Inconsequential, irrational, incorrect criticisms of PR, ER=EPR

Luboš Motl, May 20, 2015

Insights by Papadodimas and Raju (PR) and those by Maldacena and Susskind (ER=EPR) have advanced our understanding of quantum gravity in the following way:

We know that there is some Hilbert space of microstates, especially black hole microstates (or two black holes’ microstates), for example from AdS/CFT. It may be described e.g. by using the energy eigenstate basis. This Hilbert space is fixed and our goal – they have started to fulfill – is to become masters of this Hilbert space. What does it mean?

It means to be able to embed, localize, or identify operators (or even operator algebras) on/into this Hilbert space that may be associated with particular measurements that we may do in the real world. In practice, it means to see how the bulk field operators (and the black hole or wormhole interior operators are the really hard ones) are embedded in the space of black hole microstates.

In this approach, quantum mechanics is primary because a Hilbert space pre-exists and the field operators have to «adapt» to it. This is very different from the approach in which a Hilbert space is produced out of local fields, e.g. as the Fock space. In such an approach, the spacetime would be primary and fundamental. Such an approach is probably inapplicable in quantum gravity.

Both PR and ER=EPR have been said to violate the postulates of quantum mechanics. Moreover, Polchinski and Marolf have argued that there was something wrong with ER=EPR because the «generic» entangled states are not linked to a smooth geometry (the latter claim is right but the suggestion that it means a problem for ER=EPR is not).

The criticisms «PR, ER=EPR are not correct QM» actually boil down to the critics’ misunderstanding how quantum mechanics describes Nature. They don’t like PR, ER=EPR because it links some self-evidently quantum phenomena with phenomena that the critics love to describe classically. In reality, all of Nature is described quantum mechanically and the classical reasoning is always wrong, at least at the fundamental level.

For example, an anti-quantum zealot named Florin Moldoveanu wrote:

Oopsie… My bad. Honest typo. Thanks for pointing it out. The idea [ER=EPR] was actually proposed even earlier by Cristi Stoica who commented above. I did not like it then (sorry Cristi), and I do not like it now. I do not object of this idea in the context of AdS/CFT but pushing it into ordinary quantum entanglement. It is «too realist» IMHO, like constructing models of the aether with gears and springs.

I don’t think that Cristi Stoica should get credit for ER=EPR – please correct me by a reference if you think I have overlooked something. Also, it’s nonsensical to approve ER=EPR in AdS/CFT but not in quantum theories of dynamical spacetimes in general. If something this qualitative holds in AdS/CFT, it almost certainly has to hold in general. After all, an almost flat space may be embedded in AdS/CFT, too.

But the main factual criticism is that ER=EPR is «too realist». Oh, really?

There is nothing «realist» about ER=EPR. Why does Mr Moldoveanu think that ER=EPR makes a «realist» assertion about entanglement? I don’t see into every corner of his brain but I still think that the answer is rather obvious. It’s because he thinks that wormholes – Einstein-Rosen bridges, in this way – have to be described by a realist (i.e. intrinsically classical) theory because he thinks about the spacetime geometry classically.

But that is simply not the case and Maldacena’s and Susskind’s analyses implicitly refuse all these thoughts. In our quantum world, the question whether there is a wormhole somewhere – much like any other question about the «state of the physical objects» – doesn’t have any objective answer that would exist without any measurements. Instead, such a question – much like any question about the physical objects – has to be answered by a measurement, a process from which an observer learns some information.

A measurement has to be associated with a Hermitian linear operator on the Hilbert space. Its eigenvalues give us the list of possible answers that the measurement may yield. And when we decompose the state vector into the eigenvectors of the operator, the squared absolute values of the coefficients give us the probabilities that the outcome of the measurement will be the corresponding eigenvalue. This paragraph and the previous paragraph summarize the whole framework of quantum mechanics. Everything else we have to add is just some equations defining the operators – their commutation relations and/or the Heisenberg equations of motion that dictate the time evolution (and the latter is really a special example of the former). That’s it. When these things exist – a Hilbert space, some operators, and the identification of the complex amplitudes as probability amplitudes – we have a perfect quantum mechanical theory.

Maldacena and Susskind identify an empty Einstein-Rosen bridge connecting two identical black holes with some perfectly entangled state with I write schematically as

\[ \ket{MS} = \sum_{i\in B} \ket{e_i} \otimes \ket{e_i}\]

where the factor before/after the tensor product sign describes the state of the first/second black hole, respectively. \(B\) is a basis of the single black hole Hilbert space. You may construct the projection operator on this state

\[ P_{MS} = \ket{MS}\bra{MS}\]

Great. This is a projection operator, a linear Hermitian operator with eigenvalues \(0,1\). There exists a measurement linked to this operator. If the result of the measurement is \(1\), then you know that you are looking at the empty Einstein-Rosen bridge. If the result of the measurement is \(0\), then you are not.

A general state that you have before the measurement is a linear superposition of (\(c_0\) times) some vector \(\ket{\alpha_0}\) whose \(P_{MS}\) eigenvalue is \(0\); and (\(c_1\) times) some vector \(\ket{\alpha_1}\) whose eigenvalue is \(1\). In this general case, \(|c_1|^2\) gives you the probability that you will observe an empty Einstein-Rosen bridge. I assumed \(|c_0|^2+|c_1|^2=1\), a normalization condition for the state vector.

(In principle, the ER-like states are non-orthogonal to, and therefore not mutually exclusive with, particular unentangled states of the two black holes. In practice, the probability that \(\ket{MS}\) «is» a generic non-entangled state of the two black holes is just \(\exp(-2S)\) or so, basically zero, so macroscopic nearly unentangled black holes are approximately mutually exclusive with the low excitations of highly entangled ER bridges. But in principle, and for pairs of small holes, one can’t determine which topology is realized – there is not even a linear projection operator that would correspond to such a question.)

Someone who knows quantum mechanics has used this sequence of ideas, rules, and operations – question, measurement, operator, spectrum, decomposition, probabilities as answers – thousands of times. But most people still don’t think in this, quantum mechanical way. Whenever you forget to shout at them for a second, they love to replace these correct ways to deal with Nature by some classical misfits – by assuming that the Universe has certain dynamical properties independently of the measurements. But it never does.

The point of ER=EPR is that operators such as \(P_{MS}\) act on the same Hilbert space of two black holes as local field operators describing the geometry in the two black holes’ regions. In other words, the same Hilbert space may be recycled in various ways and visually different operators – and operator algebras – may be embedded into this \(H_1\otimes H_2\) Hilbert space of two black holes. Let me mention that in the wormhole interpretation of the Hilbert space, you may embed not only \(P_{MS}\) but also the local field operators describing the interior of the Einstein-Rosen bridge. You must be careful that such field operators are only well-defined (with a fixed definition) on a «relatively small» subspace of \(H_1\otimes H_2\) – that’s the state dependence I will return to.

There is absolutely nothing non-quantum about the fact that we may embed different operators, or operators based on two «very different pictures» what is going on. These two pictures are complementary to each other. The operators in them are not mutually commuting. For example, \(P_{MS}\) refuses to commute with almost all field operators \(\phi_i(x,y,z,t)\) describing the background with two independent black holes. But these commutators’ being nonzero is nothing against quantum mechanics, either. Quite on the contrary, quantum mechanics only starts once the commutators are nonzero. Nonzero commutators are the flagship, defining feature of all of quantum mechanics!

Generic entangled states are not smooth geometries. Is it a problem?

Marolf and Polchinski have correctly observed that generic highly entangled states can’t correspond to nice, smooth geometries. But they have implicitly suggested that this represented a problem for ER=EPR and that’s simply untrue. Let us «generically rotate» the Maldacena-Susskind state:

\[ \ket{MS’} = \sum_{i,j\in B} U_{ij}\cdot \ket{e_i} \otimes \ket{e_j}\]

The original state \(\ket{MS}\) is obtained for \(U_{ij}=\delta_{ij}\) but here we replaced this tensor connecting the two black holes with a general unitary matrix \(U_{ij}\) acting on the Hilbert space \(H_1\). This has an obvious interpretation that makes all these analyses much simpler than Polchinski’s and Marolf’s convoluted language. What does \(U\) do? It’s simple. Before we glue the two black holes by a bridge, we twist or scramble their microstates by the operation \(U\).

Now, when \(U\) describes e.g. a geometric rotation of the black hole event horizon, the object we get is very nice and smooth: The bridge is twisted. If two people fall into two black holes at fixed spherical coordinates \(\theta,\phi\) which are the same for both, they may meet in the black hole but they will see their pal at another point \(\theta,\phi\) on the two-sphere, thanks to the twist.

There exist other simple geometric operators \(U\) with a similarly straightforward interpretation. And you may also consider \(U\) as the exponential of some local operators, e.g. some creation operators in the wormholes’ interior, and the addition of such a matrix may add some particles from one observer’s viewpoint.

But if you consider the truly generic unitary matrices \(U\) on the space of the microstates \(H_1\), it’s clear that all the information on the «throat» is hopelessly scrambled and randomly delocalized before the two black holes are glued. The resulting picture won’t be a smooth geometry because \(U\) is nothing else than a monodromy acting on the microstates corresponding to a «round trip» around the bridge (two points above the two event horizons may be connected either by an outside trajectory, or through the ER interior).

So if \(U\) admits no smooth geometric interpretation, the monodromy won’t be smooth, either (because it’s the same thing), and if the monodromy around a cycle is not geometric, the whole spacetime arrangement can’t be interpreted as a smooth geometry.

It’s this simple. You don’t need contrived arguments involving the correlation functions by Polchinski and Marolf.

But there is nothing wrong about the non-smoothness of the generic entangled states. It doesn’t indicate any violation of ER=EPR. The equation ER=EPR still holds. But in the case of the generic \(U\), what we have is

contrived ER = contrived EPR

The Einstein-Rosen bridge is twisted by a non-geometric monodromy which makes it unsmooth and «contrived». But the new point I will make in this paragraph is that even from the viewpoint of generic quantum mechanics, the entangled EPR state will be equally contrived! It is actually extremely difficult to prepare a similar generic perfectly entangled state of two black holes.

Why is it so hard? It’s hard because all gadgets we may employ to prepare initial states do operate within some geometric framework! All building blocks of the apparatuses are ultimately described by locations and speeds and fields and shapes in a spacetime. We only know some geometric information about the degrees of freedom which is evolving into some other geometric information. This is why in practice, we will only be able to prepare the entangled state with a simple, geometric \(U\), and the preparation will be similar to a pair creation of black holes, anyway.

In principle, we may prepare any initial state of the two black holes, including maximally entangled states with generic nongeometric matrices \(U\). However, the (very large) machine that we need to prepare such an initial state will have to deal with the degrees of freedom of (and between) the two black holes in such a fine way that its natural observables will heavily fail to commute with the individual two black holes’ interior field operators. The effective field theory description will involve «chaotic noise» both in the one-bridge and two-holes versions. We will be dealing with the microstates at the most microscopic general level – and to understand the details of such operations, the effective field theory description will be useless.

There are other ways to see that something is contrived about the general entangled state. We are inevitably entangling energy eigenstates \(\ket{e_i}\) of the first black hole with superpositions of energy eigenstates of the other black holes with slightly different energies. Energies generate the evolution in time and because they don’t coincide, there won’t be a way for the two sides of the throat to agree about a time coordinate.

None of these observations weakens ER=EPR in any way. The observations only imply that some ER or EPR are more natural and simple to prepare while others are not. But an Einstein-Rosen bridge with lots of scrambling of the data added before the gluing is still a connection between the two black holes’ interiors. From a quantum information viewpoint, it is still a non-traversable wormhole.

State dependence. Is it a problem?

No.

Here I just want to review a few obvious words explaining why the state dependence in no way contradicts the postulates of quantum mechanics. The state dependence only says that local operators (mathematical entities associated with different measurements) in the spacetime are only well-defined on subspaces of the Hilbert space that differ from some reference state – or reference states – by a finite, not exponentially large, number of local changes or operations.

This is a situation we know from almost all real-world situations. The creation operators of phonons in a diamond crystal are only well-defined assuming that the electrons and carbon nuclei are arranged in a way that basically resembles a diamond crystal. On the subspaces of the Hilbert space microstates that resemble graphite or carbon gas, the diamond phonon operators are ill-defined.

(Similarly, you may define the operator measuring the color of the first pixel on your smartphone’s screen. Once you melt your smartphone, and in most high-energy states, the smartphone is either melted or evaporated, the operator linked to the pixel becomes ill-defined.)

The same is true for the creation and annihilation field operators on top of two very different black hole microstates – those that differ by the action of very many local operators. What may «look» different than in the diamond example is that two different microstates macroscopically «look» the same, as an equally empty black hole. But this is just a macroscopic illusion. Microscopically, they are as incompatible as the «diamond» and the «carbon gas».

Black hole complementarity. Is it a problem?

No. The criticisms discussed in this section are not specifically criticisms of PR and ER=EPR. Instead, these criticisms suggest that the whole black hole complementarity is inconsistent with quantum mechanics. This gloomy opinion is unjustifiable.

There is a general criticism of the black hole complementarity which says that the degrees of freedom B near the event horizon can’t be entangled both with A inside the black hole and C outside the black hole due to the monogamy of entanglement.

Monogamy is a universal rule but a paradox would only exist if A and C were carrying disjoint information – if they were different degrees of freedom (so that the full Hilbert space would be a tensor product \(H_A\otimes H_C\otimes H_{\rm rest}\)). In black hole complementarity, they’re either the same or, more precisely, overlapping. Some factor in the Hilbert space \(H_{AC}\) is recycled to allow both the A operators as well as the C operators but these two groups don’t commute with each other. If that’s so, the AB and BC entanglement are just the same entanglement written using two different bases of the A=C Hilbert space, or this claim is at least true for a «part» of the entanglement.

The key condition that is satisfied in the black hole background and that makes the black hole complementarity consistent is that A and C are degrees of freedom that geometrically can’t belong to the same causal diamond. What we actually need is the fact that due to the causal restrictions of the underlying spacetime, no observer may acquire the information both from the measurements of A and the measurements of C. It’s because the future light cone of A doesn’t overlap with the future light cone of C.

But that means that there won’t be any observer who would view both the A measurements and the C measurements as relevant – in fact, as something he is able to learn at all. The information about the system that an observer has is always shaped primarily by the «latest» measurements or observations that he got familiar with.

Even though A and C may belong to spacelike-separated regions, their nonzero commutator isn’t a problem because the two kinds of measurements can’t disturb one another, thanks to the non-existent «common future» of A and C. An observer always knows which measurement he did was the latest one. In fact, he knows the full chronology of his measurements, and that’s enough for the quantum mechanical predictions of the following measurement to be derived exactly in the same way as in the simplest undergraduate examples that you know so well.

Of course, the non-realist character of quantum mechanics is essential for the consistency here – much like in the rest of quantum mechanics. A «realist» person could say that the measurements done on C – measurements of operators not commuting with those in A – modify the predictions for the measurements in A, and vice versa. In this way, A and C would constantly disturb each other superluminally, and in some ill-defined way. But this opinion is a flawed artifact of the «realist» (classical) thinking. A measurement is always just a process from which a particular observer gets some information; that’s why a «measurement» in quantum mechanics is always linked to an observer and his perspective, it always has a «subjective» character. There can’t be any «objective» definition of what a measurement is.

So whatever happens in C, isn’t a measurement for the observer living in A, and vice versa! The processes in the other, complementary, spacelike-separated region are not measurements because the observer isn’t getting and can’t be getting (for causal reasons) information from these processes. In other words, the wave function never «collapses» when the observer isn’t making a measurement; he isn’t making a measurement if he’s not getting any information; and if a process takes place in a causally disconnected region, he can’t be getting information for causal reasons. So there’s no «collapse» due to these faraway processes! And that’s why the processes in the spacelike separated regions don’t «spoil» each other despite the non-vanishing commutators.

You may claim that the causal structure of the spacetime is just approximate – after all, we just said that the operators in A and C don’t commute with each other. For this reason, you might ask: Isn’t it strange for the consistency of quantum mechanics (the argument in the previous paragraphs) to rely on this causal structure which isn’t too exact or fundamental, anyway?

The answer is that indeed, you don’t need to rely on this causal structure. But once you declare the causal structure of the spacetime to be fundamentally non-existent (and you want to go «beyond» it), you have to do it consistently. You have to abandon the parameterization of the Hilbert space in terms of the local fields in general. If you do so, you will deal with a full Hilbert space of microstates that may be precisely identified just like the operators acting on this full Hilbert space but none of these operators will have an exact association with loci in the spacetime. And in this description, it will be manifest that the operators that used to be associated with the A and C loci act on the same portion of the Hilbert space.

Can the existence of the black hole interior operators produce any inconsistency at all?

No, it cannot. The black hole interior is just a «dead end». Cut it away along the event horizon. If you agree that the theory describing the exteriors of the black holes is consistent, the addition of the interior won’t cause any new problems because the (perhaps generalized) field operators in the black hole interior(s) may be calculated via the Heisenberg equations from the initial conditions on the event horizon(s). The information contained in the operators on the event horizon(s) is sufficient for all predictions made inside the black hole(s) and no contradiction between the predictions for the interior with the observations made outside the black hole may exist simply because the observer inside the black hole has no chance to get out (by definition of the event horizon and the interior) – and the external observers have no chance to learn about the measurements done in the interior.

The whole line of reasoning that one may find contradictions that involve comparisons of the operators inside and outside is just plain wrong because no observer may see both A and C – but all kosher quantum mechanical discussions of the information about Nature are linked with an observer. You may discuss quantum mechanics of the full Hilbert space in which both A and C are embedded but if you discuss it exactly, there won’t be any canonical assignment of general operators as particular functions of «something in A» and «something in C». In the most accurate treatment of the microstates, you always lose the link between the operators and particular regions of the spacetime. You will have to use more accurate operators that don’t see the (bulk) geometry clearly, like the operators that are simple in the CFT description of AdS/CFT. This lost connection with the (bulk) spacetime geometry isn’t a sign of the breakdown of quantum mechanics. It is a symptom of the emerging and, in some sense, approximate character of the (bulk) spacetime geometry.