Superluminal signalling witness for quantum state reduction

Abstract

Models for quantum state reduction address the quantum measurement problem by suggesting weak modifications to Schrödinger's equation that have no observable effect at microscopic scales, but dominate the dynamics of macroscopic objects. Enforcing linearity of the master equation for such models has long been used as a way of ensuring that modifications to Schrödinger's equation do not introduce a possibility for superluminal signalling. In large classes of quantum state reduction models, however, and in particular in models employing correlated noise, formulating a master equation for the quantum state is prohibitively difficult or impossible. Here, we formulate a witness for superluminal signalling that is applicable to generic quantum state reduction models, including those involving correlated as well as uncorrelated noise. Surprisingly, application of the witness to known models described by linear master equations shows that these may still admit superluminal signalling, unless a particular locality condition is obeyed. In contrast, we show that the witness introduced here provides a necessary and sufficient condition for excluding superluminal signals under all circumstances. We further apply the witness to several models driven by physical, correlated noise, where linear master equations are not analytically obtainable, and find that they allow for superluminal signalling. We suggest how specific correlated-noise models may be able to avoid it, and that the witness introduced here provides a stringent guide to constructing such models.

Figures (2)

• Figure 1: The quantity $\mathbb{E}_\xi\left[ \left| \alpha_{j=0}(\xi_t,t) \right|^2 \right]$ as a function of time for various DQSR models, evaluated using the dynamics specified in Eq. \ref{['SUVdynamics']}. Any deviations from the initial value signify a non-zero value for the witness of Eq. \ref{['martingale']}, and thus a possibility for superluminal signalling. The evolutions in the main panel all start from $\alpha(0)=\cos(\pi/6)$. A constant value of the witness indicates that the possibility of SLS is precluded. The green lines labeled $\xi_0$ show the limit of zero correlation time encountered in CSL as well as SUV models in the white-noise limit. The red and blue lines labeled $\xi_\infty$ indicate the opposite limit of infinitely correlated (constant) noise in the two-state SUV model, with noise values sampled either from the steady state of an Ornstein-Uhlenbeck (OU) process, or from spherical Brownian Motion (SBM). The left inset shows evolution with the same value for $G/J$, but for an initial state with $\alpha(0)=\cos(\pi/8)$. Comparing the main panel and the left inset shows that OU noise necessarily causes SLS even for long times, because for fixed ratio $G/J$ the witness does not return to its initial value for all possible initial conditions. In contrast, SBM noise does allow SLS to be avoided at long times. However, a non-constant value of the witness, and hence SLS, cannot be avoided at short times and constant noise values. The grey line in the right inset shows that a model with purely stochastic state evolution ($J=0$) will violate the no-signalling condition at all times. All expectation values shown are numerically averaged over $10^6$ implementations of the noise.
• Figure 2: The quantity $\mathbb{E}_\xi\left[ \left| \alpha_{j=0}(\xi_t,t) \right|^2 \right]$ as a function of time for the SUV model with correlated SBM noise. Any deviations from the initial value signify a non-zero value for the witness of Eq. \ref{['martingale']}, and thus a possibility for superluminal signalling. The lowest (blue) curve is for correlation time $J\tau=0.05$, the middle (green) one for $J\tau=0.2$, and the top (orange) line for $J\tau=1$. For each evolution, the value of $J/G$ was adjusted to ensure the emergence of Born's rule at long times. The inset shows that the time $Jt_{\text{max}}$ at which the maximum deviation of the witness from its initial value occurs, shifts to later times for larger correlation times. Despite this, it always stays well below the collapse time $Jt_{\text{coll}}$, defined as the characteristic decay time of the off-diagonal expectation value $\mathbb{E}_\xi\left[ \alpha_{j=0}(\xi_t,t) \alpha_{j=1}(\xi_t,t) \right]$. All expectation values shown are numerically averaged over $10^5$ implementations of the noise.