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My text shows (sections 0.2 and 0.3) that the joint "state space" of a system composed of two subsystems with $k$ and $l$ "bits of information", respectively, requires $kl$ bits to fully describe it. A critical consequence of this is that the $k+l$ bits in each of the individual subsystems are not sufficient to "span" their joint space, so that there must be states that are "entangled" (formally, cannot be expressed by the tensor product of the states of the subsystems). That much I (think) I understand.

But the text then seems to argue that this is a property that is exclusive to quantum systems. Is that true? Certainly there are classical systems where the state of one subsystem depends on the state of another. Doesn't any joint system that requires conditional probabilities to describe need "extra bits" beyond those necessary to describe the individual subsystems?

Perhaps I'm missing some subtlety about what constitutes a "sub system" or some unstated assumption about how systems are broken into parts. Perhaps a classical system that is "entangled" by conditional probabilities isn't thought of as consisting of valid "subsystems" in the same way that quantum system is.

Is "entanglement" unique to quantum systems? Are conditional probabilities just a kind of "classical entanglement"?


Please be pedagogical. This is not meant to be a deep or advanced mathematical question, just a simple conceptual and definitional one. Just imagine I'm having basic probability explained to me (without reference to quantum vs. classical, or even physics). If one struck out all occurrences of the word "quantum" from the linked text (sections 0.2. and 0.3), wouldn't that be a perfectly valid part of such an explanation?

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"But the text then seems to argue that this is a property that is exclusive to quantum systems." Which paragraph or line are you referring to in the linked file? (I have it in front of me, so just giving me the starting words will do.) –  Glen The Udderboat Feb 23 '13 at 20:23
    
@Gug: Second paragraph of 0.3: "From this viewpoint, the situation with quantum systems is extremely paradoxical." –  raxacoricofallapatorius Feb 23 '13 at 20:24
    
@Gugg: Note that I'm, not taking the linked course, but a much more elementary one, so I'm looking for a super-basic answer. –  raxacoricofallapatorius Feb 23 '13 at 20:25
    
@Gugg: This text (p. 38; section 3.2) also seems to be making the same point, though perhaps by omission. –  raxacoricofallapatorius Feb 23 '13 at 20:29
    
That would suggest that you think that there are only two kinds of systems: classical and quantum. And you could be right, for all I know. :) If that is indeed the case, then check Dirac's description of Quantum superposition of which quantum entanglement is a form. –  Glen The Udderboat Feb 23 '13 at 20:35
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3 Answers

You are correct that the description of a classical probability distribution of a joint system requires $kl$ parameters. There indeed is a difference between classical and quantum systems in this sense, but it is more subtle.

Every classical probability distribution can be described as a probabilistic mixture of deterministic states. For these deterministic states (extreme points of the space of probability densities), the description complexity of a joint system can be described by $k+l$ bits of information.

Every quantum density matrix can be described as a probabilistic mixture of pure states. For these pure states (extreme points of the of density matrices) the description requires $kl$ parameters.

Thus, classical probabilistic systems can be described in terms of probability distributions over more fundamental objects: deterministic states. These deterministic states only require $k+l$ parameters.

Quantum mixed states (the quantum analogue of probability distributions) can also be described in terms of probability distributions over more fundamental objects: pure states. However, these pure states now require $kl$ parameters.

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Note that I wasn't as clear as I should have been in the original question, where I rely too much on the context of the passage I was citing. There, "entanglement" is effectively defined to mean nothing more than "unfactorable" — i.e., requiring more than $k+l$ bits. Hence my note at the end about what I am seeking: "If one struck out...". If "entanglement" is formally defined differently (e.g. requiring something specific about the substates, such as being physically distant) then my question isn't really about entanglement, but about this specific (factorability and dimensionality) property. –  raxacoricofallapatorius Feb 24 '13 at 15:43
    
If you are talking only about pure states, "entanglement" and "unfactorable" are equivalent notions. Another question about that was also asked recently. –  Peter Shor Feb 24 '13 at 16:31
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Is "entanglement" unique to quantum systems?

Yes, unequivocally. As exemplified by Bell's inequality, there is a superluminal aspect to entanglement that creates experimental results that cannot be achieved fully by using any combination of fully classical mechanisms. The "hidden variable" models of quantum entanglement are just another way of postulating the question: "Is entanglement actually ordinary conditional probability in which certain variables are inaccessible to any known form of experimentation?"

Ringing the Bell

Until John Bell developed his famous inequality, no one has a way to test that idea in the laboratory. Ironically, Bell -- who was a strong supporter of Einstein's views on quantum mechanics -- was rooting for a hidden variables result, even though his name is now almost universally associated with proving the opposite case to be true.

Bell's inequality made it possible to accumulate solid experimental evidence on whether or not hidden variables, and thus ordinary conditional probabilities, were sufficient to explain quantum behaviors. The experimental outcome, which by now is very solid indeed, is that no combination of hidden variables can produce the spectra of correlations seen with quantum entanglement. In particular, if you analyzed one end of an entangled pair (e.g. entangled spin polarizations), the wave function representing the other entangled end of that pair is "instantly" updated with information about the range of possible options open to it when it is in turn analyzed. These updates are the source of the inequality in Bell's equation.

Fracturing "Now"

The nature of this entanglement-enabled "update" is quite curious.

In traditional conditional probabilities, an event that produces a correlated pair -- e.g. two arrows pointing oppositely on a dial to indicate the 100% certainty detection polarizations -- is an event that has already happened and cannot be erased. Consequently, no transfer of any kind of information is ever needed between the members of the pair; both simply contain "hidden arrows" that translate into real probability curves that when analyzed using the local setting of some detector.

Although entanglement is usually explained in terms of instantaneous "resetting" of the remote member of the pair, there is actually a simpler and more self-consistent way to understand what is going on. The first and most critical point is this: A quantum entangled pair is by definition one that has left no information record anywhere in the universe on exactly how its original entangling event occurred. That's unavoidably for a quantum scenario, since the instant such information comes into existence, that aspect of the experiment becomes classical and no longer follows quantum rules.

Now think about that for a moment. I am being neither flippant or metaphorical when I ask this question: If no record of how the original entangling event took place exists anywhere in the universe, has it really occurred yet? Causality cannot be affected in the past by what occurs to the system now, for the simple reason that by definition no conflicting history of the event exists anywhere else in the universe.

Quantum Cheats

Insisting that such unresolved quantum systems have well-defined pasts is a very Hamiltonian perspective, that is, one that insists that every component of the system have a well-defined "now" state. The Lagrangian quantum methods first proposed by Dirac, then ironically abandoned by him after Richard Feynman and Freeman Dyson took them up with a vengeance, are much more forgiving. They permit the final quantum resolution of events deep in the classical past to be remain ragged and even chaotically unresolved at multiple levels of scale. For quantum events hidden away in quiet corners of the universe, some resolutions of "how did it happen" for quantum systems can in principle remain unresolved literally for eons of classical time. For a quantum system, this high priority on immediate classical-style resolution simply does not matter. Such systems will instead remain superimposed and entangled for as long as needed, specifically until they are forced by some interaction with the information-rich classical universe to "explain" how it will ensure the absolute and universal conservation of parameters that include mass-energy, angular momentum (spin or polarization), charge, or any of the lesser known conservation rules. And then they cheat: They simply make up a history on the spot, one that always ensures that all various conservation rules really do get followed.

It is entirely self-consistent, then, to think of the act of detecting one member of an entangled pair not just as sending a "reset" instantaneously to its partner, but as deciding the original entangling event took place. And even if you don't like the idea itself... well, it turns out that it's a great way to keep it clear in your head how the conditional probabilities of an entangled event will differ from those of an otherwise similar classical conditional probability event. The first detection of an entangled event pair decides what that original event looked like... and the results of the other member of the pair must then work with that "new" past. Incidentally, I should note that when entangled particles have space-like separation, there is also a nice and necessary symmetry by which either event can be viewed as being the one that "sets" the original event. The detection spectra work out to be the same under with either interpretation. (However, if you want to have fun looking for possible oddities, both theoretically and experimentally, that's a good area to explore.)

Are conditional probabilities just a kind of "classical entanglement"?

No, because classical conditional probabilities do not include any kind of information transfer between the two entities.

Classical Emulation of Quantum Entanglement

However, if you are dead set on doing it, you can create a very slow and cumbersome classical analog to the conditional probabilities that are characteristic of quantum entanglement. The main thing you need to do is create your own fully classical "hidden channel" for resetting the other member of the pair after a detection takes place. That update channel must be kept protected and hidden, and the remote member of the pair cannot be allowed to be inspected or updated until the update arrives. Needless to say, the result can be almost unfathomably slow if you try to do very much of this, an inverse reflection of the speed of quantum computers (and also precisely why Feynman first proposed using quantum computers to study such systems, long before @PeterShor electrified everyone by showing that such computers could do far more than just simulations of quantum events.

Deep Patterns

One last tangent that I just have to mention: I find it absolutely fascinating that the hidden channels that I just described for simulating quantum entanglement correspond remarkably closely to the concept of atomic transactions in relational databases. As with simulating quantum events, these ACID constraints result in astronomical slow-downs if applied to networks that cover large areas. Such traditional databases thus correspond surprisingly well to classical attempts to emulate quantum systems, and as with such simulations work well only when physically localized. Conversely, new highly distributed database have BASE features that lose immediate coherency in favor of locality of processing. They correspond quite well to classical systems.

There appear to be deep information patterns that transcend many levels not just of physics, but of how technology itself is forced to evolve to reach new levels of capability. Quite fascinating, that.

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The crux of my question is: "If one struck out all occurrences of the word "quantum" from the linked text (sections 0.2. and 0.3), wouldn't that be a perfectly valid part of such an explanation?" It's about the range of possible states and whether even classical systems include states that cannot be described with $k+l$ bits, but require more (and whether "more' $=kl$). –  raxacoricofallapatorius Feb 24 '13 at 2:18
    
Heh! I'll check the links (multiplexing a bit though)... –  Terry Bollinger Feb 24 '13 at 2:25
    
Thanks. I'm guessing that the answer is "sort of", but that in fact, such systems wouldn't just be "entangled" (note the quotes; $\equiv$ needs the extra bits) but they also would not really be separable into subsystems anyway: that quantum systems are indeed unique in that such "entanglement" persists even for systems where the subsystems are "separable" (e.g. by physically isolating them from one another). –  raxacoricofallapatorius Feb 24 '13 at 2:31
    
Ouch... Well, I'll ramble instead: I agree that Sec 0.3 can "mostly" be re-interpreted in terms of classical correlations. However, if you go classical then: (1) correlation just means that very real bits have all been set to the same value in the past; (2) every bit is locally independent of every other bit; and (3) you have no superposition of states. All very finite and simple. For entangled correlation: (1) You have rules that specify valid combinations of bits, none of which actually exist as real values yet; (2) they are all non-local; (3) all valid combination co-exist at once. –  Terry Bollinger Feb 24 '13 at 3:19
    
For me, as an experimentalist, all this is a lot of navel gazing. Entanglement is a part of the quantum mechanical world which essentially says: there is no definite state and you can only calculate the probability of finding a specific state at (x,y,z,t). Entangled pairs are by definition a solution of a particular quantum mechanical equation with a probability of finding a specific state. If one measures one of the particles of the pari pair the probability is 1 that the other obeys the conservation laws, since QM solutions obey the conservation laws. What's all this fuss about? –  anna v Feb 24 '13 at 5:11
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Adding CW example of what I think I'm looking for, roughly. Feel free to adapt as a non-CW answer of your own if this is on the right track. I'll delete this once I have an accepted answer.

Yes. The description applies equally well to classical systems. But it begs the question of what "subsystem" means in that case. If you can't analyze the components of the system independently while preserving "entanglement", in what sense can they be considered distinct?

What makes a quantum mechanical system different is that things that are clearly distinct (at least in every sense we are familiar with) — they can even be in distant locations — can still exhibit entanglement, whereas a classical system would loose its entanglement as soon as its components were "disconnected".

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