Friday, April 4, 2014

The Quantum Cloning Wars Revisited

Cloning isn't so much in the news anymore these days, as the novelty of Dolly has worn off and cloning of farm animals and pets has become common place. But a different form of cloning--the cloning of quantum information, that is--is still very much discussed. 
Dolly the cloned sheep. (Pining for the fjords)
There are a number of fundamental rules that we abide by in quantum physics. That probabilities are given by the square of the wave function's amplitude, for example (Born's rule). Or, that quantum interference is destroyed if you try to know what's really goin' on (the lesson of the double-slit experiment).

Or, that quantum physics is linear.

It really is. What this means is that if you have quantum wavefunctions $\psi$ and $\phi$ and an operator $U$ acting on the sum of them, then you get

$U(\psi+\phi)=U\psi +U\phi$.

Pretty innocuous-looking, right? But the consequences of this little harmless statement are mighty. This linearity implies that quantum states cannot be cloned. They can't be Dolly. They can't be Xeroxed. Thou shalt not clone quantum states. How is linearity going to legislate that?

Well, let's first state what cloning of a quantum state would look like. Could we clone a state such as $\phi$? 

(I don't have to tell you what $\phi$ is to answer the question, as you will understand in a minute-and-a-half). 

If you know what $\phi$ is, then yes, you can clone this state! (Wait for the apparent contradiction to be resolved before firing off your email).

Cloning in quantum mechanics means that if you start with a state $\phi$, then after cloning you have the state $\phi\phi$: two copies of $\phi$. It is actually possible to design an operator $U$ that does precisely this:

$U\phi 0=\phi\phi$

(What happens here is that the second state "0" is turned into $\phi$ by $U$. $U$ literally measures $\phi$, and then uses this knowledge of what $\phi$ is to turn 0 into a copy of that $\phi$. So you can clone any known state.

"What do you mean by 'known'?"

Well you see, for $U\phi$ to be $\phi\phi$ (and so that we could turn 0 into $\phi$), you must already have known the basis that $\phi$ was prepared in. Because otherwise, the measurement would have changed $\phi$. 

But here comes the rub. Imagine your state is $\phi+\psi$. In general, these are not in the same basis. Let's try to clone that one:

When we apply this operator $U$ to $\phi +\psi$, look what happens because of linearity:

$U(\phi+\psi)=U\phi+U\psi=\phi\phi+\psi\psi$    (1). 

Neato. But that is not the cloning of $\phi+\psi$. That would have been $(\phi+\psi)(\phi+\psi)$. That's not the same as Eq. (1). 

So it is possible to clone any one particular (known) state, but it is not possible to clone superpositions of states (so-called "non-orthogonal states"). So in general, cloning is impossible. That's the no-cloning theorem, due to Bill Wootters and Wojciech Zurek [1], as well as Dennis Dieks [2], who discovered the theorem independently in 1982. 

It turns out that if you could clone quantum states, all hell would break loose in the universe. This is because the quantum mechanics of entanglement is really quite powerful: Einstein called entanglement "spooky action-at-a-distance". Two entangled quantum states can seemingly "communicate" over large distances. Distances so large that if they were communicating, then this would have to occur at a speed larger than light. And you can see how Einstein would seriously object to such a proposition. Strenuously. He would knock you over the head, is what he would do. 

I say "seemingly", because even though entities A and B (often dubbed "Alice" and "Bob") that share an entangled state would obtain the same exact measurement results if they proceeded to measure the shared state (even if they were in different galaxies), it turns out that these measurements cannot be used for communication. Their measurement devices show the same result, but because this result is a random number, no information can be sent. Nada. Sleep well, Albert. 

Unless, that is, Alice or Bob (or both) could make copies of their entangled state. If they could do that, well then they could use those measurements to communicate superluminally, as was discovered by the Science Fiction writer Nick Herbert, as it turns out. (It was this conjecture that ultimately led to the formulation of the no-cloning theorem.) So, all hell would break loose if quantum cloning would be allowed, because then you could communicate superluminally, which means you could travel backwards in time, kill a great-grandparent and leave the universe in the mother of all time-paradoxes.

So quantum cloning is out, and all because of the linearity of quantum mechanics, and all that's a good thing.

Or is it?

The reason I'm asking is.... black holes, of course. You may have heard about the commotion I caused by announcing that black holes do not destroy information, because that information is copied just before it disappears in the black hole's abyss. Copied, I hasten to add, by this process called "stimulated emission" that must accompany absorption, as this ubiquitous gentleman named Albert E. told us about in 1917. But if I send into the black hole a quantum state $\phi$ rather than the classical "001010010001", wouldn't the black hole then violate the sacrosanct no-cloning theorem? The one that if you violate it should make the fabric of space time melt?

That is a question worth investigating in detail, which I have done in an article I wrote in 2006, and which is currently under review. Yes, it is another one of those.

Here's what we discovered (my collaborator in this was the same same Greg Ver Steeg that collaborated with me in the 2004 article--published ten years later--and whose blog I'm linking here):

Black holes are quantum cloning machines.

How is this possible? Well, it is possible because black holes aren't perfect. Cloners, that is. It turns out that you are allowed to clone quantum states somewhat. You can do it as long as you are sloppy. That imperfect cloning is possible was discovered by Buzek and Hillery in 1996 [3]. They showed that you can design quantum cloning machines that take a quantum state $\phi$ and transform it--not into another quantum state--but into a density matrix $\rho$ that is fairly close to $\psi$. How close you ask? This is measured by the fidelity of the cloning machine $F$, which is just the expectation value of the density matrix $\rho$ within the initial quantum state $\phi$:

$F=\langle \phi|\rho|\phi\rangle$    (1)

$F=1$ means you created a perfect cloning machine where $\rho=| \phi \rangle \langle \phi|$. If you were able to do this, then of course you already know what $\phi$ was (a "known" state, one that you had previously measured). And perfect cloning of known states is allowed, because in that case you are cloning classical, rather than quantum information. You're standing next to the copier feeding sheets to the machnie. That's right: the difference between classical and quantum information is simply whether or not you know what kind of a state you have on your hands!

Come to think of it, that's really the difference between classical and quantum mechanics right there, in a nutshell. Forget all this $\hbar\to 0$ nonsense, that is not what makes quantum different from classical. It is whether you deal with orthogonal states or not. And if your quantum states are not orthogonal, then quantumness is how non-orthogonal your state is with respect to your measurement basis.

How well can you quantum-clone then? What's the highest achievable fidelity? This question was answered by Gisin and Massar a year after quantum cloning machines were invented [6]. They found that the optimal fidelity for a machine that tries to make two copies from a single unknown quantum state is $F=5/6$. That's not bad, right? Actually, they got an even more impressive result: they showed that if you had $N$ identically prepared quantum states (prepared by someone other than you, because you remember that you are not to know what kind of quantum state you are trying to clone), and you want to create $M$ copies of this states ($M$ sort-of-copies, that is), then the best you can do with an optimal universal cloning machine is

$F=\frac{M(N+1)+N}{M(N+2)}$    (2)

The "universal" here means that this cloning machine will achieve the fidelity no matter what the initial state is. There are cloning machines that can do better for some states, and worse for others. These are called "state-dependent" cloning machines.

Now, back to black holes. What kind of cloning machines are they? What's their fidelity? This, it turns out, depends on how reflective the black hole is.

"Reflective? Aren't black holes supposed to be black, ergo non-reflective?"

Actually, not necessarily. Black holes can reflect stuff, in particular if you hurl something at the black hole somewhat at an angle (that is, not straight on, as in the figure below)

Black holes are actually surrounded by a potential barrier, which we can choose to model by a semi-transparent mirror surrounding the black hole, with reflectivity $1-\alpha$. If you read the post about the quantum capacity of black holes, then you remember this reflectivity.

So, let's first consider black holes that perfectly reflect radiation. I called them "white holes" earlier, even though this is not exactly how white holes are defined in the literature. But I really don't care, because I believe that there is a fundamental relationship between black holes and white holes that involves time-reversal, or else flipping the inside and the outside of the holes. Keep on reading if that intrigues you.

So, nothing can enter a white hole, while stuff from inside the white hole makes it out unhindered. But this white hole is very different from a mirror, because besides the reflection, it also stimulates the emission of radiation in reponse to the stuff that is reflected. And these stimulated states are the clones of the incoming states. This means of course that after you send in the quantum state $\phi$, the black hole returns two almost-clones (one from the reflection, and one from stimulated emission). And guess what: the fidelity of these clones is $F=5/6$. The white hole is an optimal universal quantum cloning machine!

Now that I impressed you with this statement, let me quickly make it even more impressive. It turns out that the black hole isn't just a $1\to 2$ cloner. Because the stimulated emission of radiation produces an arbitrary number of "copies", the white hole is a $1\to M$ cloner if you need it to be. In fact, it will be an $N\to M$ cloner if you want. And it will perform this feat with the optimal fidelity of Gisin and Massar. That's Equation (2) above.

Now, if you're not duly impressed, this is perhaps because you think that white holes aren't that interesting. But you should keep in mind that in Hawking's original formulation, black holes did not absorb anything either, as paradoxical as that sounds. Also, mirrors are not universal quantum cloners. You need the stimulated emission effect to make this possible.

Now, let us look beyond the horizon. Even though the white hole perfectly reflects the quantum states you fling at it, there is actually stuff beyond the horizon. Indeed, as described in the quantum capacity blog post, there are anti-clones behind the horizon.

"Anti-clones? Are you just all-out kidding me now?"

"Anti-clone" is a good word. But it really is a thing. Anti-clones are the stimulated "twin" of the clone outside the black hole. They must be there, because you can't just stimulate a copy without violating a bunch of conservation laws, like particle number, momentum, and whatever else characterizes the thing you send in. You've got to do this in twins: particle-anti-particle, clone and anti-clone.

"What is the fidelity of the anti-clone? Is it 5/6 also?"

The answer is no. That would be bad as I'll discuss. That anti-clone has a fidelity of 2/3.  Strange you think? Not after I'll tell you what this number represents. In fact, the fidelity of the anti-clones behind the horizon (any number $M$ of them, in fact), given that somebody sent in $N$ identically prepared copies (that were all reflected, of course), is independent of $M$ and given by

$F=\frac{N+1}{N+2}$ .   (3)

So, that's as good as you can hope to reconstruct an arbitrary quantum state using the anti-clones behind the horizon.

That's actually a very interesting result, because it happens to be Pierre Simon Laplace's "rule of succession" that he derived in the 18th century. Not in the context of black holes, mind you, but in the context of our sun.
Pierre-Simon Laplace (1745-1827)
Source: Wikimedia

This is the probability that the sun will rise tomorrow given that you have observed it to rise $N$ previous times. It assumes a "prior" that posits that both the sun rising as well as not rising are possible outcomes of your "experiment", and these are added to the $N$ total observations. Thus, out of $N+2$ events, $N+1$ have the sun rising. This correspondence is surprising, because the fidelity (3) is in fact the probability to correctly estimate the state of a quantum two-state system (while the random variable "sun rising" is classical) using $N$ classical measurements only, as was shown by Massar and Popescu in 1995 [5]. Somebody should investigate this curious coincidence, that in my view is not a coincidence at all.

All right then, to sum it up: For a white hole, the clone fidelity is the optimal 5/6, and the anti-clone fidelity (behind the curtain) is 2/3, which is the best quantum state reconstruction you can do with classical means (like, measurements).

What about perfectly absorbing black holes? I'll make that short and sweet given all that we learned. The fidelity of clones outside is 2/3, while the fidelity of the anti-clones inside is 5/6.

"It's just the opposite from the white hole situation! It is as if the inside and the outside of the black hole had been flipped!"

I told you so, didn't I. Yes, sitting inside of a black hole (if your turbine engines allow you to sit), you despair that none of your signals make it outside. Your signals are turned back towards you, as if they were reflected. As if you were looking at a white hole horizon.

To wrap this meandering post up, Greg and I did calculate the cloning fidelity (for the clones outside the horizon) for arbitrary reflectivity. It turns out that this fidelity is very close to the optimal one as long as the black hole is not too tiny (see the figure below).
Cloning fidelity as a function of the number of clones produced, for moderately-sized black holes, and different black hole absorptivities. 
So, black holes are almost optimal quantum cloners. Who knew?

Actually, this possibility was discussed briefly as early as 1990, according to Lenny Susskind in his book "The Black Hole War". Indeed, Susskind writes that he proposed (in front of Sid Coleman and Stephen Hawking) that the problem would be solved if "the region just outside the horizon is occupied by a lot of tiny invisible Xerox machines" [6, p. 227]. But he then immediately retreated from this idea, because he thought it would violate the no-cloning theorem. Which we now know it does not.

Susskind later revived the idea in his "black hole complementarity" proposal, claiming that somehow information would both fall into the black hole and be reflected at the horizon, but that the no-cloning theorem would not be violated because nobody would ever know (as you can't make an experiment both inside and outside of the black hole). This idea is, as I'm sure you can now see as clear as daylight, based on a profound misunderstanding of quantum cloning, and in particular its relation to stimulated emission of radiation.

Finally, given that Pierre-Simon Laplace occupies such an interesting place in this post, I ought to note in passsing that he is the one who invented (discovered?) the special function now called "Spherical Harmonics", which plays such a fundamental role in quantum physics (as part of the wavefunction of the hydrogen atom).  Welcome home, Laplace!

Note added June 2015: The subject matter discussed in this blog post is in Ref. [7]. The paper is Open Access, so you can go ahead and read it right now!


[1] W. K. Wootters and W. H. Zurek, A single quantum cannot be cloned. Nature 299, 802 (1982) 
[2] D. Dieks, Communication by EPR devices. Phys. Lett. A 92, 271 (1982). 
[3] V. Buzek and M. Hillery, Quantum copying: Beyond the no-cloning theorem. Phys. Rev. A 54, 1844 (1996).
[4] N. Gisin and S. Massar, Opimal quantum cloning machines. Phys. Rev. Lett. 79, 2153–2156 (1997). 
[5] S. Massar and S. Popescu,  Optimal extraction of information from finite quantum ensembles. Phys. Rev. Lett. 74, 259–1263 (1995).
[6] L. Susskind, The Black Hole War. Back Bay Books, 2008.
[7] C. Adami and G. Ver Steeg, Black holes are almost optimal quantum cloners, J. Phys. A 48,  23FT01 (2015).