A New Definition of Quantum Superposition

Abstract: The usual description of the superposition of two (pure quantum) states is ambiguous, since the binary operation of summation in a Hilbert space does not pass down to the quotient projective space. Even though Dirac noted this as early as 1930, it is often asserted that the superposition is a binary operation acting on two states with a value that is a unique state. The goal for this note is to motivate a rigorous, geometrical definition of the superposition of states in the setting of complex projective space, which has been argued elsewhere to be the natural geometric phase space for quantum theory. The upshot is that the new definition of the superposition of two pure states, viewed as two distinct points in the projective space, is the unique (complex) line on which those two points lie. Finally, a comparison is given between superposition and expansion in an orthonormal basis.

• The paper introduces a new geometric definition of quantum superposition, framing it as the unique projective line passing through two pure states.
• It critiques the conventional Hilbert space formulation by showing that linear combinations yield ambiguous state representations in projective space.
• The approach aligns quantum superposition with complex projective geometry, clarifying measurement outcomes and supporting generalizations to Grassmannians.

Geometric Reformulation of Quantum Superposition

Introduction

The principle of superposition is central to quantum theory, yet its standard formulation—as the summation of quantum states in a Hilbert space—suffers from fundamental ambiguities due to the passage from Hilbert space to projective Hilbert space. The manuscript "A New Definition of Quantum Superposition" (2606.15607) addresses these issues by presenting a rigorous, unambiguous, and geometrically motivated definition of superposition, reinterpreting it within the context of complex projective geometry, which is increasingly argued to be the natural phase space for quantum theory.

Critique of the Standard Hilbert Space Perspective

The conventional identification of the superposition of two quantum states with a specific normalized linear combination is inherently flawed. Summation in the Hilbert space does not descend to a unique operation in the projective Hilbert space, as all non-zero scalar multiples represent the same physical state. While Dirac had noted this ambiguity as early as 1930, subsequent literature often overlooks it, implicitly assuming a uniquely defined "superposed" state for any pair, which is not mathematically justified.

The projective Hilbert space $\mathbb{CP}(\mathcal{H})$ represents pure states as rays rather than individual vectors; a physical state is unaffected by multiplication with a nonzero complex scalar. For any two distinct pure states (non-colinear rays), the set of all their non-trivial linear combinations forms a two-dimensional subspace (a complex projective line). The traditional approach wherein arbitrary linear combinations are normalized to produce new "superposed" states is therefore ambiguous, as this process does not select a unique ray but rather generates the entire line they span.

Geometric Definition of Superposition

The manuscript introduces a new, strictly geometric definition: the superposition of two distinct pure states corresponds to the unique complex projective line on which both states lie in $\mathbb{CP}(\mathcal{H})$. Explicitly, for pure states represented by projective points $x_1$ and $x_2$, their superposition is the full set of rays in the two-dimensional Hilbert subspace spanned by their representatives, or equivalently, the projective line passing through $x_1$ and $x_2$. This definition, rooted in the axioms of projective geometry, removes the arbitrariness inherent in the choice of coefficients in Hilbert space linear combinations and aligns with the foundational geometrical properties of $\mathbb{CP}(\mathcal{H})$.

This construction generalizes to any projective space, abstracting away from specific Hilbert space features, and leads to a binary operation (modulo the diagonal) on projective points with values in the Grassmannian of projective lines. The definition thereby establishes superposition as a structural property of the geometric lattice of subspaces, not tied to expansion coefficients or normalization procedures.

Superposition Versus Basis Expansion

A significant portion of the paper clarifies the distinction between superposition and basis expansion. The expansion of a unit vector $\psi$ in an orthonormal basis yields unique expansion coefficients, but the associated state in projective geometry is not uniquely determined by the probabilistic weights $|\langle e_\alpha, \psi \rangle|^2$ alone due to the non-uniqueness of phase relations. Consequently, the identification of a pure state as "the superposition" of basis states is misleading unless restricted to a particular projective line.

Furthermore, after quantum measurement or in quantum statistical scenarios, replacing a pure state with a mixed state characterized by the coefficients' moduli squared corresponds to a statistical mixture on $\mathbb{CP}(\mathcal{H})$, not a geometrical superposition per the new definition. This emphasizes that mixing and superposing are mathematically and physically distinct.

The manuscript illustrates this distinction in the case of a two-dimensional system (for example, spin $\mathbb{CP}(\mathcal{H})$0), highlighting the continuum of distinct quantum states that yield identical measurement probabilities in a fixed basis—further emphasizing the limitation of associating "superposition" with a particular vector or statistical mixture.

Theoretical and Practical Implications

This geometric redefinition has significant implications both for the conceptual foundation of quantum theory and for the interpretation of quantum phenomena:

• Mathematical Foundation: Anchoring superposition in projective geometry fully respects the physical indistinguishability of global complex phases and clarifies the structure of possible quantum states generated by superposition.
• Interpretational Clarity: The new definition prevents common confusions, such as reifying abstract mathematical constructions (basis expansions or mixtures) into ontological claims about quantum systems, thereby refining the language and reasoning available to quantum foundations.
• Compatibility with Quantum Geometry: The construction is immediately compatible with geometric reformulations of quantum mechanics [see also (Sontz, 17 May 2026)], suggesting that theorems of projective geometry have direct analogs in quantum theory. Extension to complex projective Grassmannians and their possible physical significance is identified as a promising direction.
• Generalization: The approach generalizes to any cardinality, associating the superposition of any collection of (sub)spaces with their smallest projective span.

Conclusion

The manuscript provides a precise, geometric definition of quantum superposition, resolving foundational ambiguities arising from Hilbert space-based descriptions. By defining superposition as the unique projective line determined by two states, it realigns the mathematical formalism with established geometric axioms and clarifies its physical and interpretational consequences. The approach facilitates an overview between quantum theory and complex projective geometry, inviting further exploration into geometric formulations of quantum dynamics, the role of Grassmannians, and the connection between projective properties and quantum probabilities. Such developments have the potential to influence both the conceptual understanding and the mathematical toolkit of quantum foundations and quantum information theory.