The membership problem for constant-sized quantum correlations is undecidable

Abstract

When two spatially separated parties make measurements on an unknown entangled quantum state, what correlations can they achieve? How difficult is it to determine whether a given correlation is a quantum correlation? These questions are central to problems in quantum communication and computation. Previous work has shown that the general membership problem for quantum correlations is computationally undecidable. In the current work we show something stronger: there is a family of constant-sized correlations -that is, correlations for which the number of measurements and number of measurement outcomes are fixed -such that solving the quantum membership problem for this family is computationally impossible. Thus, the undecidability that arises in understanding Bell experiments is not dependent on varying the number of measurements in the experiment. This places strong constraints on the types of descriptions that can be given for quantum correlation sets. Our proof is based on a combination of techniques from quantum self-testing and from undecidability results of the third author for linear system nonlocal games.

Many bounded versions of undecidable problems are NP-hard

“…Moreover, since ρ n (B) ⩾ 0 if and only if P ⊗n (χ n ) ⩾ 0 , it follows BTSP is NP-hard by applying Theorem 2 together with the fact that BMPO is NP-hard. We refer to Appendix C. 6 for more details on the reduction.…”

“…Many problems in quantum information and quantum many-body physics are undecidable. This includes the spectral gap of physical systems [1,2], membership problems for quantum correlations [3][4][5][6][7], properties of tensor networks [8][9][10], measurement occurrence and reachability problems [11,12], and many more [13][14][15][16][17]. In addition, other problems are believed to be undecidable, such as detecting quantum capacity [18], distillability of entanglement [12], or tensor-stable positivity [14].…”

Correlations in the EPR State Observables

“…The above considerations might give the incorrect impression that identifying and classifying the quantum correlations of any state is straightforward. Actually, from a technical point of view this is often not true [ 9 , 33 , 34 , 35 ]. Moreover, determining all states that comply with certain specified correlation traits can be even harder [ 36 ].…”

A Universal Representation for Quantum Commuting Correlations

“…While these sets are known to be convex and satisfy C q (n, k) ⊆ C qc (n, k), a detailed description of their geometry has only been obtained in certain restrictive scenarios (e.g., [6]). Indeed, a recent preprint [5] shows that the problem of determining whether or not a given probability distribution belongs to the set of quantum correlations is undecidable. Consequently, new descriptions of the quantum correlation sets beyond their original definitions are valuable.…”