We study the short-time behavior of the probability distribution $\mathcal{P}(H,t)$ of the surface height $h(x=0,t)=H$ in the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimension. The process starts from a stationary interface: $h(x,t=0)$ is given by a realization of two-sided Brownian motion constrained by $h(0,0)=0$. We find a singularity of the large deviation function of $H$ at a critical value $H=H_c$. The singularity has the character of a second-order phase transition. It reflects spontaneous breaking of the reflection symmetry $x\leftrightarrow -x$ of optimal paths $h(x,t)$ predicted by the weak-noise theory of the KPZ equation. At $\left|H\right|\gg |H_c|$ the corresponding tail of $\mathcal{P}\left(H\right)$ scales as $-ln\mathcal{P}\sim {\left|H\right|}^{3/2}/t^{1/2}$ and agrees, at any $t>0$, with the proper tail of the Baik-Rains distribution, previously observed only at long times. The other tail of $\mathcal{P}$ scales as $-ln\mathcal{P}\sim {\left|H\right|}^{5/2}/t^{1/2}$ and coincides with the corresponding tail for the sharp-wedge initial condition.

Hardy and Unruh constructed a family of non-maximally entangled states of pairs of particles giving rise to correlations that cannot be accounted for with a local hidden-variable theory. Rather than pointing to violations of some Bell inequality, however, they pointed to apparent clashes with the basic rules of logic. Specifically, they constructed these states and the associated measurement settings in such a way that the outcomes satisfy some conditionals but not an additional one entailed by them. Quantum mechanics avoids the broken ‘if …then …’ arrows in such Hardy–Unruh chains, as we call them, because it cannot simultaneously assign truth values to all conditionals involved. Measurements to determine the truth value of some preclude measurements to determine the truth value of others. Hardy–Unruh chains thus nicely illustrate quantum contextuality: which variables do and do not obtain definite values depends on what measurements we decide to perform. Using a framework inspired by Bub and Pitowsky and developed in our book Understanding Quantum Raffles (co-authored with Michael E. Cuffaro), we construct and analyze Hardy–Unruh chains in terms of fictitious bananas mimicking the behavior of spin-$\frac{1}{2}$ particles.

We use Bub's (2016) correlation arrays and Pitowksy's (1989b) correlation polytopes to analyze an experimental setup due to Mermin (1981) for measurements on the singlet state of a pair of spin-$\frac12$ particles. The class of correlations allowed by quantum mechanics in this setup is represented by an elliptope inscribed in a non-signaling cube. The class of correlations allowed by local hidden-variable theories is represented by a tetrahedron inscribed in this elliptope. We extend this analysis to pairs of particles of arbitrary spin. The class of correlations allowed by quantum mechanics is still represented by the elliptope; the subclass of those allowed by local hidden-variable theories by polyhedra with increasing numbers of vertices and facets that get closer and closer to the elliptope. We use these results to advocate for an interpretation of quantum mechanics like Bub's. Probabilities and expectation values are primary in this interpretation. They are determined by inner products of vectors in Hilbert space. Such vectors do not themselves represent what is real in the quantum world. They encode families of probability distributions over values of different sets of observables. As in classical theory, these values ultimately represent what is real in the quantum world. Hilbert space puts constraints on possible combinations of such values, just as Minkowski space-time puts constraints on possible spatio-temporal constellations of events. Illustrating how generic such constraints are, the equation for the elliptope derived in this paper is a general constraint on correlation coefficients that can be found in older literature on statistics and probability theory. Yule (1896) already stated the constraint. De Finetti (1937) already gave it a geometrical interpretation.

We study the short-time behavior of the probability distribution $\mathcal{P}(H,t)$ of the surface height $h(x=0,t)=H$ in the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimension. The process starts from a stationary interface: $h(x,t=0)$ is given by a realization of two-sided Brownian motion constrained by $h(0,0)=0$. We find a singularity of the large deviation function of $H$ at a critical value $H=H_c$. The singularity has the character of a second-order phase transition. It reflects spontaneous breaking of the reflection symmetry $x \leftrightarrow -x$ of optimal paths $h(x,t)$ predicted by the weak-noise theory of the KPZ equation. At $|H|\gg |H_c|$ the corresponding tail of $\mathcal{P}(H)$ scales as $-\ln \mathcal{P} \sim |H|^{3/2}/t^{1/2}$ and agrees, at any $t>0$, with the proper tail of the Baik-Rains distribution, previously observed only at long times. The other tail of $\mathcal{P}$ scales as $-\ln \mathcal{P} \sim |H|^{5/2}/t^{1/2}$ and coincides with the corresponding tail for the sharp-wedge initial condition.