The structure of the space of the physical states

Zhihao Ma, Sen Zhu

Quantum states play a fundamental role in quantum physics; so it is necessary to study intrinsic structure of quantum states. In this paper, we study topological structure and measurable structure of state space of a quantum system, and find that almost all physical important properties on quantum states coincide.

K. Kong Wan

This introductory chapter sets out a general structure of physical theories which is applicable to both classical and quantum mechanics. We start with measurable properties of a given physical system, be it classical or quantum. These properties are called observables . We then introduce a definition of the state of a physical system in terms of measured values of a sufficiently large set of observables. A theory to describe the system should consists of four basic components: Basic mathematical framework This comprises a set of elements endowed with some specific mathematical structure and properties. In mathematics such a set is generally known as a space . Description of states States are described by elements of the space in the chosen mathematical framework. For this reason the space is called the state space of the system. Description of observables Observables are to be described by quantities defined on the state space. The description should yield all possible values of observables. The relationship between observables and states should be explicitly stated. The following two cases are of particular interest: 2 For a deterministic theory like classical mechanics a state should determine the values of all observables. For a probabilistic theory like quantum mechanics a state should determine the probability distribution of the values of all observables. Description of time evolution (dynamics).

Matteo Bellori

This paper proposes a structural principle that explains how physical reality emerges from quantum configuration space. While quantum mechanics accurately describes systems as superpositions of possible configurations, it does not by itself explain why only certain configurations persist as physical reality under interaction. We introduce stability as a necessary condition for persistence, derived from the requirement that physical systems must produce consistent relational outcomes under repeated interaction. Configurations that fail to satisfy this requirement cannot be empirically identified and therefore cannot constitute elements of physical reality. Within this framework, quantum states define structured configuration spaces, interaction imposes consistency constraints, and stability acts as a selection mechanism. Measurement is reinterpreted as stability-based selection rather than collapse or branching. A concrete analysis of a decoherence scenario demonstrates how incompatible correlation structures fail to persist. The framework distinguishes between state-level stability, governing the persistence of configurations, and sequence-level stability, governing coherence across change. While the present work focuses on physical persistence, this distinction provides a foundation for extending the same principle to identity and higher-level structures. This work does not modify the formalism of quantum mechanics, but provides a structural account of how probabilistic configuration spaces give rise to persistent physical reality.

Ayush Chambyal, Brijesh, Rakesh Sharma

Quantum state tomography (QST) scales exponentially in both measurement and computational cost, making full reconstruction impractical for multi-qubit systems. Existing approaches attempt to reduce this complexity, but do not explicitly restrict the operator space based on physically relevant correlations. We develop a structured QST framework in which the density matrix is reconstructed using a restricted set of observables in a Gibbs representation. The Structured Gibbs Quantum State Tomography (SG-QST) is built by progressively including local, nearest-neighbor, and global correlations. Benchmarking on three, four, and five-qubit. GHZ states shows that comparable fidelity can be achieved with significantly fewer parameters by restricting the operator space to physically relevant observables. These results demonstrate that physically motivated operator-space restriction enables efficient and interpretable quantum state reconstruction.