Wormhole and Entanglement (Non-)Detection in the ER=EPR Correspondence

TL;DR

The paper investigates the ER=EPR proposal by proving that wormhole geometry cannot be unambiguously detected by any general-relativistic measurement, mirroring the non-observability of entanglement in quantum mechanics. An explicit setup in the maximally extended AdS-Schwarzschild spacetime shows that a single observer cannot distinguish the presence of a nontrivial ER bridge, due to topological censorship and local isometries. When multiple observers act in concert, they can determine a parameter labeling a particular ER-bridge (via tidal Weyl measurements that map to a geometry label $\alpha$), but this does not realize a projection operator onto the entire family of wormhole geometries. Consequently, ER=EPR remains compatible with linear quantum mechanics, preserving state-independence of observables, while clarifying the precise sense in which entanglement corresponds to spacetime topology. The work motivates further exploration of ER=EPR in non-AdS spacetimes and its implications for firewalls and topology change in quantum gravity.

Abstract

The recently proposed ER=EPR correspondence postulates the existence of wormholes (Einstein-Rosen bridges) between entangled states (such as EPR pairs). Entanglement is famously known to be unobservable in quantum mechanics, in that there exists no observable (or, equivalently, projector) that can accurately pick out whether a generic state is entangled. Many features of the geometry of spacetime, however, are observables, so one might worry that the presence or absence of a wormhole could identify an entangled state in ER=EPR, violating quantum mechanics, specifically, the property of state-independence of observables. In this note, we establish that this cannot occur: there is no measurement in general relativity that unambiguously detects the presence of a generic wormhole geometry. This statement is the ER=EPR dual of the undetectability of entanglement.

Figures (4)

• Figure 1: The maximally extended AdS-Schwarzschild geometry, with Kruskal-Szekeres coordinates $T,X$ and lightcone coordinates $U,V$ indicated. Of course, the singularity actually appears as a hyperbola in $T,X$. This diagram is a conformally-transformed sketch to indicate the general relationship among the coordinates; see Ref. Fidkowski:2003nf for more discussion. Regions I through IV are defined by Eq. (\ref{['eq:KS_coords']}).
• Figure 2: (left) The state $|\psi_0\rangle$, corresponding to a wormhole geometry where the ER bridge intersects the boundary at $T=0$. (right) The family of states $|\psi_\alpha\rangle$, $\alpha>0$, for which the ER bridge intersects the boundary at $T>0$.
• Figure 3: The procedure described in the text for detecting a wormhole. Alice and Bob emerge from the white hole portion of the AdS-Schwarzschild geometry, then meet again inside the black hole.
• Figure 4: Unlike in Fig. \ref{['fig:Alice_Bob_1']} above, here the geometry is shifted to some $|\psi_\alpha\rangle$ for some sufficiently large $\alpha\neq 0$; Alice and Bob hit the singularity before they can meet and are therefore unable to verify the existence of a wormhole.