**2. Teleportation Algorithm as a Quantum Channel and** *N***-Redundant Teleportation Problem**

#### *2.1. Quantum Teleportation as a Quantum Channel*

Traditional quantum teleportation algorithm developed originally in [10] has become a central procedure in quantum information theory. This process uses an entangled resource in the form of the Bell state <sup>|</sup>*β*00 <sup>=</sup> <sup>√</sup><sup>1</sup> 2 (|00 + |11). Experimentally, such an entangled state becomes difficult to create and to sustain. For this reason, it could arrive imperfect to the process. Thus, considering a general variation of this resource in the form of the general state <sup>|</sup>*χ* <sup>=</sup> <sup>∑</sup><sup>3</sup> *i*=0 <sup>√</sup>*pi* <sup>|</sup>*βi*, where <sup>|</sup>*βi* is a short notation for the Bell basis |*β*0 = |*β*00, |*β*1 = |*β*01, |*β*2 = |*β*11 and |*β*3 = |*β*10. The traditional teleportation algorithm running under this resource (instead the perfect case with *p*<sup>0</sup> = 1 and *p*<sup>1</sup> = *p*<sup>2</sup> = *p*<sup>3</sup> = 0) becomes a quantum channel whose output expression in terms of Kraus operators is given by [23]:

$$
\Lambda[\rho] = \sum\_{i=0}^{3} p\_i \hat{\sigma}\_i \rho \hat{\sigma}\_i^{\dagger} = \sum\_{i=0}^{3} p\_i \sigma\_i \rho \sigma\_i \tag{1}
$$

with !*σ<sup>i</sup>* <sup>=</sup> *<sup>σ</sup><sup>i</sup>* if *<sup>i</sup>* <sup>=</sup> 0, 1, 3 and !*σ*<sup>2</sup> <sup>=</sup> *<sup>i</sup>σ*2. *<sup>ρ</sup>* <sup>=</sup> <sup>|</sup>*ψ ψ*<sup>|</sup> is the state to teleportate (in the current work we will restrict the analysis to pure state cases, despite our outcomes can be extended to mixed states [8]). This formula, regarding teleportation algorithm as a communication channel will be discussed at the end of the article in terms of possible and current available experimental developments for its implementation. It means Kraus operators are *Ki* <sup>=</sup> <sup>√</sup>*piσi*. In the terms stated before, we are interested to assess the corresponding fidelity of the process as function of the *pi* values under several schemes. It has the form of a Pauli channel [24] and it has been recently studied to characterize its properties under indefinite causal order and measurement [8] exhibiting notable properties and symmetries of communication enhancement as function of the parameters *pi*. In the current approach, the set {*pi*|*i* = 0, 1, 2, 3} plays an additional role because it is associated with the quantum resource |*χ*.

In the current article, we will use the fidelity to measure the channel performance:

$$\mathcal{F}(\rho, \Lambda[\rho]) = \left[ \text{Tr}\left( \sqrt{\sqrt{\rho} \Lambda[\rho] \sqrt{\rho}} \right) \right]^2,\tag{2}$$

because we will restrict to the case when *<sup>ρ</sup>* is a pure state *<sup>ρ</sup>* <sup>=</sup> <sup>|</sup>*ψψ*|, then <sup>√</sup>*<sup>ρ</sup>* <sup>=</sup> *<sup>ρ</sup>*. Those facts still give the easier formula: F(*ρ*, Λ[*ρ*]) = *ψ*|Λ[*ρ*]|*ψ* = Tr(*ρ*Λ[*ρ*]). Then, in the following, we will express the fidelity briefly as F<sup>Λ</sup> ≡ F(*ρ*, Λ[*ρ*]).

#### *2.2. N-Redundant Quantum Teleportation*

In this section, we will study the effect on the fidelity of imperfect teleportation as it was previously depicted. For such reason, we first consider a set of identical and redundant *N* teleportation channels in a definite causal order as a composition of the depicted channel in (1). In addition, we consider for the sake of simplicity that each channel is identical to others in the redundant application:

$$\Lambda(\bigcirc\_{N} \Lambda)[\rho] \equiv \Lambda[\Lambda[\dots \Lambda[\rho] \dots \dots]] = \sum\_{i\_1, \dots, i\_n = 0}^{3} p\_{i\_1} \dotsm p\_{i\_n} \sigma\_{i\_N} \dotsm \sigma\_{i\_1} \rho \sigma\_{i\_1} \dotsm \sigma\_{i\_N}.\tag{3}$$

If *<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>p</sup>*, with 0 <sup>≤</sup> *<sup>p</sup>* <sup>≤</sup> <sup>1</sup> <sup>3</sup> for simplicity (to avoid the increasing parameters involved), we have gotten the expressions for the corresponding fidelity F*N*<sup>Λ</sup> ≡ Tr(*ρ*(*N*Λ)[*ρ*]) for the first five cases of redundant sequential applications of teleportation (assuming *ρ* is a pure state), getting:

$$\mathcal{F}\_{\bigtriangledown} = \|1 - 2p\|\tag{4}$$

$$\mathcal{F}\_{\square\_2 \Lambda} \quad = \ 1 - 4p + 8p^2 \tag{5}$$

$$\mathcal{F}\_{\square\_{3}\Lambda} = \begin{array}{c} 1 - 6p + 24p^2 - 32p^3 \end{array} \tag{6}$$

$$\mathcal{F}\_{\square\_4 \Lambda} \quad = \ 1 - 8p + 48p^2 - 128p^3 + 128p^4 \tag{7}$$

$$\mathcal{F}\_{\square \text{jk}\Lambda} = \left[1 - 10p + 80p^2 - 320p^3 + 640p^4 - 512p^5\right] \tag{8}$$

Interestingly, those outcomes are independent from the state to teleport (a consequence from the symmetric simplification *p*<sup>1</sup> = *p*<sup>2</sup> = *p*<sup>3</sup> = *p* and the algebraic properties of Pauli operators). Such cases can be computationally developed to get last outcomes (and other for larger cases). Figure 2 exhibits the behavior of such applications as function of *<sup>p</sup>*. The gray zone sets the middle point F1<sup>Λ</sup> <sup>=</sup> <sup>2</sup> <sup>3</sup> of fidelity F1<sup>Λ</sup> <sup>∈</sup> [ <sup>1</sup> <sup>3</sup> , 1] for the case *N* = 1 as a reference (as it was remarked in [7]). The single case *<sup>N</sup>* <sup>=</sup> 1 sets the expected outcome about the effect of *<sup>p</sup>* on F1<sup>Λ</sup> giving the worst value for *<sup>p</sup>* <sup>=</sup> <sup>1</sup> 3 . For *N* > 1, the outcome becomes as it could be expected, each application of a new teleportation worsens the output state teleported. Despite this, there are certain recoveries for *p* = <sup>1</sup> <sup>3</sup> , useful only for the lowest values of *<sup>N</sup>*. A convergent value <sup>F</sup>*N*→<sup>∞</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> appears (it corresponds to the behavior of total depolarization for the channel, *<sup>ρ</sup>*out <sup>≡</sup> (*N*Λ)[*ρ*] = *<sup>σ</sup>*<sup>0</sup> <sup>2</sup> ). The cases *<sup>p</sup>* <sup>=</sup> <sup>1</sup> <sup>4</sup> coincide for all *N* because for *N* = 1 the total depolarized state *<sup>σ</sup>*<sup>0</sup> <sup>2</sup> is obtained, then any further application of the teleportation cannot worsen the outcome.

**Figure 2.** Sequential fidelity as function of the number *N* of channels being applied, and *p* is the deformation strength in |*χ*.
