Separable Ontology in Quantum Theory

• Separable ontology in quantum theory is defined as the rigorous characterization of convex combinations of product states that capture only classical correlations.
• The automorphisms preserving separability include local unitary transformations, partial transpositions, and subsystem permutations, ensuring the invariant nature of separable states.
• These structural insights have practical implications for entanglement detection, optimization of quantum measures, and the robust design of quantum protocols.

A separable ontology for quantum theory concerns the rigorous characterization of the structure, automorphisms, and physical implications of the set of quantum states that can be written as convex combinations of product states. These are the so-called separable states within the full convex set of quantum states on finite or infinite-dimensional Hilbert spaces. A central focus is the mathematical and operational classification of transformations that preserve this set, their structural consequences for quantum information theory, and their role in the foundational interpretation of quantum measurements, entanglement, and locality.

1. Structural Definition of Separable States

In a bipartite system with Hilbert space $\mathcal{H}_m \otimes \mathcal{H}_n$, a separable state is a convex combination of pure product states,

$$\rho = \sum_k p_k\, \rho_k^{(A)} \otimes \rho_k^{(B)},$$

where $\rho_k^{(A)} \in D_m$ and $\rho_k^{(B)} \in D_n$ are density matrices on the respective subsystems, and $p_k \geq 0$, $\sum_k p_k = 1$. The set $S$ of separable states is a convex subset of all density matrices on $\mathcal{H}_m \otimes \mathcal{H}_n$, with extreme points being the pure product states.

Separable states possess an ontological significance: they represent states with only classical correlations between subsystems. Any convex combination of such states remains unentangled, and hence, under a separable ontology, the physical properties may be consistently attributed to the local subsystems.

2. Automorphisms Preserving Separable Ontology

The core mathematical result is the complete characterization of linear automorphisms of Hermitian matrices that preserve the set of separable states (Friedland et al., 2010). Such an automorphism $Y$ must be generated by a combination of the following "natural" operations:

Operation Type	Formal Description	Physical Significance
Local Unitary Transformations	$Y(A \otimes B) = (UAU^*) \otimes (VBV^*)$ for unitary $U$, $V$	Changes of basis in each subsystem (local symmetry), preserving all local structures
Partial Transposes	$$PT_j(A_1 \otimes \cdots \otimes A_k) = A_1 \otimes \cdots \otimes A_j^T \otimes \cdots \otimes A_k$$	Partial transpose in a subsystem, crucial for separability criteria
Permutations of Tensor Factors	$$S(A_1 \otimes \cdots \otimes A_k) = A_{T(1)} \otimes \cdots \otimes A_{T(k)}$$ for some permutation $T$	Swapping indistinguishable subsystems

Combining these, the full automorphism group $G(n_1,\ldots,n_k)$ of separable states is generated. The most general $Y$ preserving $S$ is of the form: $$V(A_1 \otimes \cdots \otimes A_k) = V_1(A_{T(1)}) \otimes \cdots \otimes V_k(A_{T(k)})$$ where each $V_j$ involves a unitary conjugation or transpose, and $T$ is a permutation respecting subsystem dimensions.

This result is encapsulated in Theorem 3 (bipartite) and Theorem 5 (multipartite) of (Friedland et al., 2010).

3. Preservation of Separable Ontology and Extreme Points

These automorphisms preserve both the structure of the convex set $S$ and its extreme points (pure product states). They cannot introduce new entangled states, ensuring that the separable "ontology"—the view that quantum reality can be constructed in terms of unentangled, independently attributed subsystems—is robust under the full class of symmetry operations allowed by quantum theory.

Operationally, this means any linear transformation that leaves all separable states invariant must be built exclusively from local basis changes, partial transpositions, and permutations, and no further "exotic" automorphism exists.

This is significant for tasks such as entanglement detection, as all standard (and most physically relevant) separability criteria are invariant under these symmetries.

4. Physical and Information-Theoretic Implications

Entanglement Detection and Separability Criteria

The invariance of separable states under the above automorphisms provides the foundation for separability tests—such as the Peres criterion, which uses partial transposition—and ensures that entanglement is a basis-independent property; separability does not depend on the specific representation but is an intrinsic feature of the joint density matrix.

Simplification of Optimization Problems

Many entanglement measures, e.g., the relative entropy of entanglement, are defined via optimization over separable states. The symmetry allows one to reduce such optimizations, for instance, by choosing a canonical basis or simplifying the numerical search domains.

Product Numerical Range

Let $T \in M_{mn}$ (the space of $m \times n$ matrices). The product numerical range is

$$W^{\otimes}(T) = \{ \operatorname{Tr}(TZ) : Z \in P_m \otimes P_n \}$$

where $P_m$ and $P_n$ are the sets of pure states on $\mathcal{H}_m$ and $\mathcal{H}_n$. Theorem 8 (and its multipartite generalization, Theorem 9) shows that only automorphisms of the type above preserve the product numerical range. Therefore, the action of these automorphisms not only maintains the set of separable states but also the physical observables (numerical ranges) that capture how the system responds to measurements restricted to product states.

This directly informs the analysis of quantum stability, decoherence, and the operational structure of error correction.

5. Operational Role in Quantum Protocols and Computation

In the context of quantum information protocols, understanding which operations preserve separability is essential for designing processes where entanglement must not be introduced, for example, in secure classical communication over quantum networks. The automorphism group characterized above provides the full machinery for describing quantum operations that maintain local classicality, in both theoretical design and experimental implementation.

Moreover, in quantum computation, protocols relying on pure product inputs (or aiming to prevent entanglement for certain steps) are constrained by this automorphism group; any deviation will potentially introduce entanglement and alter the computational or communication resource landscape.

6. Broader Ontological and Foundational Context

From a foundational perspective, the result reinforces the "separable ontology" as a mathematically precise and physically significant substructure of quantum theory. It affirms that all symmetries and state transformations compatible with separability are among the expected operations—change of local basis, transpose (reflecting time-reversal or antiunitary symmetry), and subsystem permutation—mirroring intuition from the theory of open quantum systems and composite measurements.

This structural rigidity implies that if separability is to play an ontological role in the interpretation of quantum theory (e.g., in discussions surrounding locality or the nature of quantum correlations), then only these natural automorphisms are admissible. Any attempt to generalize beyond this group would inevitably permit entanglement generation, thereby stepping outside the separable ontology.

7. Summary

• The set of separable states is a convex set whose automorphism group is strictly generated by local unitaries, partial transposes, and permutations of identical subsystems.
• These natural automorphisms form the complete linear group preserving separability, and no additional automorphisms exist.
• The structure ensures that bases, measurement orderings, and other representational choices have no impact on separability, grounding both conceptual clarity and operational consistency in quantum information theory.
• The product numerical range is preserved only under these automorphisms, linking the algebraic classification with observable properties of quantum systems.
• These results not only provide a rigorous underpinning for the concept of separable ontology but also supply actionable tools for entanglement testing, quantum protocol design, and foundational analysis of composite quantum systems (Friedland et al., 2010).