Written by: Christoph Adami

Primary Source: Spherical Harmonics

Quantum entanglement lies at the very heart of quantum mechanics, and it is what makes quantum physics different from classical physics. It is clear, as a consequence, that I won’t be able to make you understand quantum entanglement if you have never studied quantum mechanics. If this is truly your first exposure, you should probably consult the Wiki page about quantum entanglement, which is quite good in my view.

Whether or not Bob can do this depends on whether the *quantum channel capacity* of the black hole is finite, or whether it vanishes. If the capacity is zero, then Bob is out of luck. The best he can do is to attempt to reconstruct Alice’s quantum state using classical state estimation techniques. That’s not nothing by the way, but the “fidelity” of the state reconstruction is at most 2/3. But I’m getting ahead of myself.

Let’s first take a look at this channel. I’ll picture this in a schematic manner where the outside of the black hole is at the bottom, and the inside of the black hole is at the top, separated by the event horizon. Imagine Alice sending her quantum state in from below. Now, black holes (as all realistic black bodies) don’t just absorb stuff: they reflect stuff too. How much is reflected depends on the momentum and angular momentum of the particle, but in general we can say that a black hole has an absorption coefficient 0≤α≤1, so that α2 is just the probability that a particle that is incident on the black hole is absorbed.

*but*at the event horizon, the pairs would annihilate back, and nobody would be the wiser. Indeed, such vacuum fluctuations happen constantly everywhere in space. But if it happens at the horizon, then one of the particles could cross the horizon, while the other (that has equal and opposite momentum), speeds away from it. That now looks like the black hole radiates. And it happens at a fixed rate that is determined by the mass of the black hole. Let’s just call this rate β2.

*spontaneous emission*does not depend on how many particles Alice has sent in. In fact, you get this radiation whether or not you throw in a quantum state. These fluctuations go on before you send in particles, and after. They have absolutely nothing to do with |ψ⟩in. They are just noise. But they are (in part) responsible for the fact that the reflected quantum state ρout is not pure anymore.

But I can tell you that if this were the whole story, then physics would be in deep deep trouble. This is because you cannot recover even *classical* information from this channel if α=1. Never mind quantum. In fact, you could not recover *quantum* information even in the limit α=0, a perfectly reflecting black hole! (I have not shown you this yet, but I will).

This is not the whole story, because a certain gentleman in 1917 wrote a paper about what happens when radiation is incident on a quantum mechanical black body. Here is a picture of this gentleman, along with the first paragraph of the 1917 paper:

Albert Einstein in 1921 Einstein’s 1917 article (source: Wikimedia) “On the quantum theory of radiation” |

*with the same exact quantum numbers as the incoming radiation.*

Here’s the figure from the Wiki page that shows how stimulated emission makes “two out of one”:

Quantum “copying” during stimulated emission from an atom (source: Wikimedia) |

**In other words, all black bodies are quantum copying machines!**

Actually, now that you mention it, yes it is, and the law is much more stringent than the law against classical copying (of copy-righted information, that is). The law (called the no-cloning theorem) is such that it cannot–ever–be broken, by anyone or anything.

Note that the clone behind the horizon has a bar over it, which denotes “anti”. Indeed, the stimulated stuff beyond the horizon consists of anti-particles, and they are referred to in the literature as *anti-clones*, because the relationship between ρout and ρ¯out is a quantum mechanical NOT operation. (Or, to be technically accurate, the best NOT you can do without breaking quantum mechanics.) That the stimulated stuff inside and outside the horizon must be particles and anti-particles is clear, because the process must conserve particle number. We should keep in mind that the Hawking radiation also conserves particle number. The total number of particles going in is n, which is also the total number of particles going out (adding up stuff inside and outside the horizon). I checked.

[Note: Kamil informed me that for channels that are sufficiently “depolarizing”, the capacity can in fact be calculated, and then it is zero. I will comment on this below.]

First: α=0. In that case the black hole isn’t really a black hole at all, because it swallows nothing. Check the figure up there, and you’ll see that in the absorption/reflection column, you have nothing in black behind the horizon. Everything will be in front. How much is reflected and how much is absorbed doesn’t affect anything in the other two columns, though. So this black hole really looks more like a “white hole“, which in itself is still a very interesting quantum object. Objects like that have been discussed in the literature (but it is generally believed that they cannot actually form from gravitational collapse). But this is immaterial for our purposes: we are just investigating here the quantum capacity of such an object in some extreme cases. For the white hole, you now have two clones outside, and a single anticlone inside (if you would send in one particle).

**Technical comment for experts:**

A quick caveat: Even though I write that there are two clones and a single anti-clone after I send in one particle, this does not mean that this is the actual number of particles that I will measure if I stick out my particle detector, dangling out there off of the horizon. This number is the *mean expected number* of particles. Because of vacuum fluctuations, there is a non-zero probability of measuring a hundred million particles. Or any other number. The quantum channel is really a superposition of infinitely many cloning machines, with the 1-> 2 cloner the most important. This fundamental and far-reaching result is due to Kamil.

*Unruh channel*that also appears in a quantum communication problem where the receiver is accelerated, constantly. The capacity looks like this:

Quantum capacity of the white hole channel as a function of z |

Actually, it is a somewhat big deal, because I can tell you that if it wasn’t for that blue stimulated bit of radiation in that figure above, you couldn’t do the reconstruction at all!

*another*perfect copy behind the horizon? And then have TWO?”

*into*the black hole (what is known as the

*complementary channel)*is actually zero because (and this is technical speak) the channel into the black hole is

*entanglement breaking*. You can’t reconstruct perfectly from a single clone or anti-clone, it turns out. So, the no-cloning theorem is saved.

Now let’s come to the arguably more interesting bit: a perfectly absorbing black hole (α=1). By inspecting the figure, you see that now I have a clone *and* an anti-clone behind the horizon, and a single clone outside (if I send in one particle). Nothing changes in the blue and red lines. But everything changes for the quantum channel. Now I can perfectly reconstruct the quantum state *behind* the horizon (as calculating the quantum capacity will reveal), but the capacity in front vanishes! Zero bits, nada, zilch. If α=1, the channel from Alice to Bob is entanglement breaking.

Now let’s calm down and ponder what this means. First: Bob is out of luck. Try as he might, he cannot have what Alice had: the same entanglement with R that she enjoyed. Quantum entanglement is lost when the black hole is perfectly absorbing. We have to face this truth. I’ll try to convince you later that this isn’t really terrible. In fact it is all for the good. But right now you may not feel so good about it.

But there is some really good news. To really appreciate this good news, I have to introduce you to a celebrated law of gravity, the equivalence principle.

The principle, due to the fellow whose pic I have a little higher up in this post, is actually fairly profound. The general idea is that an observer should not be able to figure out whether she is, say, on Earth being glued to the surface by 1g, or whether she is really in a spaceship that accelerates at the rate of 1g (g being the constant of gravitational acceleration on Earth, you know: 9.81 m/sec2). The equivalence principle has far reaching consequences. It also implies that an observer (called, say, Alice), who falls towards (and ultimately into) a black hole, should not be able to figure out when and where she passed the point of no return.

The horizon, in other words, should not appear as a special place to Alice at all. But if something dramatic would happen to quantum states that cross this boundary, Alice would have a sure-fire way to notice this change: she could just keep the quantum state in a protected manner at her disposition, and constantly probe this state to find out if anything happened to it. That’s actually possible using so-called “non-demolition” experiments. So, unless you feel like violating another one of Einstein’s edicts (and, frankly, the odds are against you if you do), you better hope nothing happens to a quantum state that crosses from the outside to the inside of a black hole in the perfect absorption case (α=1).

Fortunately, we proved (result No. 3) that you can perfectly reconstruct the state behind the horizon when α=1, that this capacity is non-zero. And that as a consequence, the equivalence principle is upheld.

This may not appear to you as much of a big deal when you read this, but many many researchers have been worried sick about this, that the dynamics they expect in black holes would spell curtains for the equivalence principle. I’ll get back to this point, I promise. But before I do so, I should address a more pressing question.

*inside*of the black hole, otherwise the equivalence principle is hosed. And if it reconstructable inside, then you better hope it is not reconstructable outside, because otherwise the no-cloning theorem would be toast.

Adventurous Alice encounters a firewall? Credit: Nature.com |

Of course we would like to know the quantum capacity for an arbitrary α, which we are working on. One result is already clear: if the transmission coefficient α is high enough that not enough of the second clone is left outside of the horizon, then the capacity abruptly vanishes. Because the black hole channel is a particular case of a “quantum depolarizing channel”, discovering what this critical α is only requires mapping the channel’s error rate p to α.

This becomes even more amusing if you keep in mind that (eternal) black holes have white holes in their past, and white holes have black holes in their future.

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