Assessing the dynamical assumptions in Tsirelson inequality tests of non-classicality in harmonic oscillators

Abstract

"Macrorealism" posits that a system possesses definite properties at all times and that we can discover these properties, in principle, without disturbing the system's subsequent behaviour. The Leggett-Garg inequalities are derived under these assumptions and are readily violated by standard quantum mechanics, thereby providing a scheme to test whether demonstrably macroscopic systems can exhibit quantum coherence. Unfortunately, Leggett-Garg tests suffer from the difficult to avoid clumsiness loophole - the difficulty of proving that sequential measurements have not inadvertently disturbed the system. The recently uncovered Tsirelson inequality is derived from the simple dynamical assumption of uniform precession, obeyed by many classical systems, and requires only single-time measurements. However, Tsirelson inequality violations could be explained by a macrorealistic system that merely breaks the dynamical assumption, rather than genuine quantum behaviour. By carrying out a quantum-mechanical analysis of the Tsirelson inequality in the harmonic oscillator, we develop a protocol to rule out this possibility by assessing generalised conditions of uniform precession. We show that various measures of uniform precession, some of which are related to Leggett-Garg quantities, are satisfied well enough that the presence of quantum-mechanical interference terms must be implied. We derive several incidental mathematical results relating to violating states of Tsirelson's inequality, concerning dwell time, crossing number and probability currents, and also consider a group theoretic analysis of the Tsirelson operator.

Figures (10)

• Figure 1: A classical particle oscillating in a quadratic potential is one example of a system undergoing uniform precession. The dynamical assumption is that $Q$ cannot have the same sign at all three times, irrespective of where in its trajectory the particle starts at time $t=0$.
• Figure 2: Classical phase-space depiction of the system's dynamics for three possible measurement times. If the system is initialised in the green region for one of the experimental runs, then it will have a probability of $2/3$ of being found on the right side of the origin at a randomly-chosen measurement time $t\in\{0,T/3,2T/3\}$. Conversely, the probability will be $1/3$ if the particle begins in a blue region. As such, the ensemble average, a convex sum of these probabilities, is bounded between $1/3$ and $2/3$. This is equivalent to Eq. \ref{['The Tsirelson inequality!']} up to a scale factor. (Adapted from Zaw et al.PhysRevA.106.032222.)
• Figure 3: Unlike the blue classical trajectory, which follows the expected sinusoidal dynamics, the erratic red trajectories, still macrorealistic, either dwell for too long on one side of the origin or oscillate too rapidly, and will violate the Tsirelson inequality since they are measured on the same side three times in a row.
• Figure 4: Representation of all states in the subspace described by Eq. \ref{['convenient subspace']}, where the axes represent the (real) coefficients. (a) A colour map is used to represent the value of the Tsirelson quantity when each of these states is taken to be the initial state. The Zaw–Scarani state, labelled $|\Psi_Z\rangle$, has the greatest violation in this subspace and lies in the centre of the upper-bound-violating region. (b) Only the violating (non-classical) regions are highlighted, showing that each upper violation is paired with a lower violation.
• Figure 5: The Tsirelson operator in the energy eigenbasis with any finite energy cut-off can be interpreted as the adjacency matrix of a simple weighted graph. This graph is illustrated with a cut-off at $n=29$, where the nodes are energy eigenstates and the edges represent non-zero matrix elements of the Tsirelson operator. The colour represents the value of the matrix element.