Entanglement = Wormholes


One of the most enjoyable and inspiring physics papers I have read in recent years is this one by Mark Van Raamsdonk. Building on earlier observations by Maldacena and by Ryu and Takayanagi. Van Raamsdonk proposed that quantum entanglement is the fundamental ingredient underlying spacetime geometry.* Since my first encounter with this provocative paper, I have often mused that it might be a Good Thing for someone to take Van Raamsdonk’s idea really seriously.

Now someone has.

I love wormholes. (Who doesn’t?) Picture two balls, one here on earth, the other in the Andromeda galaxy. It’s a long trip from one ball to the other on the background space, but there’s a shortcut:You can walk into the ball on earth and moments later walk out of the ball in Andromeda. That’s a wormhole.

I’ve mentioned before that John Wheeler was one of my heros during my formative years. Back in the 1950s, Wheeler held a passionate belief that “everything is geometry,” and one particularly intriguing idea he called “charge without charge.” There are no pointlike electric charges, Wheeler proclaimed; rather, electric field lines can thread the mouth of a wormhole. What looks to you like an electron is actually a tiny wormhole mouth. If you were small enough, you could dive inside the electron and emerge from a positron far away. In my undergraduate daydreams, I wished this idea could be true.

But later I found out more about wormholes, and learned about “topological censorship.” It turns out that if energy is nonnegative, Einstein’s gravitational field equations prevent you from traversing a wormhole — the throat always pinches off (or becomes infinitely long) before you get to the other side. It has sometimes been suggested that quantum effects might help to hold the throat open (which sounds like a good idea for a movie), but today we’ll assume that wormholes are never traversable no matter what you do.

Alice and Bob are in different galaxies, but each lives near a black hole, and their black holes are connected by a wormhole.

Love in a wormhole throat: Alice and Bob are in different galaxies, but each lives near a black hole, and their black holes are connected by a wormhole. If both jump into their black holes, they can enjoy each other’s company for a while before meeting a tragic end.

Are wormholes any fun if we can never traverse them? The answer might be yes if two black holes are connected by a wormhole. Then Alice on earth and Bob in Andromeda can get together quickly if each jumps into a nearby black hole. For solar mass black holes Alice and Bob will have only 10 microseconds to get acquainted before meeting their doom at the singularity. But if the black holes are big enough, Alice and Bob might have a fulfilling relationship before their tragic end.

This observation is exploited in a recent paper by Juan Maldacena and Lenny Susskind (MS) in which they reconsider the AMPS puzzle (named for Almheiri, Marolf, Polchinski, and Sully). I wrote about this puzzle before, so I won’t go through the whole story again. Here’s the short version: while classical correlations can easily be shared by many parties, quantum correlations are harder to share. If Bob is highly entangled with Alice, that limits his ability to entangle with Carrie, and if he entangles with Carrie instead he can’t entangle with Alice. Hence we say that entanglement is “monogamous.” Now, if, as most of us are inclined to believe, information is “scrambled” but not destroyed by an evaporating black hole, then the radiation emitted by an old black hole today should be highly entangled with radiation emitted a long time ago. And if, as most of us are inclined to believe, nothing unusual happens (at least not right away) to an observer who crosses the event horizon of a black hole, then the radiation emitted today should be highly entangled with stuff that is still inside the black hole. But we can’t have it both ways without violating the monogamy of entanglement!

The AMPS puzzle invites audacious reponses, and AMPS were suitably audacious. They proposed that an old black hole has no interior — a freely falling observer meets her doom right at the horizon rather than at a singularity deep inside.

MS are also audacious, but in a different way. They helpfully summarize their key point succinctly in a simple equation:

ER = EPR

Here, EPR means Einstein-Podolsky-Rosen, whose famous paper highlighted the weirdness of quantum correlations, while ER means Einstein-Rosen (sorry, Podolsky), who discovered wormhole solutions to the Einstein equations. (Both papers were published in 1935.) MS (taking Van Raamsdonk very seriously) propose that whenever any two quantum subsystems are entangled they are connected by a wormhole. In many cases, these wormholes are highly quantum mechanical, but in some cases (where the quantum system under consideration has a weakly coupled “gravitational dual”), the wormhole can have a smooth geometry like the one ER described. That wormholes are not traversable is important for the consistency of ER = EPR: just as Alice cannot use their shared entanglement to send a message to Bob instantaneously, so she is unable to send Bob a message through their shared wormhole.

AMPS imagined that Alice could distill qubit C from the black hole’s early radiation and carry it back to the black hole, successfully verifying its entanglement with another qubit B distilled from the recent radiation. Monogamy then ensures that qubit B cannot be entangled with qubit A behind the horizon. Hence when Alice falls through the horizon she will not observe the quiescent vacuum state in which A and B are entangled; instead she encounters a high-energy particle. MS agree with this conclusion.

AMPS go on to say that Alice’s actions before entering the black hole could not have created that energetic particle; it must have been there all along, one of many such particles constituting a seething firewall.

Here MS disagree. They argue that the excitation encountered by Alice as she crosses the horizon was actually created by Alice herself when she interacted with qubit C. How could Alice’s actions, executed far, far away from the black hole, dramatically affect the state of the black hole’s interior? Because C and A are connected by a wormhole!

The ER = EPR conjecture seems to allow us to view the early radiation with which the black hole is entangled as a complementary description of the black hole interior. It’s not clear yet whether this picture works in detail, and even if it does there could still be firewalls; maybe in some sense the early radiation is connected to the black hole via a wormhole, yet this wormhole is wildly fluctuating rather than a smooth geometry. Still, MS provide a promising new perspective on a deep problem.

As physicists we often rely on our sense of smell in judging scientific ideas, and earlier proposed resolutions of the AMPS puzzle (like firewalls) did not smell right. At first whiff, ER = EPR may smell fresh and sweet, but it will have to ripen on the shelf for a while. If this idea is on the right track, there should be much more to say about it. For now, wormhole lovers can relish the possibilities.

Eventually, Wheeler discarded “everything is geometry” in favor of an ostensibly deeper idea: “everything is information.” It would be a fitting vindication of Wheeler’s vision if everything in the universe, including wormholes, is made of quantum correlations.

*Update: Commenter JM reminded me to mention Brian Swingle’s beautiful 2009 paper, which preceded Van Raamsdonk’s and proposed a far-reaching connection between quantum entanglement and spacetime geometry.

48 thoughts on “Entanglement = Wormholes

  1. I’ve always been a fan of entanglement = wormholes. I really like the point about no-signaling being in some ways equivalent to the inability to traverse the wormhole. But this also leads me into a paradox: if the worm hole is not traversable, then how can Bell violations occur? I suppose I’m going to have to catch up to the authors in their understanding of those spacetime geometries!

    Is there a similar sort of thing going on with using wormholes to make CTCs? If I remember correctly the wormhole to CTC story didn’t seem to fully work, which is good, because there is at least some reason to believe that CTCs lead to crazy beyond quantum behavior as well.

    (On a slightly tangential note I recently read The Merchant and the Alchemist’s Gate by Ted Chiang, which involves time machines, but makes a very cute point about how even if the machines do not lead to paradoxes, they could nonetheless lead to interesting “effects”. Highly recommended.)

    Finally I always loved John Baez’s http://arxiv.org/abs/quant-ph/0404040 which points out many of the similarities between quantum mechanic’s Hilbert space and one of the natural constructs in geometries with non-trivial topologies.

    • The story about using wormholes to create CTCs has two parts. First the wormhole has to be traversable; which requires violation of the weak energy condition (e.g. via the Casimir effect). Then we create the CTC by moving the wormhole mouths, exploiting the idea that a moving clock ticks slower than a stationary clock. The second step might fail even if the first succeeds, because of strong back action arising as the CTC appears (as suggested by Hawking in his formulation of the “Chronology Protection Conjecture”).

      It is a good question about Bell inequalities. I don’t know how to think about the Bell inequality violation from the wormhole viewpoint.

    • Dave Bacon – I’m glad you liked my “quantum quandaries” paper on the rich analogy between quantum mechanics and topology. Ever since I heard about the Maldacena-Susskind paper, I’ve been thinking I may have left out a section on entanglement and wormholes. Now Jamie Vicary and I are trying to write a paper that begins to fills in that missing section. We’ll see how it goes.

      • I can’t wait to see what the outcome of your investigation is. I once made the mistake of trying to figure out how to use spin networks as the basic building blocks to come up with a way to get quantum nonlocality. John can attest that the talk I gave at his group meeting about that nearly qualified me as complete and total crazy!

  2. The wormhole manifest itself with gravitational lensing – the entanglement not. Anyway, I do perceive the idea, that the process of observation manifest itself with some space-time thread connecting the observer and observed object a surprisingly naive.

  3. Is there a place which explains the statement “quiescent vacuum state in which A and B are entangled”, i.e., “vacuum = highly entangled” on the level of a physics student? (Lecture notes, book, paper..)

    • This is the key to the derivation of Hawking radiation — if we cut space in half, the vacuum of quantum field theory is highly entangled across the cut. That’s why when we trace out the degrees of freedom behind the horizon of a black hole we obtain a highly mixed density operator for the degrees of freedom outside the black hole.

      So you could look at any book that derives Hawking radiation. (Maybe other readers have recommendations.) Or you could look at my lecture notes on quantum field theory in curved spacetime here. Try in particular Chapter 3 of the Ph 236 notes, where it is explained why a uniformly accelerated observer in the Minkowski vacuum sees a thermal bath.

        • first we would have to observe hawking radiation. It would be natural in the sense, that it exists in nature – once it is observed. But there are other good ways to produce entangled particles, especially in solid state physics this is a hot topic right now.

  4. > It has sometimes been suggested that quantum effects might help to hold the throat open (which sounds like a good idea for a movie)

    For some unknown reason I expected a link to a different genre.

  5. I’m also a longtime fan of Van Raamsdonk’s intriguing paper, but (without actually having read it, mind you) I’m less certain about the ideas of Maldacena and Susskind. Ultimately, an entangled state is just a superposition of ordinary correlated states, and I don’t see how to make this picture work in a non-linear theory such as GR, where a sum of solutions in general isn’t a solution again. I mean, if an entangled state is something connected by a wormhole, what is a separable state, in this scenario? Wouldn’t it also have to be some solution of GR?

    Also, how should one think about genuine multipartite entanglement in this picture? I mean, I suppose I could see how to model a W state, as three systems any two of which are connected by a wormhole, but what about a GHZ state? In the W state case, if you loose a qubit, the remaining two will still be entangled—still sit at the mouths of some wormhole joining them. But in the GHZ case, the same operation results in a separable state, so it seems that something I do to my qubit *over here* collapses a wormhole *way over there*, which at least intuitively seems strange from a GR point of view. Are there any solutions of GR having the necessary ‘three-partite’ wormhole structure?

    Anyway, I should probably try and read the paper before running off my mouth…

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  9. To be frank I know very little about this subject. But what surprises me is that in the whole argument anti de Sitter space plays a prominent role. If I understand all tests of GR verify Einstein universe with small positive Lambda. How can you draw serious conclusions from a model based on supposed anti de Sitter universe (negative Lambda?). Is there any reason that interior of black hole could be anti de Sitter?
    What is so special about anti de Sitter space? Thanks.

  10. I am exploring this from a slightly different angle. The argument is that there is a fundamental imprecision in the classical states, and it is the imprecision is what is needed to be preserved. The relevant point is much of this is ultimately traceable back to issues associated with our concept of well-ordering. In any case, the states and associated probabilities of the density matrix must always lack precision for sufficiently large systems. Pure states must always contain error bars. So the truthfulness of whether one is in a pure state is only valid up to the level of precision allowed by the measurement. This plagues any statement about initial purity as well.

    This suggests the entanglement experience by early and late radiation can only be viewed as an approximation, e.g. the monogamy isn’t perfect. I believe this is consistent with earlier discussion about complexity.

    If Alice has freedom to communicate with Bob about what she sees because what she sees is effectively new information superposed with the old information. Bob can now feel free to talk to Carrie after Alice’s foolish act because his interaction with Carrie can be freely superposed with his prior interactions with Alice. This is possible indefinitely because the error always allows for additional precision based on the discovery of new facts. The averages change over time.

    Feel free to comment or criticize.

    http://www.fqxi.org/community/forum/topic/1864

  11. There’s a new paper that just appeared on the arXiv, “The Holographic Dual of an EPR pair as a Wormhole” (http://arxiv.org/abs/1307.1132) by Jensen and Karch, where they explicitly construct the AdS-dual of an entangled quark-antiquark pair (in the large-N limit of Super Yang-Mills). I think this clarifies the correspondence somewhat; in my previous comment, I had not realized that EPR/ER are dual descriptions of one another in the sense of a holographic correspondence.

    What I’d still like to see, however, would be an explicit discussion of a Bell experiment under this holographic duality; if it is possible to violate Bell inequalities on the AdS-side of the description, it would seem that the wormhole joining the ‘particles’ is not sufficient to describe the correlation between them—after all, it isn’t traversable. But then, how do we not simply have two entangled wormhole throats?

    Also, again, what about GHZ states? Are there multi-throated wormholes such that if you ‘cut off’ one throat, the whole hole disappears? It would seem that this is what you’d need to reproduce the features of its entanglement.

    And what’s the dual description of non-entangled states (there’s actually a paper co-authored by van Raamsdonk (here: http://arxiv.org/abs/1204.1330) that seems to consider this question, but I haven’t read it yet)? There’s presumably a dual to the state |01> and to the state |10>. The superposition of those states is of course |01> + |10>, but its dual can’t be the superposition of the duals of |01> and |10>, because of the nonlinearity of GR. So how does this construction work? How does the superposition of two states bring about the existence of a wormhole in the dual? I wish I understood enough about holography in order to tackle these questions…

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  15. I think you are all missing the bigger picture.

    What if Alice is just a terrible human being? And there’s poor Bob, unable to get out. I mean, he’s trapped in a black hole with Alice!! For, well, eternity.

    That’s an awfully long time to be trapped with a wretch like Alice!

    How do we save the Bobster?

  16. And not to belabor the point but that’s when Bob takes to drinking. Bob’s not a nice guy when he’s been drinking, you know? He’s an angry drunk.

    And of course, the would have remembered to bring some drink along because that was the point of the whole affair, wasn’t it? Get a little lubed up, relax those inhibitions.. Let’s face it, right? Those two were probably a bit bothered and worked up enough to leap into this freakin’ thing in the first place. It is such irony that they are plummeting toward their doom and yet remain unable to close the deal!

    It is any wonder that Bob turned to drink???

  17. You can see where this is going, of course. Alice now creates a firewall at Bob’s end and with all that flammable liquor and with Bob being in a stupor because of Alice’s wretchedness! Well, it doesn’t take much imagination to see that when Bob finally meets the black hole core, he won’t care one iota!

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  20. If entanglement really equals wormhole then wouldn’t it mean wormholes are being created trivially any moment anywhere? My naive understanding from the quantum computation perspective is that whenever we do a control-something gate, i.e. whenever we correlate one qubit with another in any way, an entanglement forms. It happens so much that we need to clean up this entanglement by some unitary transformation afterwards so that it doesn’t break our computation. In addition to that, the decoherence process is due to a pure state getting correlated with the chaotic bath around, and whenever we do a projection measurement we get ourselves correlated with the system getting measured, so this is really happening anytime anywhere. I could be wrong with my understanding above, because if it’s right then wormholes could be created trivially, which is kind of counter-intuitive. Well when hasn’t physics been counter-intuitive anyways :)

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  22. 1) If the Maldacena/Susskind wormhole is true, spacetime inside and outside a black hole is the same and there is no meaning to say that there is a black hole.
    2) Swingle/van Raamsdonk spacetime creation by entanglement is analogous to saying that a photon which occupied infinite probability spacetime (suggested by the double slit experiment), at measurement/entanglement, is made to occupy a newly created determinate spacetime. A more generalized statement is that entanglement mediates conversion between probability and determinate spacetime.
    3) This generalization could be a link between general relativity and quantum mechanics, explaining not only “spooky action at a distance” but many paradoxes.

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  25. This ER = EPR equation is amazing. One might have thought that ER would have connected their two concepts from _the same year_ rather than it taking 68 years, for others to do!

    • It would be interesting to know if they were thinking along these lines, but you have to remember that the EPR paper is sort of famous for…missing the deep point about quantum probability and quantum correlations that we had to wait for Bell and Kochen-Specker to uncover. The EPR paper is interesting to read (http://prola.aps.org/abstract/PR/v47/i10/p777_1) in that from a more modern perspective it sort of feels, how do I put this, “wrong?” ;)

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  27. Remember I wrote this in 1974 almost 40 years ago. See David Kaiser’s “How the Hippies Saved Physics” about me and my associates back then. We were way ahead of the pack.
    I first proposed the ER = EPR connection in the 1975 book Space-Time and Beyond E.P. Dutton co-authored with Fred Alan Wolf and artist Bob Toben – First edition. p. 134 “Each part of space is connected to every other part through basic units of interconnection, called wormholes. Signals move through the constantly appearing and disappearing (virtual) wormhole connections, providing instant communication between all parts of space. These signals can be likened to pulses of nerve cells of a great cosmic brain that permeates all parts of space. This is a point of view motivated by Einstein’s general theory of relativity in the form of geometrodynamics. A parallel point of view is given in the quantum theory as interpreted by Bohm. In my opinion this is no accident because I suspect that general relativity and quantum theory are simply two complementary aspects of a deeper theory that will involve a kind of cosmic consciousness as the key concept. Bohm writes of a “quantum interconnectedness”:
    “… However there has been too little emphasis on what is, in our view, the most fundamentally different new feature of all, i.e., the intimate interconnection of different systems that are not in spatial contact … the well known experiment of Einstein, Podolsky and Rosen … Recently interest in this question has been stimulated by the work of Bell …” D. Bohm & B. Hiley …

    PS we now know that the wormholes need not pinch off because of dark energy and that means a more general quantum theory with signal nonlocality as described by Antony Valentini, e.g.
    Subquantum Information and Computation

    Antony Valentini
    (Submitted on 11 Mar 2002 (v1), last revised 12 Apr 2002 (this version, v2))
    “It is argued that immense physical resources – for nonlocal communication, espionage, and exponentially-fast computation – are hidden from us by quantum noise, and that this noise is not fundamental but merely a property of an equilibrium state in which the universe happens to be at the present time. It is suggested that ‘non-quantum’ or nonequilibrium matter might exist today in the form of relic particles from the early universe. We describe how such matter could be detected and put to practical use. Nonequilibrium matter could be used to send instantaneous signals, to violate the uncertainty principle, to distinguish non-orthogonal quantum states without disturbing them, to eavesdrop on quantum key distribution, and to outpace quantum computation (solving NP-complete problems in polynomial time).”

  28. Following several of the present discussions
    it seems to me that there’s the
    theoretical possibility of this scenario:

    A entangled with B across
    an event horizon of a black hole,
    A inside.

    B reflects a state change of A unaffected by
    GR effects, e.g., time dilation.

    For instance,
    two state changes of A separated by
    one second in A’s frame show with
    B as two state changes of B separated by
    one second in B’s frame.

    Does this scenario indeed exist?

    A mere computer scientist would enormously appreciate
    the mercy of a naive reply by a sophisticated physicist :)

  29. Beginning in the late 1990-th I formulated a theory of wormhole spaces in arXiv: 0910.4017, Wormhole Spaces: the Common Cause for the BH Entropy-Area Law, the Holographic Principle and Quantum Entanglement; and, as a consequence a theory of emergent quantum theory in arXiv: 1205.1619, Quantum Theory as emergent from an undulatory translocal Sub-Quantum Level.

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