# Industry 4.0 Quantum Strategic Organizational Design Configurations. The Case of Two Qubits: One Reports to One

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## Abstract

**:**

## 1. Introduction

## 2. QSOD Circuit—Two-Qubit Organizational Design Configuration

## 3. Case Study

- Figure 2a describes the alignment state of agent B, $P(B=|0\rangle )$, for different values of conditioned alignment probability between agents A and B, $x=P(B=|1\rangle |A=|0\rangle ),y=P(B=|1\rangle |A=|1\rangle )\in [0,1]$, being the alignment probability of agent A$P(A=|0\rangle )$ = $1-P(A=|1\rangle )$ = 50%.
- Figure 2b describes the alignment state of agent B, $P(B=|0\rangle )$, for different values of conditioned alignment probability between agents A and B, $x=P(B=|1\rangle |A=|0\rangle ),y=P(B=|1\rangle |A=|1\rangle )\in [0,1]$, being the alignment probability of agent A$P(A=|0\rangle )$ = $1-P(A=|1\rangle )$ = 75%.
- Figure 2c describes the alignment state of agent B, $P(B=|0\rangle )$, for different values of conditioned alignment probability between agents A and B, $x=P(B=|1\rangle |A=|0\rangle ),y=P(B=|1\rangle |A=|1\rangle )\in [0,1]$, being the alignment probability of agent A$P(A=|0\rangle )$ = $1-P(A=|1\rangle )$ = 90%.
- Figure 2d describes the alignment state of agent B, $P(B=|0\rangle )$, for different values of conditioned alignment probability between agents A and B, $x=P(B=|1\rangle |A=|0\rangle ),y=P(B=|1\rangle |A=|1\rangle )\in [0,1]$, being the alignment probability of agent A$P(A=|0\rangle )$ = $1-P(A=|1\rangle )$ = 99%.
- Figure 2e describes the alignment state of agent B, $P(B=|0\rangle )$, for different values of conditioned alignment probability between agents A and B, $x=P(B=|1\rangle |A=|0\rangle ),y=P(B=|1\rangle |A=|1\rangle )\in [0,1]$, being the alignment probability of agent A$P(A=|0\rangle )$=$1-P(A=|1\rangle )$ = 99.99%.

## 4. Discussion

**R**systematically. We will begin by discussing Figure 3, which describes the change in the alignment probability of the agent B described by the function $f(x,y)=P(B=|0\rangle )$, with increasing values of the relative alignment probability of B, depending on A, given by $y=P(B=|1\rangle |A=|1\rangle )\in [0,1]$. Before we do so, a gentle reminder for the reader that taking into consideration Bayes’s theorem, this probability can be expressed with Equation (5):

**R1**. In general, we can say that the probability of alignment of agent B, $f(x,y)=P(B=|0\rangle )$, oscillates consistently around the value 0.5 as a harmonic underdamped oscillator for different values of $z=P(A=|1\rangle )$, which is the equilibrium state of the system. This is plausible.

**R2**. At the scale represented in Figure 3a we observe that the angular frequency of this oscillator changes for different values of $y=P(B=|1\rangle |A=|1\rangle )$ and therefore we can separate the behavior of the function in three different regions, marked in Figure 3a, and depicted in Figure 3b in detail.

**R3**. For values of $y=P(B=|1\rangle |A=|1\rangle )\in (0.2,1]$, the probability of alignment of agent B, $f(x,y)=P(B=|0\rangle )$, oscillates consistently around 0.5 with a minimal amplitude for all values of $z=P(A=|1\rangle )$.

**R4**. For values of $y=P(B=|1\rangle |A=|1\rangle )\in (0.1,0.2]$, the probability of alignment of agent B, $f(x,y)=P(B=|0\rangle )$, oscillates consistently around 0.5 with an exponential decay that consistently increases with $1-z=P(A=|0\rangle )$.

**R5**. For values of $y=P(B=|1\rangle |A=|1\rangle )\in [0,0.1]$, the probability of alignment of agent B, $f(x,y)=P(B=|0\rangle )$, oscillates consistently around 0.5. The oscillation presents an exponential decay for $1-z=P(A=|0\rangle )\in [0.5,0.9)$, and presents no decay for values of $1-z=P(A=|0\rangle )>0.9$.

**R6**. However, the most striking observation of all is that if we increase the mapping of the circuits by a factor of 10, as shown in Figure 4b in which we make a mapping of the $y=P(B=|1\rangle |A=|1\rangle )\in [0,0.1]$ with [0.01] intervals for $P(A=|1\rangle =0.1$, we observe the same behavior as with the mapping of the $y=P(B=|1\rangle |A=|1\rangle )\in [0,1]$ with [0.1] intervals for $P(A=|1\rangle =0.1$ shown in Figure 4a. The results

**R1**–

**R5**are valid for both mapping intervals. The parameters (exponential decay, displacement, amplitude, and phase) of the signals represented in Figure 4 are very similar. Only the frequency is inversely proportional to the mapping interval, hence scaling the oscillation shape, and suggesting that the two graphics depict a similar process. This means that the behavior of this system is fractal. This has powerful implications, which we discuss in the conclusions.

## 5. Conclusions, Limitations and Further Steps

**R1**. In fact, the probability of alignment of the original agent always oscillates around the random state.

**R2, R3, R4**, and

**R5**we also observe that there is an exchange of energy between the original agent and the added agent, so that the alignment probability of the original agent can be positively influenced for low levels of intersection between the alignment probability of the original agent and the added agent. This could be interpreted to mean that the original agent can benefit from the presence of its new partner as long as the new partner provides process information. In other words, if the added agent is able to explain some of the variability in the value creation process that could not previously be explained by the original agent, then the asymptotic stability probability of the original agent will increase. The immediate consequence of this reading is that to add hierarchy levels to strategic design models of organizations, it is necessary to ensure the asymptotic stability of the lower agents before implementing a stable aggregation. It is important to highlight at this point that, in the context of QSOD, a hierarchy does not only describe the rather classical hierarchical relationship between agents but rather a reporting relationship. This concept is more inclusive as it includes the relationship between an agent and his/her boss, but also the relationship between an agent and his/her customer, or the relationship between an agent and a supplier. Our work helps, therefore, model interactions between organizational process owners and these interactions can potentially be scaled to any organizational context, including small and medium enterprises. This finding is very powerful for industry leaders, as well as for Strategic Organizational Design scholars because it imposes a severe constraint to ensure a sustainable and stable growth of Industry 4.0 organizations.

**R6**are profound and reveal the essence of what some call the fractal organization. The interaction of two process owners reveals an energy interchange that oscillates with more or less amplitude depending on certain parameters—the conditional alignment probabilities. However, what is really striking is that, independently of the granularity in which these parameters are observed, the oscillation always follows the same pattern. Such pattern is expressed by the results

**R1**–

**R5**and represents the cornerstone of the bilateral interaction under study. Fractality in this QSOD context allows for a quantification of these complex dynamics and its pervasive effect offers robustness and resilience to the two-qubit interaction.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

QSOD | Quantum Strategic Organizational Design |

KPI | Key Performance Indicators |

## References

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**Figure 1.**(

**a**) Case study framework, (

**b**) quantum simulation of strategic organizational design (QSOD)—Bloch sphere.

**Figure 2.**Results obtained for $P(B=|0\rangle )$ for different values of the no–alignment probability of agent $A,z=P(A=|1\rangle )$. (

**a**) $P(A=|1\rangle )=0.50$. (

**b**) $P(A=|1\rangle )=0.25$ (

**c**) $P(A=|1\rangle )=0.1$ (

**d**) $P(A=|1\rangle )=0.01$ (

**e**) $P(A=|1\rangle )=0.0001$.

**Figure 3.**Summary of results of $P(B=|0\rangle )$. (

**a**) Summary of results of $P(B=|0\rangle )$ as a function of $y=P(B=|1\rangle |A=|1\rangle )$. (

**b**) Detail of results of $P(B=|0\rangle )$ with $y=P(B=|1\rangle |A=|1\rangle )\in [0,0.3]$.

**Figure 4.**Detail of results of $P(B=|0\rangle )$ for $P(A=|1\rangle =0.1$. (

**a**) Detail of results of $P(B=|0\rangle )$ with $y=P(B=|1\rangle |A=|1\rangle )\in [0,1]$. (

**b**) Detail of results of $P(B=|0\rangle )$ with $y=P(B=|1\rangle |A=|1\rangle )\in [0,0.1]$.

Qubit | Interpretation | Equation |
---|---|---|

$|{\Psi}_{A}\rangle $ | The conditional probability $z=P(A=|1\rangle )$ of qubit $|{\Psi}_{A}\rangle $ to be in no–alignment translates into the rotation angle ${\theta}_{z}$. | ${\theta}_{z}=2arctan\sqrt{\frac{z}{1-z}}$ |

$|{\Psi}_{B}\rangle $ | The conditional probability $x=P(B=|1\rangle |A=|0\rangle )$ of qubit $|{\Psi}_{B}\rangle $ to be in no–alignment depending on qubit $|{\Psi}_{A}\rangle $ translates into rotation angle ${\theta}_{x}$. | ${\theta}_{x}=2arctan\sqrt{\frac{x}{1-x}}$ |

The conditional probability $y=P(B=|1\rangle |A=|1\rangle )$ of qubit $|{\Psi}_{B}\rangle $ to be in no–alignment depending on qubit $|{\Psi}_{A}\rangle $ translates into rotation angle ${\theta}_{y}$. | ${\theta}_{y}=2arctan\sqrt{\frac{y}{1-y}}$ |

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**MDPI and ACS Style**

Villalba-Diez, J.; Benito, R.M.; Losada, J.C. Industry 4.0 Quantum Strategic Organizational Design Configurations. The Case of Two Qubits: One Reports to One. *Sensors* **2020**, *20*, 6977.
https://doi.org/10.3390/s20236977

**AMA Style**

Villalba-Diez J, Benito RM, Losada JC. Industry 4.0 Quantum Strategic Organizational Design Configurations. The Case of Two Qubits: One Reports to One. *Sensors*. 2020; 20(23):6977.
https://doi.org/10.3390/s20236977

**Chicago/Turabian Style**

Villalba-Diez, Javier, Rosa María Benito, and Juan Carlos Losada. 2020. "Industry 4.0 Quantum Strategic Organizational Design Configurations. The Case of Two Qubits: One Reports to One" *Sensors* 20, no. 23: 6977.
https://doi.org/10.3390/s20236977