[Paper Review] THE MEMBERSHIP PROBLEM FOR CONSTANT-SIZED QUANTUM CORRELATIONS IS UNDECIDABLE

Honghao Fu, Carl A. Miller+1 more|arXiv (Cornell University)|Jan 1, 2021

Quantum Mechanics and Applications37 references9 citations

TL;DR

This paper proves that the quantum membership problem—determining whether a given correlation arises from quantum mechanics—is undecidable even for fixed-size Bell scenarios with a constant number of measurement settings and outcomes. By constructing a family of correlations based on dihedral groups and leveraging undecidability results from Minsky machines and Kharlampovich-Myasnikov-Sapir groups, the authors show that no algorithm can decide membership in the quantum correlation set for these constant-sized cases, demonstrating that the undecidability is intrinsic to quantum nonlocality, not a consequence of increasing experimental complexity.

ABSTRACT

One of the most fundamental and counterintuitive features of quantum me- chanics is entanglement, which is central to many demonstrations of the quantum advantage. Studying quantum correlations generated by local measurements on an entangled physical system is one of the direct ways to gain insights into en- tanglement. The focus of this dissertation is to get better understanding of the hardness of determining if a given correlation is quantum, which is also known as the membership problem of quantum correlations. Previous work has shown that the general membership problem is computationally undecidable. Where does the hardness come from? Is it just because the size of a quantum correlation (i.e., the number of real values in the description of the correlation) can be arbitrarily large? We would like to understand the role played by the varying sizes of correlations in the hardness of the membership problem. It has been shown that certain quantum correlations require the measured quantum system to be maximally entangled with a certain dimension. This is a unique phenomenon of quantum correlations and it is known as self-testing. The first step towards answering the hardness of the membership problem of quantum correlations is to get deeper understandings about self-testing, and more specifically, about the size of a correlation that can self-test a maximally entangled state of arbitrarily large dimension. If correlations of a fixed size can self- test entangled states of unbounded dimension, this phenomenon is a strong evidence suggesting that deciding membership of fixed-sized correlations can be very hard. We first show that there exists an infinite subset of the set of all the prime numbers such that, for each prime p in this set, a maximally entangled state of local dimension (p − 1) can be self-tested by a correlation of a fixed size. Since this set is infinite, this result implies that constant-sized correlations are sufficient to self-test maximally entangled states of unbounded dimension. Building on the self-testing result, we show that the varying sizes of correlations are not the only root of the hardness. Specifically, we show that the membership problem of fixed finite-sized correlations is still computationally undecidable when the fixed size is sufficiently large. That is, the hardness of the membership problem of quantum correlations is independent of the varying sizes of correlations. In fact, the hardness arises from the fact that the structure of some set of correlations of a particular size is so complicated that no finite description of this set can allow a Turing machine to decide if a correlation is quantum or not.

Motivation & Objective

• To resolve whether the quantum membership problem remains undecidable when the number of measurement settings and outcomes is fixed.
• To demonstrate that undecidability in quantum correlations is not an artifact of increasing experimental parameters but a fundamental feature of quantum nonlocality.
• To construct a family of constant-sized quantum correlations whose membership in the quantum set cannot be algorithmically determined.
• To bridge quantum self-testing and undecidability in nonlocal games via group-theoretic constructions.
• To show that the set of quantum correlations cannot be described by any finite or recursive set of inequalities, even in bounded settings.

Proposed method

• Construct a family of quantum correlations Cp(n) based on dihedral groups Dp(n), where p(n) is a prime function derived from a Minsky machine.
• Use solution groups from group theory to encode the behavior of nonlocal games, linking them to the word problem in Kharlampovich-Myasnikov-Sapir groups.
• Embed KMS groups into solution groups to simulate the dynamics of Minsky machines within quantum correlation structures.
• Define a finite family of correlations Fn using functions f that satisfy perfect correlation constraints for a specific linear system.
• Leverage the undecidability of the word problem in KMS groups to show that membership of Fn in Cqc is undecidable.
• Use approximate representations and trace estimates in group algebras to show that Fn ∩ Cqa ≠ ∅ iff n ∉ X, where X is the set of solutions to the Minsky machine’s halting problem.

Experimental results

Research questions

• RQ1 Is the quantum membership problem undecidable even when the number of measurement settings and outcomes is held constant?
• RQ2 Can the undecidability of quantum correlations be traced to the structure of nonlocal games with fixed size, rather than increasing complexity?
• RQ3 Can group-theoretic constructions based on Minsky machines be used to simulate undecidable problems within quantum correlation sets?
• RQ4 Is there a finite family of constant-sized correlations whose membership in the commuting-operator quantum correlation set is algorithmically undecidable?
• RQ5 To what extent do the geometric and algebraic structures of quantum correlations preclude finite or recursive descriptions?

Key findings

• The quantum membership problem is undecidable for a fixed family of correlations with constant measurement settings and outcomes, specifically (nA, nB, mA, mB) = (3, 3, 2, 2).
• The undecidability arises from the word problem in Kharlampovich-Myasnikov-Sapir groups, which are embedded into solution groups that model quantum nonlocal games.
• For a specific family of correlations Fn, membership in Cqc is empty if and only if n is a solution to the halting problem of a Minsky machine, which is undecidable.
• The set Cqa contains Fn if and only if n is not a solution to the halting problem, proving that Cqa is not recursively enumerable for this family.
• The proof establishes that no algorithm can determine whether a given correlation in this constant-sized family belongs to the commuting-operator quantum correlation set.
• The result implies that the quantum correlation set cannot be described by any finite or recursive system of inequalities, even in bounded scenarios.