Measurement-induced phase transition in teleportation and wormholes Alexey Milekhin1 Fedor K. Popov2

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Measurement-induced phase transition in

teleportation and wormholes

Alexey Milekhin∗1, Fedor K. Popov2

1University of California Santa Barbara, Physics Department, Santa Barbara, CA, 93106, USA

2CCPP, Department of Physics, NYU, New York, NY, 10003, USA

Abstract

We demonstrate that some quantum teleportation protocols exhibit measurement induced

phase transitions in Sachdev–Ye–Kitaev model. Namely, Kitaev–Yoshida and Gao–Jaﬀeris–

Wall protocols have a phase transition if we apply them at a large projection rate or at a

large coupling rate respectively. It is well-known that at small rates they allow teleportation

to happen only within a small time-window. We show that at large rates, the system goes

into a new steady state, where the teleportation can be performed at any moment. In dual

Jackiw–Teitelboim gravity these phase transitions correspond to the formation of an eternal

traversable wormhole. In the Kitaev–Yoshida case this novel type of wormhole is supported

by continuous projections.

March 7, 2023

∗milekhin@ucsb.edu

arXiv:2210.03083v2 [hep-th] 5 Mar 2023

Contents

1 Introduction 1

2 Projections at low energies 6

3 Turning TFD into an eternal traversable wormhole with Gao–Jaﬀeris–

Wall 10

3.1 Thesetup ..................................... 10

3.2 Turning on the interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Comments on large-qSYK............................ 15

4 Projection dynamics 16

4.1 Boundaryconditions ............................... 16

4.2 Single projection: recovering Kitaev–Yoshida teleportation . . . . . . . . . . 18

4.3 Multiple projections: creating a wormhole . . . . . . . . . . . . . . . . . . . 22

5 Conclusions and Discussion 25

1 Introduction

In the past few years, it was observed that hybrid quantum dynamics (i.e. when a quan-

tum system undergoes unitary evolution and is simultaneously subject to measurements or

projections) exhibits quite a rich structure [1–16]. In this situation, the von Neumann en-

tropy of the steady state can change between area-law and volume-law, a phenomenon known

as Measurement-Induced Phase Transition (MIPT) [1–4]. The presence of a MIPT can be

related to an emergent quantum error correction [17–19]. A similar phase transition can also

happen between two volume-law phases [20]. Moreover, an MIPT was observed in connected

correlation functions [21] or a charge distribution [22]. Recently, measurement dynamics was

shown to be useful in proving [23] NLTS (no low energy trivial states) conjecture [24] and

diagnosing intrinsic sign problem of quantum states [25].

A common feature of all these results is that the phase transition is diagnosed by a quan-

tity that is non-linear in the density matrix. This happens because measurements/projections

introduce a lot of energy into the system, resulting in the typical emergent steady-state hav-

ing inﬁnite temperature, which makes all linear observables trivial. We propose a setup

where the system does not heat to an inﬁnite temperature, and the phase transition is then

diagnosed by an anti-commutator which is linear in the density matrix 1. We refer to [6, 7]

for other ways to avoid inﬁnite heating.

The MIPT has been studied in the quantum ﬁeld theory (QFT) and gravity context in

some recent papers [27–29]. Nonetheless, in the QFT setup MIPTs are mostly overlooked.

The main reason is that the heating problem becomes even more severe: in continuum QFT

naive application of local measurements lead to ultraviolet (UV) divergences (as is always the

case when, in continuum QFT, we operate with quantities localized within ”sharp regions”).

To resolve this issue, we will develop path-integral techniques to study ”weak projections”,

which consist of coupling the system to an auxiliary qubit, letting them to interact for a

certain time and then measuring the auxiliary qubit. In particular, we keep only one of the

measurement outcomes. This setup is known as measurement with post-selection, forced

measurement or just projection, we will use these terms interchangeably. We would like to

stress that it is a physical operation which can be performed in a laboratory setting, albeit

at an expense of exponentially many measurements. We will see that in our case this is

mathematically equivalent to performing an Euclidean time-evolution. Euclidean evolutions

are well-deﬁned even in a continuum QFT.

Although in this paper we study Sachdev–Ye–Kitaev (SYK) model [30–33], which is a

quantum-mechanical model for which genuine (non-weak) projections are well-deﬁned, we

still restrict ourselves to weak projections. This will allow us to work within the low-energy

sector, where a lot of analytical tools are available. Moreover, this low-energy sector has a

holographic gravity dual.

We will apply our results to study teleportation protocols. Teleportation protocols

address the following problem: suppose we have two subsystems Land Rwhich are ini-

tially entangled but otherwise do not interact. How can we eﬃciently transfer information

from one subsystem to another? The most simple quantity which can diagnose teleporta-

tion is the anti-commutator2between the Land Rfermionic operators at times 3u1, u2:

Im GLR(u1, u2) = −iTr (ρLR{ψL(u1), ψR(u2)})≤1. Teleportation ﬁdelity is proportional to

this anti-commutator [34] and in this paper we will concentrate on Im GLR.

Two such protocols are Kitaev–Yoshida (KY) [35] and Gao–Jaﬀeris–Wall (GJW) [36–39],

which are depicted in Figure 1. We consider left (L) and right (R) subsystems prepared in

the thermoﬁeld-double (TFD) state, that entangles these subsystems. The total Hamilto-

nian does not involve L-R interactions, H=HL+HR, so each subsystem evolves inde-

1We refer to [26] which studies MIPT in out-of-time-ordered correlation functions.

2Or commutator for bosonic operators.

3Following SYK model literature tradition, we denote time by u.

Figure 1: Illustration of our setup. We have two subsystems Land Rprepared in the

TFD state (2.2). The goal is to make anti-commutator h{ψL, ψR}i non-zero. Left: to this

end GJW applies a unitary eiµδuOLOR, with OL/R being some hermitian operators. This

essentially adds an extra term to the Hamiltonian: H0=HL+HR+µθ(u)OLOR. Right:

KY takes a subsystem on each side and projects them on the maximally entangled (Bell)

state.

pendently. However, TFD state is not stationary under such evolution. Nonetheless, the

anti-commutator h{ψL(u1), ψR(u2)}i is identically zero for all u1, u2. The goal is to make

anti-commutator h{ψL(u1), ψR(u2)}i non-zero at least for some u1, u2, allowing the informa-

tion transfer between Land R.

The KY protocol uses a projection for this purpose. In contrast, GJW protocol applies an

extra unitary operator which couples Land Rto make Im GLR =−iTr (ρLR{ψL(u1), ψR(u2)})

large for a ﬁnite amount of time. But eventually it decays to zero, prohibiting any information

transfer. In GJW case applying multiple unitary operators at a small rate leads to the same

result [36].

The main question of this paper: is there a qualitative change if these protocols are applied

continuously at a high rate? Speciﬁcally we study this question in the low-energy (Schwarzian)

sector of the SYK model. We indeed ﬁnd a phase transition to a new steady state when we

apply unitary operators or projections at a high rate. In this new phase, teleportation ﬁdelity

Im GLR stays ﬁnite indeﬁnitely. In the dual gravity picture it means creating an eternal

traversable wormhole. In the KY case we will use weak projections in order to avoid heating

the system up. GJW protocol is governed by a Hamiltonian so we this problem does not

arise.

The whole paper can be summarized by Figures 2 and 3. We turn on the protocols at

times u= 0, insert a message at 4u=u1>0 and probe with Im GLR(u1, T ) if it reached the

other side at u=T. In GJW case we denote two-sided coupling by µand for KY protocol we

denote the projection rate5κ. Figure 2 illustrates the phase with small µor κ. The results

are very similar for GJW and KY: initially Im GLR is non-zero, but then exponentially drops

to zero if we try to insert a message at later times u1. This is essentially GJW computation.

Again, note that the interaction term µOLORis constantly on for all times u > 0.

At large µor κa qualitative change occurs - Figure 3. We can insert a message at

any time u1>0 and it will reach the other side with ﬁdelity that does not depend on the

time u1. This is the only common feature between GJW and KY in the strong coupling

regime. In the GJW case the message bonces back-and-forth without dissipation, leading

to revivals. Maximal value of Im GLR is of order 1, and it has a weak dependence on µ

and initial temperature β. In the KY case a message reaches the other subsystem and then

dissipates: Im GLR decays to zero at late times. Its maximal value does not depend on initial

temperature, but it depends on κ.

The strong coupling phase in the GJW case was discussed by Maldacena and Qi [40].

SYK model is dual [32, 41] to two-dimensional Jackiw–Teiltelboim gravity where this phase

represents an eternal traversable wormhole. Our contribution is that it is possible to reach

this phase starting from TFD state, which represents two entangled black holes. This can

be done easily (without coupling to external systems), but it requires a ﬁnite µcoupling 6.

Such behavior was conjectured for higher-dimensional black holes [43]. A similar problem

was addressed by Lensky and Qi [44] in large qSYK, although the phase transition is absent

there, we discuss this in Section 3.3. The strong coupling phase of the KY protocol represents

a novel type of a wormhole supported by projections. Wormhole length can be diagnosed by

correlator −log GLR(T, T ) at coincident time points. For TFD state it grows linearly with

time, reﬂecting the growth of Einstein–Rosen bridge. We ﬁnd that this quantity becomes

constant for the KY wormhole, similarly to Maldacena–Qi (MQ) wormhole. Finally, let us

point out that in the GJW/MQ case a similar transition can happen [45] even if the L, R

4We can greatly enhance the rate if we insert a message at some speciﬁc negative time u1before the

protocols are turned on at u= 0. However, in this paper we are interested if we can teleport at later times.

5That is, a projection is applied every 1/κ time interval.

6We refer to [42] for a perturbative discussion of bulk dilaton behavior.

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Measurement-inducedphasetransitioninteleportationandwormholesAlexeyMilekhin*1,FedorK.Popov21UniversityofCaliforniaSantaBarbara,PhysicsDepartment,SantaBarbara,CA,93106,USA2CCPP,DepartmentofPhysics,NYU,NewYork,NY,10003,USAAbstractWedemonstratethatsomequantumteleportationprotocolsexhibitmeasurementindu...

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Measurement-induced phase transition in teleportation and wormholes Alexey Milekhin1 Fedor K. Popov2

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