Picturing general quantum subsystems

We extend the usual process-theoretic view on locality and causality in subsystems (based on the tensor product case) to general quantum systems (i.e.\ possibly non-factor, finite-dimensional von Neumann algebras). To do so, we introduce a primitive notion of splitting maps within dagger symmetric monoidal categories. Splitting maps give rise to subsystems that admit comparison via a preorder called comprehension, and support an adaptation of the usual categorical trace. We show that the comprehension preorder precisely captures the inclusion partial order between von Neumann algebras, and that the splitting map trace captures the natural notion of von Neumann algebra trace. As a consequence of the development of these diagrammatic tools, we prove that the known equivalence between semi-causality and semi-localisability for factor subsystems extends to all (including non-factor) subsystems.

💡 Research Summary

The paper “Picturing General Quantum Subsystems” tackles a fundamental limitation in the standard categorical treatment of quantum subsystems, which traditionally relies on tensor‑product factorisation of Hilbert spaces and thus on factor (i.e. simple) von Neumann algebras. Many physically relevant situations—superselection rules, direct‑sum decompositions, and quantum‑gravity models with superposed geometries—require subsystems that are described by non‑factor (non‑simple) finite‑dimensional von Neumann algebras. The authors set out to develop a diagrammatic, process‑theoretic framework that can capture such general subsystems while remaining faithful to the algebraic structure.

The core technical contribution is the introduction of splitting maps within a dagger symmetric monoidal category. A splitting map χ : H → H_L ⊗ H_R is required only to be an isometry (χ†χ = 1_H), not a unitary. This relaxation allows χ to embed a Hilbert space that may be a direct sum into a tensor product, thereby modelling both tensor‑product and direct‑sum decompositions uniformly. The authors give concrete examples: the usual unitary factorisation, and an embedding for a direct‑sum space H = H₁ ⊕ H₂ that maps |x₁⟩⊕|x₂⟩ to |x₁⟩⊗|∅₁⟩ + |∅₂⟩⊗|x₂⟩. The image projector π_χ = χχ† plays a central role in the subsequent constructions.

Using a splitting map, the notion of χ‑locality is defined: an operator A ∈ L(H) is χ‑local if there exists a “on‑site” operator Ā ∈ L(H_L) such that A = χ†(Ā ⊗ 1_R)χ. This captures precisely the idea that A acts only on the left factor of the decomposition induced by χ. The set of χ‑local operators, denoted loc(χ), forms a sub‑algebra that coincides with the usual left‑factor algebra when χ is unitary.

The authors then introduce a preorder of comprehension on splitting maps: χ₁ ≤ χ₂ iff every χ₁‑local operator is also χ₂‑local. This preorder mirrors the inclusion order of von Neumann subalgebras. They prove that the comprehension preorder is in fact a partial order and that it is order‑isomorphic to the lattice of finite‑dimensional von Neumann subalgebras of L(H). In categorical terms, the map sending a splitting map to its associated sub‑algebra is a monotone equivalence of preorders.

A major achievement is the definition of a splitting‑map trace. For a von Neumann algebra A ⊆ L(H) with commutant B = A′, the trace Tr_B : L(H) → ⊕_i L(H_i^L) is defined by summing the ordinary partial traces over the right factors of each atomic central projector π_i of Z(A):
 Tr_B(X) = ∑i Tr{H_i^R}(π_i X π_i).
The authors show that this trace coincides exactly with the standard von Neumann trace on A, establishing that the diagrammatic trace is faithful to the algebraic one.

With these tools, the paper revisits the well‑known equivalence for factor subsystems: semi‑causality (no‑signalling from one part to the other) is equivalent to semi‑localisability (the channel can be written as a composition of a local operation on one side followed by a channel that ignores the other side). Previously this equivalence was proved only for factor algebras. By exploiting splitting maps and the associated trace, the authors extend the proof to arbitrary finite‑dimensional von Neumann algebras, i.e., to all quantum subsystems, including those arising from direct sums or superselection sectors. The result demonstrates that non‑signalling can be expressed purely in terms of compositional structure, even when the subsystems lack a simple tensor‑product factorisation.

In summary, the paper delivers:

1. A novel diagrammatic primitive—splitting maps—that uniformly represents both tensor‑product and direct‑sum decompositions of quantum systems.
2. A rigorous correspondence between splitting‑map induced subsystems and von Neumann subalgebras, including an order‑preserving equivalence of preorders.
3. A trace construction on splitting maps that reproduces the standard von Neumann trace.
4. An extension of the semi‑causality ↔ semi‑localisability equivalence from factor to general subsystems.

These contributions provide a powerful, theory‑independent language for reasoning about quantum subsystems, opening avenues for applying categorical quantum mechanics techniques to settings with superselection rules, mixed causal structures, and quantum‑gravity‑inspired models where the traditional tensor‑product picture is insufficient.