Some Ulam's reconstruction problems for quantum states

TL;DR

This paper explores the quantum analog of Ulam's reconstruction problem, providing necessary conditions for reconstructing multipartite quantum states from partial information, with applications to graph states and quantum error correction.

Contribution

It introduces new criteria for the existence of joint quantum states from marginals, especially when subsystem labels are unknown, extending the classical Ulam's conjecture to quantum states.

Findings

Necessary conditions for quantum state reconstruction from marginals.

Constraints on quantum states derived from stabilizer elements and polynomial invariants.

An answer to Ulam's reconstruction problem for generic quantum states.

Abstract

Provided a complete set of putative $k$-body reductions of a multipartite quantum state, can one determine if a joint state exists? We derive necessary conditions for this to be true. In contrast to what is known as the quantum marginal problem, we consider a setting where the labeling of the subsystems is unknown. The problem can be seen in analogy to Ulam's reconstruction conjecture in graph theory. The conjecture - still unsolved - claims that every graph on at least three vertices can uniquely be reconstructed from the set of its vertex-deleted subgraphs. When considering quantum states, we demonstrate that the non-existence of joint states can, in some cases, already be inferred from a set of marginals having the size of just more than half of the parties. We apply these methods to graph states, where many constraints can be evaluated by knowing the number of stabilizer elements of certain weights that appear in the reductions. This perspective links with constraints that were derived in the context of quantum error-correcting codes and polynomial invariants. Some of these constraints can be interpreted as monogamy-like relations that limit the correlations arising from quantum states. Lastly, we provide an answer to Ulam's reconstruction problem for generic quantum states.