A derivation of quantum theory from physical requirements (1004.1483v4)

Abstract: Quantum theory is usually formulated in terms of abstract mathematical postulates, involving Hilbert spaces, state vectors, and unitary operators. In this work, we show that the full formalism of quantum theory can instead be derived from five simple physical requirements, based on elementary assumptions about preparation, transformations and measurements. This is more similar to the usual formulation of special relativity, where two simple physical requirements -- the principles of relativity and light speed invariance -- are used to derive the mathematical structure of Minkowski space-time. Our derivation provides insights into the physical origin of the structure of quantum state spaces (including a group-theoretic explanation of the Bloch ball and its three-dimensionality), and it suggests several natural possibilities to construct consistent modifications of quantum theory.

• The paper establishes five physical axioms that uniquely reconstruct quantum mechanics, yielding both quantum theory and classical probability theory.
• It demonstrates that requirements like finiteness, local tomography, and symmetry lead naturally to the Bloch ball representation of a qubit.
• The framework implies that any extension of quantum theory must relax a key axiom, guiding future research in quantum gravity and related fields.

Derivation of Quantum Theory from Physical Requirements

The paper "A Derivation of Quantum Theory from Physical Requirements" by Lluıs Masanes and Markus P. Müller presents a novel approach to deriving the formal structure of quantum theory (QT) by identifying and imposing a set of foundational physical requirements. This approach draws parallels to how special relativity relies on simple requirements like the principles of relativity and light speed invariance to derive its formalism. The authors aim to clarify the physical origin of the quantum state spaces' structure and explore the potential for extending quantum theory by modifying these foundational requirements.

Main Contributions

The core of the paper proposes five physical requirements that uniquely specify the formalism of quantum mechanics. These requirements act as axioms from which QT can be reconstructed, reversing the typical process in quantum physics where theoretical predictions are derived from assumed mathematical postulates like Hilbert spaces, state vectors, and linear operators. The authors show that these five requirements lead to only two theories — classical probability theory (CPT) and quantum theory. The requirements are as follows:

1. Finiteness: A system that carries one bit of information must have its state characterized by a finite set of outcome probabilities.
2. Local Tomography: The state of a composite system is fully characterized by the statistics obtained by measuring each subsystem separately.
3. Equivalence of Subspaces: State spaces of lower capacity (subsystems) must be equivalent to projections of state spaces of higher capacity.
4. Symmetry: For any pair of pure states, there is a reversible transformation mapping one onto the other.
5. All Measurements Allowed: All mathematically well-defined measurements that fit the state's probability framework are physically allowable.

Results and Conclusions

The analysis reveals that these requirements are both necessary and sufficient to derive the well-known mathematical structure of QT. In particular, their analysis shows that the three-dimensional Bloch ball representation of a qubit naturally emerges under these axioms, highlighting the importance of symmetry, which implies the continuity of reversible transformations. They also underscore how QT's uniquely probabilistic nature, including its adherence to the principle of superposition, naturally arises from these physical requirements when continuity is introduced.

The research further suggests that any future theory extending or modifying QT would need to violate at least one of these five foundational requirements. This introduces pathways for theoretically exploring new quantum-like theories that can accommodate generalizations involving these basic principles.

Implications and Future Research Directions

The derivation provides a clearer understanding of QT's foundations by eschewing abstract mathematical constructions in favor of dependably comprehensible physical principles. This could inform efforts to delineate the boundaries of quantum mechanics and identify potential avenues for alternative formulations, such as those that could arise in quantum gravity.

The authors acknowledge that while QT is compatible with these requirements, a theory that generalizes QT may only necessitate relaxing continuity or symmetry conditions — suggesting a potential framework within which future innovative theories can be developed. These could have practical implications not only in fundamental physics but also in fields reliant on quantum processes, such as quantum computing and cryptography.

In conclusion, the work by Masanes and Müller offers a robust foundation for QT derived from physically intuitive axioms, inviting further exploration into the characteristics and potential generalizations of these requirements in pursuit of new theoretical developments in physics. This meticulous approach provides fertile ground for the continued investigation into the quantum mechanics underpinnings and their possible extensions.