Operational Transition Probability

• Operational Transition Probability is a measure that quantifies outcome likelihood based on initial state, effects, and the applied operation.
• It generalizes classical and quantum probability assignments by incorporating device-specific interventions and state updates.
• Applications span quantum mechanics, thermodynamics, and generalized probabilistic theories, illustrated by metrics like the Born rule and quantum random walks.

An operational transition probability is a context-dependent quantitative assignment describing the likelihood of realizing a specific outcome or state—conditional on both the initial preparation and the associated operation (measurement device, dynamical intervention, or control)—within a well-defined physical or mathematical process. It systematically generalizes the classical and quantum notions of transition amplitude or overlap, replacing or supplementing the static probability assignment with one that incorporates the effect of operations, instruments, or dynamical maps. This concept is central in quantum theory, generalized probabilistic theories, stochastic processes, and thermodynamics, underpinning both foundational and applied aspects of statistical prediction, control, and transformation.

1. Abstract Effect-Algebra and Generalized Frameworks

The most general non-classical context for operational transition probability is that of effect algebras $(E,0,1,\oplus)$ equipped with states $s:E \to [0,1]$ (additive, $s(1)=1$) and operations $J$ (convex, positive, subnormalized maps). For $a,b\in E$, the transition probability for the sequential test "a then b," given initial state $s$ and operation $J$, is defined as

$$P_{s,J}(a,b) := s(a)\,\widetilde{J(s)}(b) = \frac{s(a)\,J(s)(b)}{J(s)(1)}$$

where $\widetilde{J(s)}$ is the state updated by $J$ and normalized on the event that $J$ "clicks" ($J(s)(1)\neq 0$). This formula encodes both the a priori likelihood of $a$ in $s$ and the conditional probability for $b$ after $J$ is applied. When comparing two states $s_1,s_2$ under a common $J$, one defines a compatibility functional $P_J(s_1,s_2) := J(s_2)(1)$. Such definitions provide the basis for operational transition probabilities in quantum, classical, and generalized probabilistic frameworks, with dependence on both the states/effects and the choice of operation or instrument (Gudder, 2024).

2. Specialization to Quantum Mechanics: Hilbert Space and Instruments

In standard quantum mechanics, the effect algebra specializes to the convex set of effects $E(H) = \{A: 0\leq A \leq I\}$, with states being density matrices $\rho$ and operations $I$ given by completely positive, trace-nonincreasing maps. The operational transition probability for an initial test $A$, operation $I$, and post-update effect $B$ is

$$P_{\rho,I}(A,B) = \frac{\mathrm{Tr}(\rho A)\;\mathrm{Tr}[I(\rho) B]}{\mathrm{Tr}[I(\rho)]}$$

If $I$ is a Lüders (projective, minimally disturbing) operation $L_C(\rho) = C^{1/2} \rho C^{1/2}$, then

$$P_{\rho, L_C}(A,B) = \frac{\mathrm{Tr}(\rho A)\; \mathrm{Tr}[C^{1/2}\rho C^{1/2} B]}{\mathrm{Tr}(C\rho)}$$

For pure states $\rho = |\psi\rangle\langle\psi|$ and effects $A = B = |\phi\rangle\langle\phi|$, this reduces to the standard Born probability $|\langle\psi|\phi\rangle|^2$.

Alternate measurement protocols, such as the Holevo operation $H(a,\alpha)(\rho) = \mathrm{Tr}(\rho a)\,\alpha$ (with $\alpha$ fixed), yield

$$P_{\rho,H(a,\alpha)}(b,c) = \mathrm{Tr}(\rho b)\,\alpha(c)$$

showcasing the operational dependence: Holevo's construction discards "which effect triggered" beyond normalization, whereas Lüders' is minimally disturbing. This flexibility captures a spectrum of physically meaningful assignments, all within the operational transition probability formalism (Gudder, 2024).

3. Operational Transition Probability in Generalized Probabilistic Theories

In Generalized Probabilistic Theories (GPTs), one considers a finite-dimensional ordered vector space $V$ (state cone $V^+$, order unit $u$), with states $\Omega=\{\omega \in V^+|u(\omega)=1\}$ and effects $E$ forming the dual cone. The operational probability $p(e|\omega)$ for effect $e$ in state $\omega$ must meet structural constraints—particularly affinity (linearity in $\omega$)—to avoid superluminal signaling. Causal consistency and sharp measurements together force $p(e|\omega)$ to be affine in $\omega$, and then standard GPT-reconstruction identifies $p(e|\psi)=\tau(\psi,e)$, the quadratic transition probability (Born rule) in quantum theory: $p(a|\rho) = \mathrm{Tr}(\rho E_a)$ Thus operational transition probability in GPTs unifies classical, quantum, and hypothetical theories: it is the only assignment compatible with no-signaling, convexity, and measurement repeatability (Alegre, 14 Dec 2025).

4. Physical Interpretation and Device Dependence

Operational transition probability always refers to the entire experimental protocol: not just an initial state or effect, but also the detailed structure of the operation or instrument implementing the physical process ("J" in abstract notation). Distinct operations (e.g., Lüders vs. Holevo) yield different state updates and transition probabilities, a reflection of the experimental context. In the sequential measurement context, $P_{s,J}(a,b)$ is the joint probability of obtaining "yes" for $a$ in $s$, then "yes" for $b$ after operation $J$ fires.

This device dependence guarantees that even when initial and final effects are fixed, the operational transition probability captures not just the properties of the system, but also that of the intervention mechanism—a necessity for predictive and retrodictive tasks in foundations, control, and information theory (Gudder, 2024).

5. Operational Transition Probability Beyond Quantum Theory

Operational transition probabilities play an extensive role outside standard quantum frameworks. In open quantum random walks, the kernel

$$P_\rho(j\to i) = \mathrm{Tr}[B^i_j\,\rho_j\,(B^i_j)^*]$$

generalizes Markov transition probabilities to quantum settings, with the normalization condition ensuring a stochastic interpretation on the commutative subalgebra. When lifted to Quantum Markov Chains on trees, operational transition probabilities structure the entire Markov field, revealing non-classical phenomena such as phase transitions and entropy differentiation between coexisting phases (Mukhamedov et al., 2022).

In stochastic thermodynamics, the operational transition probability is the maximal probability of success $P_{\max}(\rho\to\sigma)$ to convert $\rho$ to $\sigma$ via thermal operations: $$P_{\max}(\rho\!\to\!\sigma) = \min_k \frac{M_k(\rho)}{M_k(\sigma)}$$ with $\{M_k\}$ being the set of thermo-majorization monotones. This operational approach renders explicit the tradeoff between success probability and resource expenditure (e.g., single-shot work of transition), and admits analogs in entanglement theory (Alhambra et al., 2015).

In operational models of general relativity, transition probabilities are encoded as contractions over duotensors and hopping metrics, specifying the likelihood of outcomes across spacetime regions joined by type surfaces and generalizing quantum probabilistic structure to diffeomorphism-invariant field theories (Hardy, 2016).

6. Key Formulas and Exemplary Cases

A summary of central formulas across frameworks:

Framework	Transition Probability Formula	Reference
Effect-algebra (abstract)	$P_{s,J}(a,b) = s(a)\,\frac{J(s)(b)}{J(s)(1)}$	(Gudder, 2024)
Kraus-operator (Hilbert space)	$$P_{\rho,I}(A,B) = \mathrm{Tr}(\rho A)\,\mathrm{Tr}[I(\rho) B]/\mathrm{Tr}[I(\rho)]$$	(Gudder, 2024)
Lüders reduction	$$P_{\rho,L_C}(A,B) = \mathrm{Tr}(\rho A)\, \mathrm{Tr}(C^{1/2}\rho C^{1/2}B)/\mathrm{Tr}(C\rho)$$	(Gudder, 2024)
Quantum overlap (pure)	$$P_{|\psi\rangle,L_{|\phi\rangle}}(|\phi\rangle,|\phi\rangle) = |\langle\psi|\phi\rangle|^2$$	(Gudder, 2024)
Thermodynamic operation	$$P_{\max}(\rho\to\sigma) = \min_k \frac{M_k(\rho)}{M_k(\sigma)}$$	(Alhambra et al., 2015)
OQRW (quantum stochastic)	$$P_\rho(j\to i) = \mathrm{Tr}(B^i_j \rho_j (B^i_j)^*)$$	(Mukhamedov et al., 2022)

These explicit expressions, always conditional upon states and operations, formalize the operational assignment of probabilities in all major paradigms.

7. Operational Transition Probability in Quantum Logic and Jordan Algebras

The operational transition probability can be characterized in quantum logics and Jordan algebras by a universal requirement: for projections $p,q$ in a suitable orthomodular poset or JBW-algebra, $\mathbb{P}(q|p) = s$ if and only if for every normal state $\mu$ with $\mu(p)=1$ one finds $\mu(q)=s$, or equivalently if the Jordan triple product satisfies $\{p,q,p\} = s p$. This algebraic perspective generalizes the Hilbert space Born rule and highlights structural features such as isoclinicity and compatibility, underlying phenomena such as the impossibility of universal cloning (Niestegge, 2021).

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Operational transition probability thus provides a unified, operationally meaningful assignment for transition likelihoods across a broad spectrum of mathematical, physical, and informational theories: quantum, probabilistic, thermodynamic, and beyond. Its formulation always encodes the dependence on both the initial data and the precise operation acting, ensuring a deeply contextual and physically interpretable probability assignment.