Teleportation with an Increasing Number of Teleportation Channels in an iNdefinite Causal Order Superposition*

Formula (15) exhibits the superposition of terms finally involving the states *ρ*, *σ*1*ρσ*1, *σ*2*ρσ*<sup>2</sup> and *σ*3*ρσ*<sup>3</sup> while they become entangled with the control state *ρc*. In the next sections, we deal with two cases of interest for the use of the teleportation algorithm under indefinite causal order.
