Geometric phases distinguish entangled states in wormhole quantum mechanics

We establish a relation between entanglement in simple quantum mechanical qubit systems and in wormhole physics as considered in the context of the $\mathrm{AdS}/\mathrm{CFT}$ correspondence. We show that in both cases, states with the same entanglement structure, indistinguishable by any local measurement, nevertheless are characterized by a different Berry phase. This feature is experimentally accessible in coupled qubit systems where states with different Berry phase are related by unitary transformations. In the wormhole case, these transformations are identified with a time evolution of one of the two throats.

Figure 1

Kruskal diagram of the eternal Schwarzschild black hole in AdS with time in the vertical and spatial coordinates in the horizontal direction. Vertical lines denote the left ($L$) and right ($R$) asymptotic boundaries where the CFTs are defined. The jagged lines are the singularities and diagonal lines represent the black hole horizon ($H$). The blue line corresponds to the wormhole dual to the original TFD state while the red one corresponds to a time-shifted wormhole. Arrows indicate the directions of time in the boundary theories.

Figure 2

Schematic representation of a wormhole corresponding to the colored lines in Fig. 1. The times $t_L$ and $t_R$ of the left and right regions are identified at the interface $I$. In the bulk a different relation is used that accounts for the sign flip of the timelike Killing vector at the horizon $H$. The Berry phase is accumulated along the closed loop printed in green.