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Robustness of Hidden-Variable Theories and Matrix Scaling

This paper refutes Aaronson's conjecture that Schrödinger's hidden-variable theory is robust against small perturbations by constructing a counterexample, while simultaneously proposing a modified version of the theory that satisfies robustness and all other desirable axioms.

Original authors: Giulio Malavolta, Harold Nieuwboer, Akshay Ramachandran, Michael Walter

Published 2026-09-15

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CC BY 4.0

Original authors: Giulio Malavolta, Harold Nieuwboer, Akshay Ramachandran, Michael Walter

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Quantum mechanics describes the behavior of the smallest particles in nature with a mathematical precision that has never failed a test, yet it leaves a deep philosophical gap. The theory tells us the probability of finding a particle in a specific place, but it refuses to say where that particle actually was before we looked. This absence of a definite path has troubled physicists for nearly a century. To fill this gap, some have proposed "hidden-variable theories," which suggest that particles do have definite positions and histories, but that these details are simply hidden from our view. The challenge is to construct a theory that keeps these hidden paths while still matching the perfect predictions of standard quantum mechanics. If such a theory could be built, it would not only satisfy our curiosity about reality but also change how we understand the limits of computation, potentially allowing computers to solve problems that are currently impossible.

For years, a specific proposal inspired by the physicist Erwin Schrödinger stood as a leading candidate for such a theory. This approach, often called the Schrödinger theory, attempts to map the history of a quantum system by adjusting a grid of probabilities until the starting and ending conditions match the rules of quantum mechanics. A crucial requirement for this theory to be useful in the real world is that it must be robust. This means that if the input data changes just a tiny bit—like a slight shift in the initial state of a particle—the resulting history should also change only by a tiny bit. If a microscopic change in the input caused a massive, unpredictable shift in the output, the theory would be useless for practical applications or for understanding how nature behaves under slight disturbances.

A team of researchers has now demonstrated that this specific Schrödinger theory is not robust. They constructed a precise mathematical example where two quantum setups are almost identical, differing by a minuscule amount that is far smaller than anything we could measure in a lab. Despite this near-perfect similarity, the theory assigned completely different histories to the two setups. In one case, the probability of a particle taking a certain path was high, while in the nearly identical second case, that same probability dropped to almost zero. The difference between the two outcomes was not a small fluctuation but a fundamental shift, proving that the theory breaks down when faced with even the slightest imperfection in the data. This finding settles a long-standing question in the field, showing that this particular version of the hidden-variable theory cannot be the final answer.

However, the story does not end with a failure. The researchers did not stop at disproving the old idea; they built a new version of the theory that fixes the problem. They introduced a simple constraint, a limit on how much probability can flow through any single path, which acts as a stabilizer. With this new rule in place, the theory became robust. Now, when the input changes slightly, the output changes only slightly, just as a reliable physical theory should. This new "capped" theory retains all the other desirable features of the original proposal, such as symmetry and consistency with quantum rules, while finally achieving the stability required for practical use.

The work also clarified the landscape of what is possible in this field. The researchers showed that while you can have a theory that is robust and symmetric, you cannot have one that is robust, symmetric, and also perfectly preserves the structure of complex, multi-part systems all at once. You must choose which properties to keep. Their new capped theory succeeds by keeping robustness and symmetry, while accepting a slightly weaker form of consistency for complex systems. This provides a complete map of the trade-offs involved, showing exactly which combinations of rules can coexist and which must be sacrificed. The result is a clearer, more honest picture of what a hidden-variable theory can and cannot be, moving the field from speculation to a defined set of possibilities.

Technical Summary: Robustness of Hidden-Variable Theories and Matrix Scaling

Problem Statement
Motivated by quantum foundations and complexity theory, this paper investigates the properties of hidden-variable theories (HVTs) as formalized by Aaronson. Specifically, it addresses the robustness of the "Schrödinger theory," an HVT inspired by Schrödinger's 1931 proposal and formalized by Aaronson. In this theory, given an initial quantum state $\rho$ and a unitary evolution $U$ , a joint probability distribution is assigned by rescaling the entrywise modulus of $U$ (via the Sinkhorn algorithm) such that the column marginals match the Born rule probabilities of $\rho$ and the row marginals match those of the final state $U\rho U^\dagger$ .

Aaronson conjectured that this mapping is robust: that inverse polynomially small perturbations in the inputs ( $\rho$ and $U$ ) lead to inverse polynomially small perturbations in the resulting joint distribution. This property is crucial for complexity-theoretic applications, particularly in characterizing the power of non-collapsing measurements (the complexity class DQP). The paper seeks to determine if this conjecture holds and, if not, whether a modified theory can satisfy robustness alongside other desirable axioms (symmetry, indifference, decomposition invariance, and commutativity).

Methodology
The authors employ a combination of matrix scaling theory, optimization on transportation polytopes, and explicit counterexample construction.

Counterexample Construction: To disprove the robustness of the original Schrödinger theory, the authors construct a specific family of inputs. They define a non-negative matrix $M_\delta$ with a specific zero-pattern and a parameter $\delta$ . They demonstrate that the Sinkhorn scaling of $M_\delta$ is highly sensitive to $\delta$ . This matrix is then "lifted" to a quantum setting by constructing a high-dimensional Hilbert space ( $N = 2n+1$ ) and defining unitaries $U_\delta$ and $U_{\delta'}$ that share the same support but differ by an exponentially small amount ( $\|U_\delta - U_{\delta'}\|_{\max} < 2^{-n}$ ). The initial state $\rho$ is chosen such that the joint distributions assigned by the Schrödinger theory differ significantly (by a constant in total variation distance) despite the inputs being exponentially close.
Optimization and Capacity Constraints: To address the failure of robustness, the authors propose a "Capped Schrödinger Theory." This variant is defined via a constrained optimization problem. Instead of minimizing the Kullback-Leibler (KL) divergence $D_{KL}(P \parallel |U|)$ over the standard transportation polytope (matrices with fixed marginals), the new theory minimizes this divergence over a polytope $C(\rho, U)$ with additional capacity constraints. Specifically, the entries of the joint distribution $P$ are bounded by $P_{ji} \leq C_{ji} = 2|U_{ji}|\sqrt{(U\rho U^\dagger)_{jj} \rho_{ii}}$ .
Proof of Robustness: The robustness of the capped theory is proven using the strong convexity of the KL divergence and the max-flow min-cut theorem. The capacity constraints ensure that the feasible sets for nearby inputs $(\rho, U)$ and $(\tilde{\rho}, \tilde{U})$ are "close" in a way that allows for the projection of solutions. By bounding the distance between the minimizers of the KL divergence on these nearby polytopes, the authors establish that the joint distribution changes only polynomially with respect to input perturbations.
Axiomatic Analysis: The paper systematically analyzes the compatibility of Aaronson's axioms. It proves new impossibility theorems showing that certain combinations of axioms cannot be satisfied simultaneously, thereby refining the landscape of possible HVTs.

Key Contributions and Results

Disproof of Robustness Conjecture: The paper provides a counterexample (Theorem 1.3) showing that the original Schrödinger theory is not robust. The authors construct a pure state and two unitaries that are exponentially close, yet the assigned joint distributions differ by at least an inverse-linear term in at least one entry and by a constant in total variation distance. This failure occurs even when the unitaries share the same minimal blocks (support), indicating the instability is inherent to the scaling procedure itself.
Proposal of Capped Schrödinger Theory: The authors introduce a modified theory (Definition 4.1) that imposes capacity constraints on the joint distribution. They prove (Theorem 1.4) that this new theory satisfies:
- Symmetry: Invariance under basis relabeling.
- Indifference: Preservation of the block structure of $U$ .
- Robustness: Stability under small perturbations.
- Distributional Product Commutativity: A specific form of commutativity for product states.
New Limitations on Axioms: The paper establishes two new impossibility results (Theorems 1.5 and 1.6):
- No HVT can satisfy both Indifference and Product Commutativity (in the strong sense of transition matrices).
- No HVT can satisfy both Indifference and Decomposition Invariance.
  These results, combined with previous work, provide a complete classification of which subsets of Aaronson's axioms can be simultaneously satisfied.

Significance
The paper resolves a main open problem posed by Aaronson regarding the stability of the Schrödinger theory. By demonstrating that the original theory is not robust, it clarifies the limitations of using this specific HVT for complexity-theoretic characterizations involving non-collapsing measurements.

The introduction of the Capped Schrödinger Theory is significant because it salvages the robustness property while retaining symmetry, indifference, and a form of commutativity. This yields a "complete picture" of the trade-offs inherent in hidden-variable theories: one can achieve robustness and indifference simultaneously, but only at the cost of decomposition invariance and strong commutativity. The work thus delineates the precise boundaries of what is possible within the axiomatic framework of HVTs, offering a refined tool for investigating the computational power of quantum systems with non-collapsing measurements.

The paper also highlights a connection between matrix scaling, entropic optimal transport with capacity constraints, and quantum foundations, suggesting that the stability of the Sinkhorn algorithm in the presence of capacity constraints is a key factor in constructing physically plausible hidden-variable models.

Technical Summary: Robustness of Hidden-Variable Theories and Matrix Scaling

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