The partial trace is commonly introduced in quantum mechanics as an algebraic operation used to define reduced states of composite systems. However, the probabilistic origin of this operation goes systematically unnoticed in the literature. Here, we show that the partial trace emerges naturally from the requirement of consistency between the Born rule for measurement probabilities and the classical marginalization of probability mass functions. Starting from the classical marginalization rule relating joint and marginal probability distributions, we impose that the reduced density operator of a subsystem must reproduce the local measurement statistics derived from the global state. We show that this requirement directly leads to the standard expression of the partial trace. From this perspective, the reduced density operator appears not as an ad hoc algebraic construction, but as a natural consequence of the probabilistic structure of quantum mechanics.
The partial trace is commonly introduced in quantum mechanics as an algebraic operation used to define reduced states of composite systems. However, the probabilistic origin of this operation goes systematically unnoticed in the literature. Here, we show that the partial trace emerges naturally from the requirement of consistency between the Born rule for measurement probabilities and the classical marginalization of probability mass functions. Starting from the classical marginalization rule relating joint and marginal probability distributions, we impose that the reduced density operator of a subsystem must reproduce the local measurement statistics derived from the global state. We show that this requirement directly leads to the standard expression of the partial trace. From this perspective, the reduced density operator appears not as an ad hoc algebraic construction, but as a natural consequence of the probabilistic structure of quantum mechanics.
This thesis takes inspiration from quantum physics to investigate mathematical structure that lies at the interface of algebra and statistics. The starting point is a passage from classical probability theory to quantum probability theory. The quantum version of a probability distribution is a density operator, the quantum version of marginalizing is an operation called the partial trace, and the quantum version of a marginal probability distribution is a reduced density operator. Every joint probability distribution on a finite set can be modeled as a rank one density operator. By applying the partial trace, we obtain reduced density operators whose diagonals recover classical marginal probabilities. In general, these reduced densities will have rank higher than one, and their eigenvalues and eigenvectors will contain extra information that encodes subsystem interactions governed by statistics. We decode this information, and show it is akin to conditional probability, and then investigate the extent to which the eigenvectors capture "concepts" inherent in the original joint distribution. The theory is then illustrated with an experiment that exploits these ideas. Turning to a more theoretical application, we also discuss a preliminary framework for modeling entailment and concept hierarchy in natural language, namely, by representing expressions in the language as densities. Finally, initial inspiration for this thesis comes from formal concept analysis, which finds many striking parallels with the linear algebra. The parallels are not coincidental, and a common blueprint is found in category theory. We close with an exposition on free (co)completions and how the free-forgetful adjunctions in which they arise strongly suggest that in certain categorical contexts, the "fixed points" of a morphism with its adjoint encode interesting information.
Abstract We advance the notion of a classical density matrix, as a classical analogue of the quantum mechanical statistical operator, and investigate its main properties. In the case of composite systems a partial trace-like operation performed upon the global classical density matrix leads to a marginal density matrix describing a subsystem. In the case of dynamically independent subsystems (that is, non-interacting subsystems) this marginal density matrix evolves locally, its behavior being completely determined by the local phase-space flow associated with the subsystem under consideration. However, and in contrast with the case of ordinary marginal probability densities, the marginal classical density matrix contains information concerning the statistical correlations between a subsystem and the rest of the system.
We present a minimal, representation-invariant route from normalized probabilistic structure to operator state description. Starting from probability assignments to admissible effects, we show how simplex-type (context-fixed) descriptions embed into the convex set of density operators as the minimal basis-invariant closure. The key representation is the trace pairing p(E) = Tr(ρE), where ρ ⪰ 0 and Trρ = 1. We also formalize composition and marginalization: correlations are encoded by joint states ρAB, while locally accessible statistics are represented uniquely by reduced states ρA = TrB(ρAB). The paper is intentionally conservative: it establishes a logically closed front-end for later developments (modular generators, CPTP-compatible dynamics, and phase structure) without introducing them here as axioms.
We investigate the conceptual relationship between decoherence and the origin of probabilistic outcomes in quantum mechanics. While decoherence theory successfully explains the suppression of interference and the emergence of classical behavior through interactions with environmental degrees of freedom, its role in providing a complete account of the probabilistic nature of quantum measurement remains limited. In this work, we argue that the suppression of interference and the origin of probabilistic outcomes constitute distinct problems that should be conceptually distinguished. By “probabilistic outcomes,” we refer to the appearance of statistical distributions in measurement results as described by the Born rule. To clarify this distinction, we analyze limiting scenarios, including isolated quantum systems and the universe treated as a closed system, where the standard decoherence mechanism becomes ineffective or inapplicable, yet probabilistic outcomes persist. These considerations are not intended as a formal proof, but rather as a means to isolate conceptual dependencies. We then revisit the Intrinsic Observability Framework (IOF), in which the Hilbert space is intrinsically decomposed into observable and hidden degrees of freedom. Within this framework, probabilistic descriptions arise from restricted access to the full quantum state, rather than from environmental interactions or dynamical processes. By contrasting decoherence with the IOF perspective, we argue that decoherence provides an account of the emergence of classicality, but does not directly address the origin of quantum probabilities. The IOF, on the other hand, offers a structural perspective at the level of pure states. This suggests that a complete understanding of quantum measurement requires recognizing the distinct roles played by these two elements.
We present a reduced density operator for electronically open molecules by explicitly averaging over the environmental degrees of freedom of the composite Hamiltonian. Specifically, we include the particle-number nonconserving (particle-breaking) interactions responsible for the sharing of electrons between the molecule and the environment, which are neglected in standard formulations of quantum statistical mechanics. We propose an unambiguous definition of the partial trace operation in the composite fermionic Fock space based on composite states in a second quantization framework built from a common orthonormal set of orbitals. Thereby, we resolve the fermionic partial trace ambiguity. The common orbital basis is constructed by spatial localization of the full orbital space, in which the full composite Hamiltonian naturally partitions into a molecule Hamiltonian, an environment Hamiltonian, and an interaction Hamiltonian. The new reduced density operator is based on the approximation of commutativity between the subsystem Hamiltonians (i.e., molecule and environment Hamiltonians) and the interaction Hamiltonian, but our methodology provides a hierarchical approach for improving this approximation. The reduced density operator can be viewed as a generalization of the grand canonical density operator. We are prompted to define the generalized chemical potential, which aligns with the standard interpretation of the chemical potential, apart from the possibility of fractional rather than strictly integer electron transfer in our framework. In contrast to standard approaches, our framework enables an explicit consideration of the electron occupancy in the environment at any level of theory, irrespective of the model used to describe the molecule. Specifically, our reduced density operator is fully compatible with all possible level-of-theory treatments of the environment. The approximations that render our reduced density operator identical to the grand canonical density operator are (i) restriction of excitations to occur within the same orbitals and (ii) assumption of equal interaction with the environment for all molecule spin orbitals (i.e., the wide-band approximation).
This work discusses simple examples how quantum systems are obtained as subsystems of classical statistical systems. For a single qubit with arbitrary Hamiltonian and for the quantum particle in a harmonic potential we provide explicitly all steps how these quantum systems follow from an overall ''classical" probability distribution for events at all times. This overall probability distribution is the analogue of Feynman's functional integral for quantum mechanics or for the functional integral defining a quantum field theory. In our case the action and associated weight factor are real, however, defining a classical probabilistic system. Nevertheless, a unitary time-evolution of wave functions can be realized for suitable systems, in particular probabilistic automata. Based on these insights we discuss novel aspects for correlated computing not requiring the extreme isolation of quantum computers. A simple neuromorphic computer based on neurons in an active or quiet state within a probabilistic environment can learn the unitary transformations of an entangled two-qubit system. Our explicit constructions constitute a proof that no-go theorems for the embedding of quantum mechanics in classical statistics are circumvented. We show in detail how subsystems of classical statistical systems can explain various ``quantum mysteries". Conceptually our approach is a straightforward derivation starting from an overall probability distribution without invoking non-locality, acausality, contextuality, many worlds or other additional concepts. All quantum laws follow directly from the standard properties of classical probabilities.