Direct-Sum Quantum Theory

• Direct-Sum Quantum Theory is an organizational framework that decomposes finite-dimensional Hilbert spaces into orthogonal direct-sum components, clarifying quantum measurement contexts and partition logic.
• It enables novel state-recovery protocols by quasi-copying amplitudes and applying outcome-blind operations to probabilistically restore quantum states.
• The framework informs quantum communication complexity and emergent locality by structuring Hamiltonian blocks and enforcing resource lower bounds in parallel computations.

Direct-Sum Quantum Theory is an organizational and conceptual framework in quantum mechanics in which the Hilbert space of a physical system, rather than being analyzed primarily in terms of its subspaces or tensor-product factorizations, is systematically decomposed into orthogonal direct-sum components. This structure underlies several independent threads of modern research, including the logic of measurement contexts, the emergence of effective locality and spacetime structure, new approaches to quantum communication complexity, and operationally nontrivial state-recovery protocols. The direct-sum perspective both generalizes and refines traditional quantum logic and is quantitatively essential in numerous foundational and applied settings (Suzuki et al., 14 Sep 2025, Pollack et al., 2018, Ellerman, 2016, 0807.1267, Touchette, 2014).

1. Direct-Sum Decomposition of Hilbert Spaces

Given a finite-dimensional Hilbert space $\mathcal{H}$, a direct-sum decomposition is a partition of $\mathcal{H}$ into orthogonal subspaces,

$$\mathcal{H} = \bigoplus_{i=1}^r \mathcal{H}_i, \qquad \mathcal{H}_i \perp \mathcal{H}_j \ \text{for } i\ne j, \qquad \sum_{i=1}^r \dim \mathcal{H}_i = \dim\mathcal{H}.$$

Each vector $|\psi\rangle\in\mathcal{H}$ has a unique decomposition as $|\psi\rangle=\sum_{i=1}^r P_i|\psi\rangle$ with $P_i$ the orthogonal projector onto $\mathcal{H}_i$ (Ellerman, 2016, Pollack et al., 2018). The set of all such direct-sum decompositions (DSDs) of $\mathcal{H}$ forms a meet-semilattice under the refinement order: $\sigma\preceq\pi$ if every block of $\pi$ is contained in some block of $\sigma$.

In contrast with the geometry of subspaces (which forms a lattice), the DSD structure is specifically tailored to encode measurement contexts, equivalence relations, and partition logic. Notably, while not every finite dimension $N$ factorizes as a tensor product (due to number-theoretic constraints), direct-sum decompositions always exist for any $N$ and any observable with an appropriate spectral partition (Pollack et al., 2018).

2. Quantum Logic and Measurement in the Direct-Sum Framework

A fundamental organizational shift of Direct-Sum Quantum Theory is the replacement of the traditional subspace lattice logic (Boolean subset logic) with a partition logic based on DSDs. In this dual logic, direct-sum decompositions are categorically dual to subspaces: under the free-vector-space functor, set partitions become direct sums in the associated vector spaces (Ellerman, 2016).

Given a self-adjoint operator $A$ on $\mathcal{H}$, the spectral theorem yields a DSD by eigenspaces: $$\mathcal{H} = \bigoplus_{\lambda\in\mathrm{spec}(A)} V_\lambda$$, each $V_\lambda$ the eigenspace for $\lambda$. Measurement in this framework treats the entire context $\pi_A = \{V_\lambda\}$, and the Born rule assigns probability $p_\lambda= \|P_\lambda\psi\|^2$ for outcome $\lambda$ (Ellerman, 2016).

The DSD logic permits fine-grained description of measurement by arbitrary self-adjoint operators, not merely projective measurements associated with individual subspaces. Atomic DSDs (two-summand) relate directly to traditional rank-1 projections but the full DSD framework natively encodes all spectral measurements.

3. Operational and Informational Phenomena: State Recovery and Erasure

The direct-sum formalism enables state-recovery protocols that are operationally distinct from classical or conventional quantum pathways. In particular, there exist scenarios where a quantum state, after an amplitude-transfer ("quasi-copying") into an orthogonal complement, undergoes measurement on that complement and is then recoverable via a fixed, measurement-outcome-blind operation (Suzuki et al., 14 Sep 2025).

Concretely, for $$\mathcal{H}'_n \otimes \mathrm{span}\{|a\rangle,|a^\perp\rangle\}$$, one prepares an initial state in $\mathcal{H}_n\ (\propto |a\rangle)$, applies a quasi-copying operation $\mathcal{K}$ distributing amplitude into $\mathcal{H}_n$ and $\mathcal{H}_n^\perp$, performs an arbitrary POVM $\{M_\nu\}$ on $\mathcal{H}_n^\perp$, and finally a fixed post-selection onto $\mathcal{H}_n$ via the projector $P$. The main theorem establishes that the original state $\rho$ is probabilistically restored (with probability $\cos^2\phi$), unaffected by the specific outcome $\nu$ of the intermediate measurement.

From the informational perspective, this process erases "which-outcome" data: the mutual information between the measurement outcome $N$ and the input $\rho$, conditioned on successful recovery, vanishes: $I(N; \rho\,|\,M=0) = 0.$ The distribution of $N$ becomes uniform, demonstrating a form of outcome erasure without ever accessing measurement records (Suzuki et al., 14 Sep 2025). This challenges standard intuitions about the necessity of fine-grained outcome dependence in quantum reversibility and side-channel erasure.

4. Locality, Space Emergence, and Hamiltonian Block Structure

Direct-sum decomposition is also central to foundational questions in quantum field theory and many-body systems, notably where tensor product locality is absent or inapplicable (Pollack et al., 2018). For a Hamiltonian $H$ and a direct-sum $\mathcal{H} = \mathcal{H}_A \oplus \mathcal{H}_B$, one writes

$H = H_A + H_B + V,$

with $H_A = P_A H P_A$, $H_B = P_B H P_B$, and the coupling $V = P_A H P_B + P_B H P_A$.

If $\|V\|\ll\min\{\|H_A\|, \|H_B\|\}$, the system exhibits "direct-sum locality": low-lying states in $\mathcal{H}_A$ or $\mathcal{H}_B$ remain localized under time evolution for times $t \ll 1/\|V\|$. This structure justifies effective superselection of sectors, robustness of emergent spacetime interpretations, and identifies when approximate classicalization of degrees of freedom is viable purely from Hilbert-space considerations.

A finite-dimensional double-well serves as illustrative example: the natural direct-sum split into left and right wells ensures the tunneling spread $\mathbb{T}$ is minimal for ground states, and "scrambled" decompositions rapidly increase delocalization. This speaks to why only specific direct-sum structures, not generic partitions, underlie emergent locality and decoherence (Pollack et al., 2018).

5. Direct-Sum Theorems in Quantum Communication Complexity

Direct-sum theorems in quantum information theory quantify the resource requirements for simultaneously solving multiple independent instances of a computational problem using quantum protocols. In the space of communication complexity, such theorems affirm that—up to lower-order corrections—the quantum (and classical) resources scale at least linearly in the number of parallel instances (0807.1267, Touchette, 2014).

If $Q^{1,\mathrm{pub}}_\varepsilon(f)$ represents the one-way entanglement-assisted quantum communication complexity (worst-case error $\varepsilon$) of a relation $f$, the one-way direct-sum result is

$$Q^{1,\mathrm{pub}}_{\varepsilon+\delta}(f^m) \geq \Omega(m\cdot Q^{1,\mathrm{pub}}_\varepsilon(f)),$$

where $f^m$ denotes $m$ parallel copies of $f$. Proofs rely on message compression, information cost additivity, and substate theorems, showing that information cost (a fully quantum extension of Shannon's mutual information) lower bounds the communication cost for protocols—and that this property is preserved under direct-sum decompositions (0807.1267, Touchette, 2014).

Extensions further cover bounded-round entanglement-assisted protocols, simultaneous message passing, and privacy trade-offs, always depending on the additivity and compressibility properties inherent to the DSD structure of the underlying Hilbert space.

6. Combinatorics, Logic, and the QM/Sets Model

DSDs have a rich internal combinatorics directly parallel to set partitions. For a vector space $V$ of dimension $n$ over $\mathbb{F}_q$, each DSD has a signature given by block sizes $(n_1,\ldots, n_m)$, with $n_1+\cdots+n_m=n$. The number of such DSDs with a given signature is

$$\frac{[n]_q!}{\prod_{k=1}^n a_k!\,([k]_q!)^{a_k}} q^{\frac12(n^2-\sum_k a_k k^2)},$$

where $a_k$ is the number of blocks of size $k$ and $[k]_q!$ the $q$-factorial (Ellerman, 2016).

Summing over all possible $m$-block signatures yields $q$-analogs of the Stirling and Bell numbers, interpolating between classical set combinatorics ($q\to1$) and finite field settings ($q=2$: QM/Sets model).

The QM/Sets model, where $V\cong\mathbb{Z}_2^n$ and pure states are nonzero subsets $S\subset U$, recovers measurement, probability, and state-update rules in a logically transparent form, connecting attributes (observables) to set partitions and their induced DSDs (Ellerman, 2016). It supplies a toy universe in which the logic of direct-sum decompositions underpins all features of quantum measurement—demonstrating both pedagogical and foundational utility.

7. Limitations, Generalizations, and Open Directions

The direct-sum formalism is universally available for finite-dimensional spaces, but crucial applications require further structure:

• Recovery protocols based on direct-sum splitting are inherently probabilistic; deterministic, outcome-blind reversal is precluded by no-cloning and information-disturbance constraints (Suzuki et al., 14 Sep 2025). Success probability is strictly less than unity whenever any nontrivial information is extracted from the system.
• Imposing physical relevance on a DSD (direct-sum locality, measurement context, etc.) demands that off-diagonal Hamiltonian blocks be suitably small or that the decomposition aligns with robust pointer subspaces.
• In the context of communication complexity, while DSD-based direct-sum lower bounds are tight in the one-way or SMP models, weak direct-sum statements and privacy tradeoffs are required for multi-round/protocol settings, often introducing sublinear corrections (0807.1267, Touchette, 2014).
• The dual nature of DSD logic in relation to projection-operator logic remains under active investigation, both for foundational quantum logic and for generalized measurement scenarios incorporating mixed observables and information flows.

The aggregate significance is that Direct-Sum Quantum Theory generalizes and enriches the structural and operational resources of quantum physics, yielding deep insights into measurement, locality, reversibility, combinatorics, and quantum information processing across a spectrum of models and applications (Suzuki et al., 14 Sep 2025, Pollack et al., 2018, Ellerman, 2016, 0807.1267, Touchette, 2014).