Spooky Action at a Distance: The No-Communication Theorem

Eliezer, I know your feelings about density matrices, but this is exactly the sort of thing they were designed for. Let ρAB be the joint quantum state of two systems A and B, and let UA be a unitary operation that acts only on the A subsystem. Then the fact that UA is trace-preserving implies that TrA[UA ρAB UA*] = ρB, in other words UA has no effect whatsoever on the quantum state at B. Intuitively, applying UA to the joint density matrix ρAB can only scramble around matrix entries within each "block" of constant B-value. Since UA is unitary, the trace of each of these blocks remains unchanged, so each entry (ρB)ij of the local density matrix at B (obtained by tracing over a block) also remains unchanged. Since all we needed about UA was that it was trace-preserving, this can readily be generalized from unitaries to arbitrary quantum operations including measurements. There, we just proved the no-communication theorem, without getting our hands dirty with a single concrete example! :-)

Scott, I am sure that would be a deeply satisfying explanation, and moreover, I would be able to find a nice concrete example by which I could explain it to all my readers, if only I knew what the hell a density matrix means, physically. Not how to define it, what it means. This information seems to have been left out of physics textbooks and Wikipedia.

I have never really had time to sit down and just study QM properly. I'm sure that the meaning of a density matrix will be completely obvious in retrospect. I'm sure that I will slap myself on the forehead for not getting earlier. And I'm sure that, once I finally get it, I will be filled with the same feeling of absolute indignation that overtook me when I realized why the area under a curve is the anti-derivative, realized how truly beautiful it was, and realized that this information had not been mentioned anywhere in my goddamned calculus textbook. Why?

Scott: I'm a bit confused by what you're doing here: TrA[UA ρAB UA*] = ρB

Specifically, I understand what a trace is, but what's TrA mean? ie, I'm guessing it's not a scalar given that it looks like multiplying (if that's the intended operation) it by the transformed density matrix of AB gives you a density matrix of B, which presumably has fewer dimensions.

Psy-Kosh: TrA just means the operation that "traces out" (i.e., discards) the A subsystem, leaving only the B subsystem. So for example, if you applied TrA to the state |0〉|1〉, you would get |1〉. If you applied it to |0〉|0〉+|1〉|1〉, you would get a classical probability distribution that's half |0〉 and half |1〉. Mathematically, it means starting with a density matrix for the joint quantum state ρAB, and then producing a new density matrix ρB for B only by summing over the A-indices (sort of like tensor contraction in GR, if that helps).

Eliezer: The best way I can think of to explain a density matrix is, it's what you'd inevitably come up with if you tried to encode all information locally available to you about a quantum state (i.e., all information needed to calculate the probabilities of local measurement outcomes) in a succinct way. (In fact it's the most succinct possible way.)

You can see it as the quantum generalization of a probability distribution, where the diagonal entries represent the probabilities of various measurement outcomes if you measure in the "standard basis" (i.e., whatever basis the matrix happens to be presented in). If you measure in a different orthogonal basis, identified with some unitary matrix U, then you have to "rotate" the density matrix ρ to UρU before measuring it (where U is U's conjugate transpose). In that case, the "off-diagonal entries" of ρ (which intuitively encode different pairs of basis states' "potential for interfering with each other") become relevant.

If you understand (1) why density matrices give you back the usual Born rule when ρ=|ψ〉〈ψ| is a pure state, and (2) why an equal mixture of |0〉 and |1〉 leads to exactly the same density matrix as an equal mixture of |0〉+|1〉 and |0〉-|1〉, then you're a large part of the way to understanding density matrices.

One could argue that density matrices must reflect part of the "fundamental nature of QM," since they're too indispensable not to. Alas, as long as you insist on sharply distinguishing between the "really real" from the "merely mathematical," density matrices might always cause trouble, since (as we were discussing a while ago) a density matrix is a strange sort of hybrid of amplitude vector with probability distribution, and the way you pick apart the amplitude vector part from the probability distribution part is badly non-unique. Think of someone who says: "I understand what a complex number does -- how to add and multiply one, etc. -- but what does it mean?" It means what it does, and so too with density matrices.

As for your pedagogical question, Eliezer -- well, the gift of explaining mathematical concepts verbally is an incredibly rare one (I wish every day I were better at it). I don't think most textbook writers are being deliberately obscure; I just think they're following the path of least resistance, which is to present the math and hope each individual reader (after working it through) will have his or her own forehead-slapping "aha!" moment. Often (as with your calculus textbook) that's a serious abdication of authorial responsibility, but in some cases there might really not be any faster way.

To echo Scott, a density matrix is a probability distribution over quantum states - over a set of basis states, specifically. But if you pick a new basis, the same density matrix resolves into a different probability distribution over a different set of quantum states. If you believe that reality does reduce to amplitude flows in configuration space, then that means that one basis, the position basis, is the real one (since it corresponds to different possible states of reality, i.e. distributions of amplitude in configuration space); you can think of a density matrix as a probability distribution, period, and you're done. Density matrices will mean trouble if and only if you want to think of various incompatible choices of basis as equally real.

I would suggest that understanding superdense coding is another test of whether one's explanation of the significance of entanglement works. There is no purely quantum communication at a distance, but there can be a quantum rider on an otherwise classical communication channel.

when I realized why the area under a curve is the anti-derivative, realized how truly beautiful it was, and realized that this information had not been mentioned anywhere in my goddamned calculus textbook. Why?

Have you seen better since? Please can anyone recommend a high quality, intuitive, get-what-it-actually-means, calculus tutorial or textbook, preferably with exercises, and if so, please could you share the link?

While I'm asking, same question for Statistical hypothesis testing. Thanks.

Eliezer; My suspicion is that a better representation might take its basic mathematical objects as local states of entanglement.

A local state of entanglement is a bit of an oxymoron. Though you could take one part of an entangled state and trace over all distant degrees of freedom. You'd end up with a "reduced density matrix". If you did that for each part separately, you'd have a set of reduced density matrices. To reconstitute the whole, you need some extra information as well, about how they fit together. In any case, that would be one way to pursue this program; a potentially more sophisticated version of searching for the "elements of reality" in the tensor factors of the global quantum state, which is a bit like using entanglement to define what is local, rather than vice versa.

If I had the time myself (and maybe I'll make the time), I would be trying to pursue this line of thought in the context of "M(atrix) theory", which I believe is a limit of string theory that reduces to point objects ("D0-branes") connected by a web of strings (the rows and columns of the capital-M Matrix correspond to the D0-branes, the matrix elements to the strings connecting them). There are many people who think that those are the fundamental degrees of freedom of string theory, and it has the Machian, particulate, geometry-independent feel that one might expect of the bottom level. You would then be trying to piece together the quantum state of the universe from reduced D0-brane density matrices, basically. But I think that if you could do this, you wouldn't need the many-worlds perspective any more. These quasilocal component quantum states would not be further reduced to amplitude distributions on little configuration spaces; they would be the ultimate states of things themselves.

Mitchell: No, even if you want to think of the position basis as the only "real" one, how does that let you decompose any density matrix uniquely into pure states? Sure, it suggests a unique decomposition of the maximally mixed state, but how would you decompose (for example) ((1/2,1/4),(1/4,1/2))?

Scott: Think of someone who says: "I understand what a complex number does -- how to add and multiply one, etc. -- but what does it mean?"

Sounds like a perfectly legitimate question to me. Feynman's excellent answer is that, in the context of QM, it means a little 2D arrow.

I say this tongue-in-cheek and completely seriously at the same time.

The idea that density matrices summarize locally invariant entanglement information is certainly helpful, but I still don't know how to start with a density matrix and visualize some physical situation, nor can I take your proof and extract back out an argument that would complete the demonstration in this blog post. I confess this is strictly a defect of my own education, but...

But still, surely you see the difference between saying "Now let this be a trace-preserving operation on this density matrix that is the outer product of the AB state," and saying "Now split up the joint amplitude distribution on A and B according to distinct states of B, and let anything whatsoever happen to the A side; since the evolution is unitary, it won't change the squared modulus of any B-group of states, hence it won't change the perceived probabilities at B."

Recovering irrationalist, no, I've never seen a good calculus textbook in my life. Admittedly my requirements are unusual: The book I've always wanted to read is "The Pure Joy of Calculus For People Who Are Good At Math", rather than "A Giant Dull Tome of Calculus For Students Who Would Rather Be Somewhere Else" or "Calculus for Nitpicking Formalists".

Scott: Thanks, that clarifies it. (As far as the, what now I look up and find to be the partial trace thing)

Scott and Eliezer: As far as what density matricies "really physically are"... well, this isn't an answer so much as a notion of what form the answer might take: Since the laws of probability and rationality are LAWS rather than "just good ideas", it isn't entirely shocking that there'd be some mathematical object th that would seem to act like the place where the territory and map meet. More to the point, the some mathematical object related to the physics that says "this is the most accurate your map can possibly be given the information of whatever is going on with this part/factor of reality."

Alternately, since our map has to be encoded physically. ie, the map isn't the teritory, but the territory contains the map, then not too shocking that there's something about reality that could tell us something semidirectly about maps.

I may just be reshuffling my confusion here. In fact, I pretty sure I am, but it does seem to me that something like what I said above is possibly "what a density matrix 'really' is"

"The idea that density matrices summarize locally invariant entanglement information is certainly helpful, but I still don't know how to start with a density matrix and visualize some physical situation, nor can I take your proof and extract back out an argument that would complete the demonstration in this blog post. I confess this is strictly a defect of my own education, but..."

From what I understand (which is admittedly not much; I could well be wrong), a density matrix is the thingy that describes the probability distribution of the quantum system over all possible states. Suppose that you have a set of quantum states A1...An. The density matrix is a way of describing a system that, say, has a 75% chance of being in A1 and a 25% chance of being in A2, or a 33% chance of being in A1 or A2 or A4, or whatever. You can then plug the density matrix into the standard quantum equations, but everything you get back will have one extra dimension, to account for the fact that the system you are discussing is described by a distribution rather than a pure quantum state.

The gist of Scott Aaronson's proof is (again, if I understand correctly): Suppose that you have two quantum systems, A and B. List the Cartesian product over all possible states of A and B (A1B1, A2B1, A3B1, etc., etc.). Use a density matrix to describe a probability distribution over these states (10% chance of A1B1, 5% chance of A1B2, whatever). Suppose that you are physically located at system A, and you fiddle with the density matrix using some operator Q. Using some mathematical property of Q which I don't really understand, you can show that, after Q has been applied, another person's observations at B will be the same as their earlier observations at B (ie, the density matrix after Q acts the same as it did before Q, so long as you only consider B).

Eliezer: "Why can't you signal using an entangled pair of photons that both start out polarized up-down? By measuring A in a diagonal basis, you destroy the up-down polarization of both photons. Then by measuring B in the up-down/left-right basis, you can with 50% probability detect the fact that a measurement has taken place, if B turns out to be left-right polarized ... the answer turns out to be simple: If both photons have definite polarizations, they aren't entangled."

You can adjust this slightly so that answer no longer applies. Start with two entangled photons A and B that we know have opposite polarizations, so they really are entangled. At A we have a detector behind a filter that can be rotated either vertically or at a 45 degree angle. This is our signal source.

At B, we use a mirror that reflects, say, vertically polarized photons and transmits horizontally polarized; then we recombine the beams from slightly different angles onto a detector. So if we were to send B photons that have gone through a vertical or horizontal filter, we get no interference pattern at the detector, but if we send it photons that went through a diagonal filter, one would show up.

Now if we put the diagonal filter on at A, we know the diagonal polarization at B, and therefore do not know the horizontal/vertical polarization, and so we get an interference pattern. If we put vertical filter on at A, we know the vertical polarization at B, and the interference pattern disappears. Thus we seem to have faster-than-light (or back in time, if you prefer) communication.

(Of course this doesn't actually work, but I think it's a lot harder to explain why in understandable terms.)

Since the laws of probability and rationality are LAWS rather than "just good ideas", it isn't entirely shocking that there'd be some mathematical object th that would seem to act like the place where the territory and map meet. More to the point, the some mathematical object related to the physics that says "this is the most accurate your map can possibly be given the information of whatever is going on with this part/factor of reality."

That's a beautiful way of putting it, which expresses what I was trying to say much better than I did.

That is a good puzzle, Jeff. I remember debating a similar experiment a few years ago on a mailing list. At the time I discovered an analysis of a related idea at http://www.flownet.com/gat/QM.pdf.

The conclusion was that you don't get interference regardless of what you do at the other end, because the paths are potentially distinguishable. There would only be interference when paths are indistinguishable. In other words, photons in the entangled state are kind of weird. I think in terms of the discussion above, you might say that such a photon needs to be described by a density matrix rather than a state vector. This makes it work a little differently than photons from simpler sources.

It's particularly annoying that nobody gave me an answer, because the answer turns out to be simple

The answer you give might be the simple answer to the question you asked, but I have trouble figuring out what you were suggesting in the first place (even given the answer!). Figuring out the question, the confusion, so that it's possible to supply the simple answer is really hard.

Similarly, I'm skeptical of your claim that your calculus text failed an easy chance to communicate insight. I've had a lot of bad experiences with textbooks, where I eventually figure it out, perhaps from another source, come back and can't see what was wrong with the book. If there is something worthy of indignation for its not being shared, why don't you share it?

Let me try, and likely fail, to communicate mathematical insight: matrices are evil. Moving to matrices involves choosing a basis. Usually, as in Scott's example, you just want a direct sum decomposition; it's more natural, and it doesn't clutter the problem with unnecessary entries or indices.

Scott: how does that let you decompose any density matrix uniquely into pure states?

Yes, that was a very bad thing I said. Because even if my density matrix is diagonal in one basis, if I nontrivially change basis it will no longer be diagonal, and so won't be interpretable as a straightforward probability distribution over the new basis states. I must retract, retreat, and retrace my steps.

Scott: Thanks. Though "eeeew" at all my typos. Anyways, there're still aspects I'm unsure of, but hopefully with more reading, playing around with the relevant mathematical dohickies, and thinking, I'll gain a better grasp.

I find myself unable to give an "interpretation-independent" account of what a density matrix is that would be any advance on what's already been said. It is, among other things, a way to represent a probability distribution over quantum states, but you can get the same density matrix from different starting points; but that is less of a problem if you have decided in advance that only certain starting points (e.g. configuration basis, position eigenstates) correspond to reality; that's what I should have said. And the response of a diehard "positionist" to Scott's challenge would be, I think, that any density matrix which cannot be reduced to a mixture of position-basis pure states could only have arisen as the reduced density matrix describing some part of a larger pure state.

I suppose the bigger question is whether the formal ability to change basis in Hilbert space, or even to work independently of any basis at all, is a principle which should be accorded the same significance as, say, special relativity. There is a curious argument for the priority of position, utilizing probability currents and "weak-valued measurements". In that paper it is expressed in the context of Bohm-like hidden variables theories, but I wonder if it can be transposed into a many-worlds perspective.

Hal Finney: "...at the time I discovered an analysis of a related idea at http://www.flownet.com/gat/QM.pdf".

That's a great link--thanks! That puzzle puzzled me for years, ever since I read about some EPR experiments in Scientific American as a kid, and wondered why they didn't just tweak the experiment a bit to make it actually interesting. That paper is the best explanation I've seen by far.

The conclusion was that you don't get interference regardless of what you do at the other end, because the paths are potentially distinguishable.

That's not quite true. The conclusion was that there actually is interference at the other end, but there are two interference patterns that cancel each other out and make it appear that there is no interference. You can apparently produce interference by bringing (classical) information back form one end of the experiment to the other, but you aren't really creating it, you are just "filtering out" interference that was already there.

Hal Finney: From that pdf

Spooky correlations between separate photons were demonstrated in an experiment at the Royal Signals and Radar Establishment in England. In this simplified depiction, a down-converter sends pairs of photons in opposite directions. Each photon passes through a separate two-slit apparatus and is directed by mirrors to a detector. Because the detectors cannot distinguish which slit a photon passes through each photon goes both ways generating an interference pattern.... Yet each photon's momentum is also correlated with its partner's. A measurement showing a photon going through the upper left slit would instantaneously force its distant partner to go

through the lower slit on the right.

I'm so glad that paper says measurement is entanglement because what I've been thinking is that consecutive photons, in a dimmed to one photon at a time two-slit experiment, are the ones interfering not the photon interfering with itself.

Also everything so far from this series says that polarization also determines the slit that is chosen.

What would happen if you had three sources on a rotating platform each taking turns firing at the two slits?

(I can't find the "rerun" version of this page, so am posting my questions here).

1. For all these types of experiments, how do they "aim" the particle so it hits its target from far away? It would seem that the experimenters would know pretty much where the particle is when it shoots out of the gun (or whatever), so would not the velocity be all over the place? In the post on the Heisenberg principle, there was an example of letting the sun shine through a hole in a piece of paper, which caused the photons to spread pretty widely, pretty quickly.

2. Does the polarization vector change as the photon moves along? It seems to be very similar to a photon's "main" wave function, as it can be represented as a complex number (and is even displayed as an arrow, like Feynman uses). But I know those Feynman arrows spin according to the photon's wavelength.

3. Finally - and this is really tripping me up - why can we put in the minus sign in the equation that you say "we will need" later, instead of a + sign? If you have two blobs of amplitude, you need to add them to get the wave function, yes? If that is not the case, I have SEVERELY misunderstood the most basic posts of this sequence.

There must be something I'm missing here. The previous post pretty definitively proved to me that the no communication clause must be false.

Consider the latter two experiments in the last post:

A transmitted 20°, B transmitted 40°: 5.8%

A transmitted 0°, B transmitted 40°: 20.7%

Lets say I'm on Planet A and my friend is on Planet B, and we are both constantly receiving entangled pairs of photons from some satellite stationed between us. I'm filtering my photons on planet A at 20°, and my friend on planet B is filtering his at 40°. He observes a 5.8% chance that his photons are transmitted, in accordance with the experiment. I want to send him a signal faster than light, so I turn my filter to 0°. He should now observe that his photons have a 20.7% chance of being transmitted.

This takes some statistical analysis before he can determine that the signal has really been sent, but the important part is that it makes the speed of sending the message not dependent on the distance, but on the number of particles sent. Given a sufficient distance and enough particles, it should be faster than light, right?

From the decoherent point of view, the no-communication theorem is fairly simple (if you are comfortable with tensor products*). Suppose that Alice and Bob are studying the two quantum systems and , whose state spaces are represented by Hilbert spaces and , respectively. Then the state space of the joint system is . Now suppose that Alice makes a measurement on** system , and Bob makes a measurement on system . These measurements are represented physically by unitary transformations and . The effect of the measurements on the joint system are therefore represented by the unitary transformations and , where and are the identity transformations on and , respectively. The key to the no-communication theorem is the observation that the transformations and commute with each other. (Either way you take the product you get .) It implies that if we do our calculations assuming that Alice did her measurement first, then we will get the same answers as if we do our calculations assuming that Bob did his measurement first. So let's do our calculations assuming that Bob measured first, as it will be easier to analyze that way.

After Bob makes his measurement, the amplitude of the universe is split up into two blobs, one corresponding to Bob recording Possible Outcome 1 and another correspondint to Bob recording Possible Outcome 2. The size of these blobs, as measured by square-integrating, is independent of anything that Alice does (since according to this formulation of the problem, Alice hasn't done anything yet). Now when Alice makes her measurement, the size of the blobs is preserved because of unitarity. Moreover (and this is the crucial point) the blob corresponding to Outcome 1 gets mapped to another blob corresponding to Outcome 1, and the blob corresponding to Outcome 2 gets mapped to another blob corresponding to Outcome 2. Thus, the final size of the blobs corresponding to the different outcomes is independent of Alice's choice, and according to the Born probabilities that means Bob's expectations about his measurement are also independent of Alice's choice.

The fact that outcomes are preserved under Alice's action is worth remarking further on. Intuitively, it corresponds to the fact that recorded measurements don't erase themselves randomly. Scientifically, it corresponds to the complicated phenomenon known as decoherence, which is much harder to describe rigorously than the no-communication theorem is. Philosophically, it corresponds to the fact about the world that the Copenhagen interpretation thinks of as an assumption, and which many-worlders think too complicated to be considered a fundamental assumption of physics.

* For those not familiar with tensor products, they are the mathematical objects Eliezer is implicitly talking about whenever he writes things like "(Human-LEFT Sensor-LEFT Atom-LEFT) + (Human-RIGHT Sensor-RIGHT Atom-RIGHT)". A working definition is that the tensor product of an M-dimensional space with an N-dimensional space is an MN-dimensional space.

** And/or a modification to system ; the composition of any number of measurements and modifications will always be represented by a unitary transformation.

A final remark: The no-communication theorem, as I've sketched it above, shows that entangled but noninteracting particles cannot be used for distant communication. It says nothing about faster-than-light communication, as it does not make the connection between the ability of particles to interact and the speed of light, a connection which requires more formalism. The fact that FTL communication is impossible is a theorem of quantum field theory, the relativistic version of quantum mechanics. The basic idea is that the evolution operators corresponding to spacelike separated regions of spacetime will commute, allowing the above argument to take place with and replaced by more realistic operators.

This post is tagged "signaling". :-)