Wormhole and entanglement (non-)detection in the ER=EPR correspondence
JHE
Wormhole and entanglement (non-)detection in the
Pasadena 0
U.S.A. 0
Ning Bao 0 1 2
Jason Pollack 0 1
Grant N. Remmen 0 1
0 Pasadena , CA 91125 , U.S.A
1 Walter Burke Institute for Theoretical Physics, California Institute of Technology
2 Institute for Quantum Information and Matter, California Institute of Technology
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, speci cally, 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.
Black Holes; Gauge-gravity correspondence; AdS-CFT Correspondence
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HJEP1(205)6
1 Introduction
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1
Introduction
quantum mechanics. In a recent paper, Marolf and Polchinski [14] have pointed out that
this nonlinearity cannot be \hidden"; if it is strong enough to remove a
rewall from a
generic state, it must lead to violations of the Born Rule visible from outside the horizon.
In this paper, we consider a di erent idea inspired by the
rewall paradox, the
ER=EPR correspondence [15], which asserts the existence of an exact duality between
Einstein-Podolsky-Rosen (EPR) pairs, i.e., entangled qubits, and Einstein-Rosen (ER)
bridges [16{18], i.e., nontraversable wormholes. This duality is supposed to be contained
within quantum gravity, which is in itself meant to be a bona
de quantum mechanical
theory in the standard sense. The ER=EPR proposal is radical, but it is not obviously
excluded by either theory or observation, and indeed has passed a number of nontrivial
checks [19{23]; if true, it has the potential to relate previously unconnected statements
about entanglement and general relativity in a manner reminiscent of the AdS/CFT
correspondence [24{26]. In a previous paper [27], we pointed out that in ER=EPR the no-cloning
{ 1 {
theorem is dual to the general relativistic no-go theorem for topology change [28, 29];
violation on either side of the duality, given an ER bridge (two-sided black hole), would lead
to causality violation and wormhole traversability.
In light of the result of ref. [14], one might be worried that ER=EPR is in danger. It is
well-known that entanglement is not an observable, in the sense we will make precise below;
we cannot look at two spins and determine whether they are in an arbitrary, unspeci ed
entangled state with one another. Yet ER=EPR implies that the two spins are connected
by a wormhole, so that the geometry of spacetime di ers according to whether or not they
are entangled. If this di erence in geometry could be observed, entanglement would become
a (necessarily nonlinear) observable as well and the laws of quantum mechanics would be
violated, contrary to the assumption that the latter are obeyed by quantum gravity.
In this paper, we show that ER=EPR does not have this issue. Unlike the modi cations
to quantum mechanics considered by ref. [14], wormhole geometry can be hidden. In
particular, we show that in general relativity no measurement can detect whether the
interior of a black hole has a wormhole geometry. More precisely, observers can check for
the presence or absence of speci c ER bridge con gurations, but there is no projection
operator (i.e., observable) onto the entire family of wormhole geometries, just as (and, in
ER=EPR, for the same reason that) there is no projection operator onto the family of
entangled states.
The remainder of this paper is organized as follows. We rst review the basic quantum
mechanical statement that entanglement is not an observable. Next we introduce the
maximally extended AdS-Schwarzschild geometry in general relativity and, using AdS/CFT,
on the CFT side. As a warmup, we rst show that no single observer can detect the
presence of a wormhole geometry. We then turn to more complicated multiple-observer setups
and show, as desired, that they are unable to detect the presence of nontrivial topology in
complete generality.
2
Entanglement is not an observable
The proof that one cannot project onto a basis of entangled states [
30
] proceeds as follows.
Assume the existence of a complete basis set of entangled states j Ei i, distinct from the
basis set of all states. A projection onto this basis could be written in the form
Note, however, that the set of all entangled states has support over the entire Hilbert space,
as the entangled states can be (...truncated)