Abstract
Since the late 90s a paradigm shift began in decision research that has implications for leadership research. Due to the limitations of standard decision theory (based on Kolmogorovian/Bayesian decision theory) scholars began to build a new theory based on the ontological and epistemological foundations of quantum mechanics. The last decade has witnessed a surge in quantumlike modeling in the social sciences beyond decisionmaking with notable success. Many anomalies in human behavior, viz, order effects, failure of the sure thing principle, and conjunction and disjunction effects are now more thoroughly explained through quantum modeling. The focus of this paper is, therefore, to link leadership with quantum modeling and theory. We believe a new paradigm can emerge through this wedding of ideas which would facilitate better understandings of leadership. This article introduces readers to the mathematical analytical processes that quantum research has developed that can create new insights in the social scientific study of leadership.
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Notes
 1.
However, we should caution readers that we would like to adapt the mathematical and logical framework of quantum theory only, rather than the physics of it. The emerging quantum like modelling in social sciences aims at that, for example, see Haven and Khrennikov (2013).
 2.
A good coverage of extant emerging literature can be found in the research hand book edited by Haven and Khrennikov (2018).
 3.
Such contextual utility models can show various effects like preference reversals, ambiguity aversion or attraction, all embedded in a single coherent framework. Our point is that a single coherent framework is critically needed in leadership decision theory also.
 4.
In quantum physics there are stylized uncertainty relations, for example, the product of momentum and position uncertainty measures are greater than or equal to h/2π, where h is Plank’s constant. In social science we can refer to the superposition description readily, for example, as in quantum decision theory, where deep uncertainty is described by the superposition of beliefs, which is defined in terms of density matrix operators (we present more detail on this formalism later).
 5.
Vaxjo interpretation of the modified formula for total probability has emerged out of efforts by scientists at Vaxjo conferences on quantum foundations since last twenty years (Khrennikov 2023).
 6.
The formula presented here is originally motivated by the superposition principle in quantum mechanics, as discussed in the paper, and this formula famously appears in the probability computation of ‘double slit experiment’in quantum physics, which is so well emphasised in Feynman lectures on physics Volume 3.
 7.
In quantum theory measurements are described by projection operators, or projection postulate, act of measurement is equivalent to projections of the initial superposed state into a definite Eigenvalue, probability of such a projection is provided by the Born’s rule. Such projection operators live in the Hilbert space of the system and are orthonormal to each other. There are other projection operators which are named as positive operators, which describes ‘unsharp’ measurements. In decision theory terms, orthogonal projection operators will project the intial belief state to a specific final state immediately after the measurement (for example immediately after a question is asked, where the act of asking question is measurement), where as a positive operator will project the initial superposed belief state into an unsharp state, for example ‘may be’ type of response.
 8.
Readers can be referred to a formal mathematical literature on the commuting and noncommuting observables or questions in decision theory.
 9.
In finance for example, there is a wide literature on soft and hard information: soft being Facebook like environment which is less verifiable and hard being Balance sheet like which is more readily verifiable.
 10.
These equations are known as Master equations in quantum theory, which describe generally how a systems state evolves over time with interactions and with the information environment embedded in the equation’s parameters.
 11.
In this regard QDT can also play a fundamental role in complexity theory, which describes society and economy as a complex dynamical system, with deep uncertainty.
 12.
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Appendices
Appendix 1
Basic Concepts in Quantum Mechanics
Quantum physics began when Max Plank and Albert Einstein proposed that energy can only be radiated and absorbed in small units, now called quanta, and that light or electromagnetic radiation are streams of massless particles called photons. The word quantum mechanics was coined in 1920s by Heisenberg, Born, Pauli, Jordan and other eminent scientists. The core structure of Quantum theory was built by 1930s, and since then scientists and philosophers have continued to develop it.
Prior to quantum physics, classical physics was tied to three principles: (1) locality, which demands that there has to be a speed limit to signaling between events in spacetime, which is challenged by entanglement, (2) causality, which demands a strict cause and effect relationship in nature, or a strict one directional arrow of time, and (3) realism which demands a subjectobject split in (an objective) nature. However, at the quantum level of reality each of these principles is violatable. Such bold new ontological insights changed the course of modern physics by challenging classical assumptions about the nature of the physical universe and even the idea of an objective reality.
To compliment this new interest a new language of mathematics and logic was developed for quantum research by such people as Heisenberg, Shrödinger, Born, and Neuman. Quantum statistics, in the form of Boson and Fermion statistics enables significant research breakthroughs by, for example, Satyen Bose and Einstein.
Theoretical advances, most notably by Richard Feynman, gradually reformulated quantum mechanics by, for example, introducing the path integral or sum over histories technique, which opened the door for quantum field theory, quantum electrodynamics, and the ‘standard model’ of particle physics, which remains the most successful model of the universe.
Here we just provide a few definitions of the basic objects in quantum mechanics, in Appendix 2 we provide a more detailed account of the mathematical structure of the theory.
Wave function: the description of a quantum state or a quantum system, a complex amplitude, whose modulus squared (square of the absolute value) provides the probability of the system to be found in a specific region. Wave function is described in a superposition of possibilities, or eigen values, until it is measured/observed. Wave function evolves over time in a deterministic manner following an equation of motion, namely Schrödinger’s equation of motion. The wave function lives in a complex normed vector space, named as Hilbert space.
Measurement: wavefunction evolves deterministically, until the experimenter measures a specific property of the system: for example, position, or velocity, or spin. Orthodox views suggest that measurement makes the wave function collapse to one of the eigen values measured/observed in the superposition. However, this process of measurement and collapse is a truly random process and is not dependent on our state of knowledge of the initial conditions of the system. Hence, randomness in quantum theory is ontological rather than epistemological.
More recently, some of the features of quantum reality such as contextuality, entanglement, and observer effects have drawn the attention of social scientists because social systems have important quantumlike features to which the logical and statistical tools of quantum physics can be applied.
Appendix 2
Basic Mathematical Tools or Concepts
We begin with a brief comparison between classical probability theory (CPT) and quantum probability theory (QPT).
The main features of classical probability theory are:

Events are represented by sets, which are subsets of Ὠ.

Sample space, sigma algebra, measure (probability)*, are the main features of the related Kolgomorov measure theory.

Boolean logic is the type of compatible logic with CPT, which allows for deductive logic, and basic operations like union and intersection of sets, DeMorgan Laws () of set theory are valid.

Conditional probability: P(a/b) = p(a and b)/p(b); p(b)>0We see conditional probability is a direct consequence of Boolean operations.

Based on the Boolean logic the set theory of probability also directs to Bell’s inequalities: P(A and B) + P(B− and C)> /= P(A and C).
The main features of Quantum Probability Theory are:

State space is a complex linear vector space: Hilbert space***; Finite/ infinite D, symbolized as H.

H is endowed with a scalar product (positive definite), norm, and an orthonormal basis, nondegenerate.

Any state can be visualized as a ray in this space.

Superposition principle: which states that a general state can be written as a linear superposition of ‘basis states’, in information theory language the basis states are 0> or 1>.

Measurement: most of the times projection postulate**.

Measurement implies projection onto a specific Eigen subspace.

Probability, updating can be visualized as sequential projections on Eigen subspaces.

NonBoolean logic is compatible with such state space structure, which means violation of commutation and associative properties.
The main features of NonBoolean Logic are:

Algebra of events is prescribed by quantum logic.

Events form an event ring R, possessing two binary operations, addition and conjunction.

P(A U B) = P(B U A) (this Boolean logic feature is invariant in Quantum logic).

P{A U (B U C)} = P{(A U B) U (A U C)} (associative, property also holds good).

A U A = A (idempotency).

P(A and B) # P(B and A) (non commutatitivity, incompatible variables).

A and (B U C) # (A and B) U (A and C) (no distributivity).
The fact that distributivity is absent in quantum logic was emphasized by Birkhoff and von Neumann. Suppose there are two events B1 and B2 that, when combined, form unity, B1 ∪ B2 = 1. Moreover, B1 and B2 are such that each of them is orthogonal to a nontrivial event A # 0, hence A ∩ B1 = A ∩ B2 = 0. According to this definition, A ∩ (B1 ∪ B2) = A ∩ 1 = A. But if the property of distributivity were true, then one would get (A ∩ B1) ∪ (A ∩ B2) = 0. This implies that A = 0, which contradicts the assumption that A # 0.
The main features of Quantumlike Modeling of Belief States are:

Bruza et al. (2015): cognitive modelling based on quantum probabilistic frame work, where the main objective is assigning probabilities to events

Space of belief is a finite dimensional Hilbert space H, which is spanned by an appropriate set of basis vectors

Observables are represented by operators (positive operators/Hermitian operators) which need not commute
$$\left[ {{\text{A}},\;{\text{B}}} \right]\;{ = }\;{\text{AB}}\;\;{\text{BA}}\; = \;0$$
Generally, any initial belief state is represented by density matrix/ operator, outer product of ψ with itself ρ =ψ ⟩⟨ ψ, this is a more effective representation since it captures the ensemble of beliefs
Pure states and mixed states
Mixed states: ∑wψ⟩ ⟨ψ, hence mixed state is an ensemble of pure states with w’s as probability weights.
Some properties of ρ: ρ = (ρ*)T, for pure states ρ = ρ^{2}, where T stands for transpose operation.
Measuring the probability of choosing one of the given alternatives, which is represented by the action of an operator on the initial belief state.
While making decision superposition state collapses to one single state (can be captured by the Eigen value equation).
Observables in QPT represented by Hermitian operators:
E(A) = Tr(A ρ), every time measurement is done one of the Eigenvalues of the A is realized.
A=∑aP spectral decomposition rule: a’s are the Eigen values and P’s are the respective projectors which projects the initial state to the Eigen subspace with a specific a
As soon as the measurement is done the state ρ’: Pi ρPi/Tr(Pi ρ).
Simultaneously updating of the agents’ belief state.
A Quick Review of Formula FOT Total Probability/Law of Total Probability (LTP), Modified in Quantum Like Set Up
First we see the LTP in classical set theory as below:
P (B and (A or C)) = P(B and A) +P(B and C)
(measure theoretic additivity)
Hence it follows:
Hence in particular if P(A or C) = 1, then P(B) = {P(BA) * P(A) + P(BC) * P(C)}, this is the LTP (law of total probability) as we know in familiar CPT(classical probability theory).
But in the QPT (quantum probability theory) additivity does not follow, which means LTP is violated since there are interference terms.
To get the modified LTP as in non Kolgomorovian QDT set up we have to go through the concept of positive valued operators (POVM) as below.
A positive operator valued measure (POVM) is a family of positive operators {Mj} such that ∑Mj = I, where I is the unit operator. It is convenient to use the following representation of POVMs:
where Vj: H → H are linear operators. A POVM can be considered as a random observable. Take any set of labels α1,..., αm, e.g., for m = 2, α1 = yes, α2 = no. Then the corresponding observable takes these values (for systems in the state ρ) with the probabilities p(αj) ≡ pρ(αj) = TrρMj = TrVjρV * j.
We are also interested in the postmeasurement states. Let the state ρ was given, a generalized observable was measured and the value αj was obtained. Then the output state after this measurement has the form: ρj = VjρV * j/(TrVjρV *j).
Both order effects and interference terms in LTP can be demonstrated using POVM.
Consider two generalized observables a and b corresponding to POVMs Ma = {V * j Vj} and Mb = {W * j Wj}, where Vj ≡ V (αj) and Wj = W(βj) correspond to the values αj and βj. If there is given the state ρ the probabilities of observations of values αj and βj have the form:
Now we consider two consecutive measurements: first the ameasurement and then the bmeasurement. If in the first measurement the value a = α was obtained, then the initial state ρ was transformed into the state
For the consecutive bmeasurement, the probability to obtain the value b = β is given by
This is the conditional probability to obtain the result b = β under the condition of the result a = α. We set p(α, β) = pa(α)p(βα).
Now since operators need not commute p(α, β) = p(β, α).
We recall that, for two classical random variables a and b which can be represented in the Kolmogorov measuretheoretic approach, the formula of total probability (FTP) has the form pb(β) = ∑ pa(α)p(βα).
Further we restrict our consideration to the case of dichotomous variables, α = α1, α2 and β = β1, β2.
FTP with the interference term for in general nonpure states given by density operators and generalized quantum observables given by two (dichotomous) PVOMs:
or by using ordered joint probabilities pb(β) = p(α1, β) + p(α2, β) + 2λβ√p(α1, β)p(α2, β). Here the coefficient of interference λ has the form: λ = Trρ{W*(β)V*(αi)V(αi)W(β) − V*(αi)W*(β)W(β)V(αi)}/2√pa(α1)p(βα1)pa(α2)p(βα2) Introduce the parameters
This parameter is equal to the ratio of the ordered joint probabilities of the same outcome, but in the different order, namely, “b then a” or “a then b”. Then,
Interference term λ = ½ {√(p(α1, β)/p(α2, β) * (γα1β 1) + √(p(α2, β)/p(α1, β) * (γα2β − 1).
In principle, this coefficient can be larger than one. Hence, it cannot be represented as λ = cosθ for some angle (“phase”) θ, cf. However, if POVMs Ma and Mb are, in fact, spectral decompositions of Hermitian operators, then the coefficients of interference are always less than one, i.e., one can find phases θ.
One important note is that such phase terms cannot always be expressed in trigonometric terms, Hyperbolic phase terms are also possible, which are typical of results obtained from decision making models (Haven and Khrennikov 2013).
Entanglement Mathematics
As we have seen throughout that quantum theory allows superposition of the basis states to form new states, many of such superpositions, but not all, poses the quality of entangled states. For example, we start with a qubit system (i.e. a system which has only two basis states 0> and 1>, where they may represent up and down states, for example in decision making models they represent belief sets of decision makers as up state or down state related to any future event), now such a system can be written in superpositions of the basis states in a number of ways.
x> = 1/√2 {00> + 11>}, this state can be called as an entangled state, since say if these qubits are given to Alice and Bob, and even they are separated light years apart, if Alice measures her system there is always a 50–50 chance of finding a 0> or 1>, however as soon as she discovers that it is determined with 100% probability that Bob has to have 0> in the first case and 1> in the second case.
Hence there is no superluminal communication happening, only that subsystems are in a random state and the system as a whole is in a pure state.
Again, another hallmark of such states is that mathematically they are not separable, in the sense that x> cannot be written as a sum over tensor products of only 0> or 1>.
Comparatively, separable states are like y> = 1/√2{00> + 01>}, in such a case Alice will always with probability 1 measure her subsystem to be in 0> but Bob still will have a 50% chance of 1> or 0>, again y> can be separated as 1/√2{0>(0> + 1>)} which means a tensor product between 0> and the superposition of 0> and 1>.
Measure of degree of entanglement: concurrence measure is a type of measure of degree of entanglement, say a general entangled state is written as: a 00> + b01> + c10> + d11>.
Then the state is maximally entangled if ad − bc = 1, and there is no entanglement if ad − bc = 0.
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Rooney, D., Patra, S. (2023). Remodeling Leadership: Quantum Modeling of Wise Leadership. In: Chakraborti, A., Haven, E., Patra, S., Singh, N. (eds) Quantum Decision Theory and Complexity Modelling in Economics and Public Policy. New Economic Windows. Springer, Cham. https://doi.org/10.1007/9783031388330_4
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