1. Introduction

2. Background

3. Methods

4. Results

5. Quantum Open Systems

5.1. Quantum Open Systems Equation Components and Explanations

The quantum open system equation has a long history outside of its application in social and information science applications. The quantum open system is an extension of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation, often shortened to the Lindblad equation [77,78]. The quantum open system is composed of a number of different parts. The equations provided in (40) through (43) are found in [14] where it was used to model temporal judgment and constructed preference.

$$\frac{d}{dt}\rho \left(t\right)=-i·\left(1-\alpha \right)·\left[H,\rho \right]+\alpha ·\mathcal{L}\left(\rho \right)$$

(40)

$$\mathcal{L}\left(\rho \right)=\sum {\gamma }_{ij}·\left(L_{ij}·\rho ·L_{ij}^{†}-\frac{1}{2}\left\{L_{ij}^{†}·L_{ij,},\rho \right\}\right)$$

(41)

$$\left\{\left(L_{ij}^{†}·L_{ij,}\right),\rho \right\}=\left(L_{ij}^{†}·L_{ij,}\right)·\rho +\rho ·\left(L_{ij}^{†}·L_{ij,}\right)$$

(42)

$$\left[H,\rho \right]=H·\rho -\rho ·H$$

(43)

The open system in equation (40) is used to describe cognition and human decision making with various applications [14,74,79]. The first part of the equation (40), $i\left[H,\rho \right]$), represents the quantum component. The second part of the equation (40), $\mathcal{L}\left(\rho \right)$), represents the classical Markov component. The weighting parameter, $\alpha$, in equation (40) provides a means to weight which element (e.g., Markov or quantum) will dominate the system. For example, when $\alpha =0$, it signifies that the system is a fully quantum regime which indicates a higher ontic uncertainty. Conversely, when $\alpha =1$, quantum dynamics no longer take place and the system model becomes Markovian. The quantum open system models begins with oscillatory behavior and due to the interaction with the environment the oscillation dissipates to a steady state as $\underset{t\to \infty }{\lim }\widehat{\rho }\left(t\right)={\widehat{\rho }}_{steady}$ [77].

Different than the two- and four-dimensional quantum and the Markov models, state representation is modeled by a density matrix, represented with $\rho$. In equation (40), a density matrix which is an outer product of the system state is represented by:

$$\rho =\psi ·{\psi }^{†}$$

(44)

Using a density operator provides a unique advantage to model human cognition and decision process because a single probability distribution is used in both quantum and the Markov components [55]. In other words, this description is the evolution of the superposition of all possible states and it is a temporal oscillation. The temporal oscillation (i.e., superposition state) can vanish transiently due to a measurement or interaction with another agent or environment. Pure quantum models cannot describe a system when the system becomes an open system, which means it is no longer isolated. When a system starts interacting with the environment, the superposition state of the system begins dissipating, which is called decoherence. Decoherence is the transition from a pure state to a classical state, particularly when a there is interaction from an environment [10,80]. The concept of decoherence is, however, controversial [65], and therefore, its interpretation is not discussed here. Pure Markov models, on the other hand, demonstrate an accumulative behavior, which fails to capture indecision represented by oscillatory behavior.

To demonstrate the evolution of the density operators, equation (44) can be written as:

$$\rho \left(t\right)=\left|\psi \left(t\right)\right.⟩⟨\left.\psi \left(t\right)\right|$$

(45)

The elements of a density operator can be written as:

$$|\psi ⟩=\sum _n{c}_n|n⟩$$

(46)

Then a density operator can be written as:

$$\begin{gathered} \begin{array}{c}\rho =|\psi ⟩⟨\psi |=\left({\displaystyle \sum _n}{c}_n|n⟩\right)\left({\displaystyle \sum _m}{c}_m^{*}⟨m|\right)=\end{array} \\ \underset{diagonalterms}{\underbrace{\sum _n{\left|c_n\right|}^2|n⟩⟨n|}}+\underset{off-diagonalterms}{\underbrace{\sum _{m\ne n}{c}_n{c}_m^{*}|n⟩⟨m|}} \end{gathered}$$

By using Equation (44) a two-state system, $|\psi ⟩=\xi |0⟩+\beta |1⟩$, can be represented with a density matrix as:

$$\begin{array}{c}|\psi ⟩⟨\psi |=\underset{diagonal}{\underbrace{{\left|\xi \right|}^2|0⟩⟨0|+{\left|\beta \right|}^2|1⟩⟨1|}}+\\ \underset{off-diagonalterms}{\underbrace{\xi {\beta }^{*}|0⟩⟨1|+{\xi }^{*}\beta |1⟩⟨0|}}\end{array}$$

(47)

The matrix representation of equation (47) with the state vector $|0⟩=\left(\genfrac{}{}{0pt}{}{1}{0}\right)$ and $|1⟩=\left(\genfrac{}{}{0pt}{}{0}{1}\right)$ can be written as:

$$\left(\begin{array}{cc}{\left|\xi \right|}^2& {\xi }^{\mathrm{*}}\beta \\ \xi {\beta }^{\mathrm{*}}& {\left|\beta \right|}^{\mathrm{*}}\end{array}\right)$$

(48)

When a system becomes perturbed, an interaction with the environment begins. By using the matrix representation in (48), the transition from a quantum state to a classical state can be represented as:

$$\underset{QuantumState}{\underbrace{\left(\begin{array}{cc}{\left|\xi \right|}^2& {\xi }^{*}\beta \\ \xi {\beta }^{*}& {\left|\beta \right|}^{*}\end{array}\right)}}\overrightarrow{interaction}\underset{ClassicalEnsamble}{\underbrace{\left(\begin{array}{cc}{\left|\alpha \right|}^2& 0\\ 0& {\left|\beta \right|}^2\end{array}\right)}}$$

(49)

Examining the evolution of equation (49), provides a comprehensive understanding of the system’s behavior. Such a full picture can be captured by the Lindblad equation shown in equation (40), which provides a more general state representation of the system, in this case, a cognitive system, by including a probability mixture across pure states:

$$\rho \left(t\right)={\displaystyle \sum _j}{p}_j·\psi ·{\psi }^{†}$$

(50)

As discussed in [74], through linearity, the density matrix in equation (50) follows the same quantum temporal evolution as in equation (40). Consequently, the density matrix captures two types of uncertainty, epistemic and ontic. Epistemic uncertainty represents an observer’s (e.g., modeler’s) uncertainty about the state of the decision maker. An ontic type of uncertainty represents the decision maker’s indecisiveness or internal uncertainty concerning the stimuli via superposition of possible states. In other words, a decision-maker’s ambiguity over evidence may be described as the vacillation of a decision maker.

For the Hamiltonian matrix (51), the diagonal elements, or drift parameters ${\mu }_Q$ controls the rate at which quantum component of the system state (superposition) shifts to an alternate delegation strength. This element is known as potential which captures the dynamics that pull back the superposition state to a higher amplitude of a basis state of a specific column. (Lower rows in the matrix represent higher delegation strength). These elements, ${\mu }_Q\left(x\right),$ are function of $x$: ${\mu }_Q\left(x\right)=a·x+b·x^2+c$. This represents the entries of the Hamiltonian matrix that pushes the decision maker’s delegation strength to the higher level of delegation strength (lower rows). In this study, for ${\mu }_Q\left(x\right)$, a simple linear function ($b=0andc=0$) is used where ${\mu }_Q\left(x\right)=a·x$.

Off-diagonal elements, diffusion rate, ${\sigma }_Q$, controls the diffusion amplitudes that captures the dynamics of flowing out from the basis state; diffusion rate induces uncertainty over different delegation strength level. In the context of trust, trust to AI will push the decision maker’s delegation state to the lower rows, whereas distrust will try pull it back to upper rows.

(51)

(52)

The drift rate ${\mu }_M$ represents the dynamics of the intensity matrix, K, (52) that pushes the delegation strength to the higher strength level; in this case lower rows in the matrix represent higher strength level. The diffusion rate is similar to the Hamiltonian; this introduces dispersion to the delegation strength, which captures some of the inconsistencies in decision making. The difference between Hamiltonian (H) and the intensity matrix (K) is that the diffusion rate in intensity matrix disperse via the probability distribution, whereas in the H case it is done via amplitudes.

The main challenge in building an open system model of a cognitive phenomenon is choosing the $\mathsf{\Gamma }$ matrix’s coefficients, ${\gamma }_{ij}$. Depending on choice, the $\mathsf{\Gamma }$ matrix is constrained by different requirements. For example, following the discussion in [74] in the case of choosing the ${\gamma }_{ij}$ to be the transition probabilities in Markov transition matrix, $T_{ij}\left(t=\epsilon \right)$, a direct connection to Markov dynamics can be achieved. Then, by choosing $\alpha =0$, equation (40) reduces to equation (41). After following the vectorized solution provided in Busemeyer et al. (2020) and Martínez-Martínez and Sánchez-Burillo, (2016) the solution of equation (41) provides:

$$\frac{d{\rho }_{kk}\left(t\right)}{dt}={\sum }_j{\rho }_{jj}·{\gamma }_{kj}-{\sum }_i{\gamma }_{ik}·{\rho }_{kk}$$

(53)

Setting up ${\gamma }_{ij}=T_{ij}\left(\epsilon \right)$ requires that the entries within each column in ${\gamma }_{ij}$must add up to one because $T\left(t\right)$ is restricted by single stochasticity. Following the discussion in [74], since a Markov process is based on the intensity matrix (K), that obeys Kolmogorov-Chapman solution $\frac{d\varphi \left(t\right)}{dt}=K·\varphi \left(t\right)$, the solution of equation (53), which is $\frac{d\varphi \left(t\right)}{dt}=(T\left(\epsilon \right)-I)·\varphi \left(t\right)$, results in incompatibility. Therefore, it may not fully capture the Markov dynamics.

As discussed in Busemeyer et al. (2020) setting the ${\gamma }_{ij}$ to intensity matrix can work as a solution. However, this introduces another challenge that requires the columns in ${\gamma }_{ij}$ to sum to zero, because an intensity matrix is a negative definite matrix. As a result, a density matrix cannot be maintained for the entire time interval. The best solution provided in [74] is to set ${\gamma }_{ij}=\left(\frac{T\left(\epsilon \right)}{\epsilon }\right)$, which provides a solution for the issues that arise in the previous two solutions. However, for very small values of $\epsilon$, the interference terms, off-diagonal terms, rapidly dissipates.

5.2. Exploratory Analysis

The experimental results provided the groundwork for tuning the parameters of the quantum open system equation. These results demonstrated that setting the ${\gamma }_{ij}$ component of the Lindblad operator equal to the intensity matrix provided the best results. With these parameters, it is now possible to model both choice and no choice conditions with one equation. The fit parameters used for the Hamiltonian matrix, intensity matrix, and alpha parameter are provided in Table 4. However, capturing the earlier timings proved difficult because of the sheer number of combinations that would have to be attempted to tune the equation further. However, the sum of squares errors (SSE) was improved by 54.8% (difference of 226 in previous modeling results). The SSE is provided in Table 5. Future work is set to develop machine learning models that can tune the parameters more efficiently. Nevertheless, a quantum open system modeling approach shows promise for modeling human behavior when interacting with AI.

Figure 11. Quantum open system modeling of preference strength vs. time for experimental results.

Figure 11. Quantum open system modeling of preference strength vs. time for experimental results.

The interplay between the Markov and quantum dynamics provides a potentially new approach for modelling decision-making dynamics in general and decision-making with AI in particular. If human decision-making does indeed follow a more quantum open systems approach, developing more optimal machine policies for interacting with humans may yield more interesting results [64]. While the results of this study appear promising, more work is still needed to flush out more details and to test the bounds of how far these techniques may generalize.

6. Discussion

7. Summary

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