Quantum reconstruction theorem recovers local information from relations alone

What remains when local statistics disappear?

A quantum state determines probabilities for measurement outcomes. For several qubits, choosing a measurement direction for each qubit produces a joint probability table: a list of how likely the different combinations of outcomes are. Change those directions, and the table changes.

Each table contains both the statistics of individual qubits and information about how their outcomes depend on one another. Lin and Jin ask whether retaining only the second kind of information, across measurement settings, can still identify the underlying quantum state.

Their theoretical answer is yes for three or more qubits, provided the state cannot be written as a tensor product across any division into two nonempty groups. This condition, called irreducibility in the paper, rules out independent blocks. It is a condition about product factorization, not simply another name for entanglement.

The work, titled “Quantum State Reconstruction from Sklar-Type Relational Data,” was submitted to arXiv on November 8, 2026. The supplied record identifies it as a preprint and does not establish peer-reviewed publication.

Defining dependence without the margins

The starting point is a perspective borrowed from Sklar’s theorem in classical probability: separate the statistics of individual variables, called marginals, from their dependence structure.

For the discrete probability tables used here, the authors treat tables as having the same dependence when one can be obtained from another by multiplying probabilities by positive factors associated with individual local outcomes, then normalizing. This operation changes local statistics while preserving the particular dependence structure under study.

For two binary outcomes, that structure is completely captured by an odds ratio. It is the product of the probabilities for the two matching-outcome combinations divided by the product for the two opposite-outcome combinations. Multiplying rows and columns by positive factors leaves this ratio unchanged.

For more qubits, one number is insufficient. The authors instead use coefficients in an expansion of the logarithm of the joint probabilities. Terms involving two or more variables collectively characterize the retained dependence structure. These are not merely the usual pairwise correlations.

Why quantum consistency restores information

For an individual classical table, dependence data alone do not determine the discarded marginals. The additional constraint comes from considering many measurement contexts together: every table must arise from the same quantum density operator, the mathematical representation of the state.

The proof develops this idea through conditioning. In a three-qubit system, fix a measurement direction and an outcome for one qubit. The conditional statistics of the other two qubits can be represented by a two-qubit density operator. Crucially, the dependence data needed for this conditional system can be obtained from the original dependence data without supplying the discarded marginal probabilities.

The authors then use a two-qubit reconstruction result established in earlier work. For a non-product two-qubit state, dependence data across all local contexts determine the state up to a simultaneous spin-flip ambiguity: two candidate states can retain identical odds ratios.

In the three-qubit argument, compatibility across different conditioning choices removes that residual ambiguity. Compatibility across measurement directions also fixes the probabilities of the conditioning outcomes—the local information that had been discarded.

The authors extend this mechanism recursively. Their general theorem gives unique reconstruction for irreducible states of three or more qubits from dependence data across all local measurement contexts.

Exact mathematics, not yet practical tomography

Zero probabilities require special care because logarithms and odds ratios can become undefined. The paper uses only regular contexts, where every table entry is strictly positive. It shows that these contexts form an open, dense, full-measure set, and that suitable algebraic identities extend by continuity. The argument does not require assigning dependence values to undefined tables or using the locations of their zeros.

The larger practical limitation is the all-context assumption. The paper does not demonstrate an experimentally usable reconstruction scheme with a limited set of measurement settings, statistical-error tolerance and feasible computation. Those remain directions for further development.

The authors suggest that invariance under local positive reweighting might eventually help with unknown relative detection efficiencies. That is a proposed advantage, not an experimentally established result. Whether an analogous reconstruction works for continuous-variable systems is also left open.

The central finding is therefore a statement about information: under the theorem’s conditions, local statistics removed separately from measurement tables are still constrained by the requirement that all those tables describe one quantum state.