Collision in the interior of wormhole

Published for SISSA by Springer Received: January 2, 2021 Accepted: February 5, 2021 Published: March 15, 2021 Ying Zhao Institute for Advanced Study, Princeton, NJ 08540, U.S.A. E-mail: Abstract: The Schwarzschild wormhole has been interpreted as an entangled state. If Alice and Bob fall into each of the black hole, they can meet in the interior. We interpret this meeting in terms of the quantum circuit that prepares the entangled state. Alice and Bob create growing perturbations in the circuit, and we argue that the overlap of these perturbations represents their meeting. We compare the gravity picture with circuit analysis, and identify the post-collision region as the region storing the gates that are not affected by any of the perturbations. Keywords: Black Holes, Models of Quantum Gravity ArXiv ePrint: 2011.06016v2 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP03(2021)144 JHEP03(2021)144 Collision in the interior of wormhole Contents 1 2 Perturbed theomofield double and quantum circuit 2.1 Bulk tensor network and quantum circuit 2.2 Epidemic model 2.3 Quantum circuit from the point of view of Alice 2.4 Quantum circuit from the point of view of Bob 2.5 Size and interior trajectory 2 2 3 4 6 8 3 Collision in the interior 3.1 The spreading of two epidemics: overlap of two perturbations in the quantum circuit 3.2 Post-collision region and the number of healthy gates in the circuit 3.3 More detailed match of time dependence 10 4 Conclusion and discussion 17 A Geodesic distance A.1 Geodesic distance 1 A.2 Geodesic distance 2 19 19 20 1 11 12 16 Introduction The Schwarzschild wormhole has been interpreted as an entangled state [1]. We assume Alice and Bob share the state, each holds one half of the entangled pairs. If each of them sends in a signal at appropriate time, the two signals can meet in the interior. On the other hand there are no boundary interactions between them. This makes ER = EPR [2] a mysterious statement. In AdS/CFT, it was argued that the bulk geometry reflects the quantum circuit preparing the boundary state [3–5]. In particular, in the black hole interior there is a unitary circuit preparing the state whose complexity gives the volume / action in the interior [6–8]. In this paper we use this quantum circuit picture to explain the “meeting” of two signals from different boundaries. When Alice and Bob send in signals, they create growing perturbations in the circuit. When both signals are sent in early enough, the two perturbations will have overlap in the quantum circuit (figure 1). Here, by “overlap” we mean that there will be a portion of a circuit in which both perturbations appear. We argue that this overlap represents the meeting of the two signals in the interior. In [9] it was argued that different kinds of gates in the quantum circuit are stored in different parts of spacetime region. Say, Alice and Bob share thermofield double. Alice –1– JHEP03(2021)144 1 Introduction <latexit sha1_base64="b3xlQ0+6IG93CSjLNrDFkTdGWlw=">AAACDHicbZDLSgMxFIbP1Futt6pLN8EiuCoZEXQjFNy4rGIv0A4lk6ZtaOZCckYYhr6C4FZfw5249R18Cx/BtJ2FbT0Q+Pj/c+DP78dKGqT02ymsrW9sbhW3Szu7e/sH5cOjpokSzUWDRyrSbZ8ZoWQoGihRiXasBQt8JVr++Hbqt56ENjIKHzGNhRewYSgHkjO0Ugt7D+SG0F65Qqt0NmQV3BwqkE+9V/7p9iOeBCJErpgxHZfG6GVMo+RKTErdxIiY8TEbio7FkAXCeNks7oScWaVPBpG2L0QyU/9eZCwwJg18uxkwHJllbyr+68Wj1EhurJfTYg4cXHuZDOMERcjnMQaJIhiRaTOkL7XgqFILjGtpf0L4iGnG0fZXshW5y4WsQvOi6tKqe39ZqdG8rCKcwCmcgwtXUIM7qEMDOIzhBV7hzXl23p0P53O+WnDym2NYGOfrF1oHmvY=</latexit> tR = 0 <latexit sha1_base64="BcEKK5z34taaoUPeX7OpzA3BDjk=">AAACDnicbZDNSgMxFIUz9a/Wv6pLN8EiiIsyI4JuhIIbl1Xsj7TDkEkzbWiSGZI7wlD6DoJbfQ134tZX8C18BNN2Frb1QuDjnHvh5ISJ4AZc99sprKyurW8UN0tb2zu7e+X9g6aJU01Zg8Yi1u2QGCa4Yg3gIFg70YzIULBWOLyZ+K0npg2P1QNkCfMl6SsecUrASo8Q3ONrDMFZUK64VXc6eBm8HCoon3pQ/un2YppKpoAKYkzHcxPwR0QDp4KNS93UsITQIemzjkVFJDP+aBp4jE+s0sNRrO1TgKfq34sRkcZkMrSbksDALHoT8V8vGWSGU2O9nOZzQHTlj7hKUmCKzmJEqcAQ40k3uMc1oyAyC4Rqbn+C6YBoQsE2WLIVeYuFLEPzvOq5Ve/uolJz87KK6Agdo1PkoUtUQ7eojhqIIole0Ct6c56dd+fD+ZytFpz85hDNjfP1C/d5m9c=</latexit> tR tR = t ⇤ <latexit sha1_base64="3xCj9+mqHGeOAjvowoiOxIOhtBc=">AAACCHicbZDLSsNAFIZPvNZ6q7p0M1gEVyUpgi4LblzWSy/QhjKZTtuhk0mYORFC6AsIbvU13Ilb38K38BGctlnY1gMDH/9/DvzzB7EUBl3321lb39jc2i7sFHf39g8OS0fHTRMlmvEGi2Sk2wE1XArFGyhQ8nasOQ0DyVvB+Gbqt564NiJSj5jG3A/pUImBYBSt9IC9+16p7Fbc2ZBV8HIoQz71Xumn249YEnKFTFJjOp4bo59RjYJJPil2E8NjysZ0yDsWFQ258bNZ1Ak5t0qfDCJtn0IyU/9eZDQ0Jg0DuxlSHJllbyr+68Wj1AhmrJfTYg4cXPuZUHGCXLF5jEEiCUZk2grpC80ZytQCZVrYnxA2opoytN0VbUXeciGr0KxWPLfi3V2Wa9W8rAKcwhlcgAdXUINbqEMDGAzhBV7hzXl23p0P53O+uubkNyewMM7XL6ZXmiM=</latexit> overlap <latexit sha1_base64="BebKjCfYcm0+h4iBZFnuAxILcSk=">AAACE3icbVDNSgMxGMzWv1r/qh69BIvgqeyKoMeCF48V7A+0tWTTbBuaTULyrbgsfQzBq76GN/HqA/gWPoJpuwfbOhAYZuaDyYRacAu+/+0V1tY3NreK26Wd3b39g/LhUdOqxFDWoEoo0w6JZYJL1gAOgrW1YSQOBWuF45up33pkxnIl7yHVrBeToeQRpwSc9NAF9gSZcglB9KRfrvhVfwa8SoKcVFCOer/80x0omsRMAhXE2k7ga+hlxACngk1K3cQyTeiYDFnHUUliZnvZrPUEnzllgCNl3JOAZ+rfi4zE1qZx6JIxgZFd9qbiv54epZZT67ycLfaA6LqXcakTYJLOa0SJwKDwdCA84IZREKkjhBrufoLpiBhCwc1YchMFy4OskuZFNfCrwd1lpebnYxXRCTpF5yhAV6iGblEdNRBFBr2gV/TmPXvv3of3OY8WvPzmGC3A+/oFbsmffA==</latexit> t wL>0 throws in a perturbation from left side at time tw (figure 3). We argued that from Alice’ point of view, the interior geometry matches the quantum circuit in figure 5. In particular, the interior region inside Alice’s entanglement wedge stores the quantum gates that are affected by the perturbation. In the first half of the paper we look at the same situation from Bob’s point of view. We argue that we can also identify the trajectory of the infalling object in the interior close to the horizon as the perturbation in the quantum circuit (figure 13). The closer the object tR is to the horizon, the larger its size is. t wR>0 In the second half of the paper we consider the collision of two infalling objects in the interior of thermofield double, one sent in from the left boundary and one from the right boundary. In the corresponding quantum circuit, the two perturbations will have overlap (figure 1). Using the epidemic model [10, 11] with two epidemics coming in at different times, we estimate the number of the healthy gates in the overlap region, and show that it matches the spacetime volume of post-collision region in the gravity picture. This paper is organized as follows. In section 2 we study the example of perturbed tR =point +1 oftRview = t⇤of both boundaries. We identify the trajectory tR theormofield double from the of infalling object close to the horizon in the interior as corresponding to the perturbation in the quantum circuit. In section 3 we study the collision of two infallling objects in the interior, one from the left boundary and the other one from the right boundary. We show that it corresponds to the overlap between the two perturbations in the quantum circuit. We find a detailed match between the volume of the post-collision region and the number of healthy gates from circuit analysi (...truncated)