Device-Independent Tests of Classical and Quantum Dimensions

We address the problem of testing the dimensionality of classical and quantum systems in a “black-box” scenario. We develop a general formalism for tackling this problem. This allows us to derive lower bounds on the classical dimension necessary to reproduce given measurement data. Furthermore, we generalize the concept of quantum dimension witnesses to arbitrary quantum systems, allowing one to place a lower bound on the Hilbert space dimension necessary to reproduce certain data. Illustrating these ideas, we provide simple examples of classical and quantum dimension witnesses.

Figure 1

Device-independent test of classical or quantum dimensionality. Our scenario features two black boxes: a state preparator and a measurement device.

Figure 2

Schematic representation of the sets of experiments achievable from classical and quantum states of given dimensions for case study 1. The set of experiments attainable from classical bits, forms the polytope ${\mathbb{P}}_{3,2}^2$ (blue region). The inequality $I_3\le 3$ (solid line), a facet of this polytope, is a “tight two-dimensional classical witness.” The set of experiments attainable from two-dimensional quantum states, i.e., qubits (green and blue regions), is strictly larger. The inequality $J_3\le \frac{3\pi }{2}$ (dashed curve) is a qubit witness; it cannot be violated by performing measurements on qubits: qutrits are required. The set of all possible experiments (blue, green, and red regions) forms the polytope ${\mathbb{P}}_{3,2}$; any point in it can be reproduced with a trit or a qutrit.