Carl Miller

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 quantum? This question is 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. Intuitively, our result means that the undecidability that arises in understanding Bell experiments is innate, and 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.

Applications of randomness such as private key generation and public randomness beacons require small blocks of certified random bits on demand. Device-independent quantum random number generators can produce such random bits, but existing quantum-proof protocols and loophole-free implementations suffer from high latency, requiring many hours to produce any random bits. Here we develop a broadly applicable framework, quantum probability estimation, for yielding efficient quantum-proof protocols. The framework is general and encompasses methods from previous works [Miller and Shi, SIAM Journal on Computing 46, 1304 (2017); Arnon-Friedman et al., Nature Communications 9, 459 (2018)]. Quantum probability estimation can adapt to changing experimental conditions, allows stopping the experiment as soon as the prespecified randomness goal is achieved, and can tolerate imperfect knowledge of the input distribution. Moreover, we demonstrate device-independent quantum randomness generation from a loophole-free Bell test with quantum probability estimation, obtaining multiple blocks of 512 random bits with an average experiment time of less than 5 minutes per block and with certified error bounded by $2^{-64}\approx 5.42\times 10^{-20}$.

Certifying the existence of entanglement between two parties is a fundamental problem in quantum information science. In this work, we develop a protocol for verifying that two parties located at specified positions share an entangled quantum state. We accomplish this by embedding the CHSH game in a quantum position verification protocol. This provides a form of entanglement testing that not only ensures that provers passing the protocol share entanglement, but that they are also located where they claim to be. This prevents parties from passing the verification test by simply forwarding the input of the verification protocol to other parties that share entanglement. The protocol has low requirements on the quantum computational abilities of honest provers---namely, it only requires the honest provers to manipulate two qubits each. It achieves security against adversaries located at incorrect positions that share at most a logarithmic amount of quantum memory with respect to the size of the classical input.

Trapdoor claw-free functions (TCFs) are immensely valuable in cryptographic interactions between a classical client and a quantum server. Typically, a protocol has the quantum server prepare a superposition of two-bit strings of a claw and then measure it using Pauli-X or Z measurements. In this paper, we demonstrate a new technique that uses the entire range of qubit measurements from the XY-plane. We show the advantage of this approach in two applications. First, building on (Brakerski et al. 2018, Kalai et al. 2022), we show an optimized two-round proof of quantumness whose security can be expressed directly in terms of the hardness of the LWE (learning with errors) problem. Second, we construct a one-round protocol for blind remote preparation of an arbitrary state on the XY-plane up to a Pauli-Z correction.