Maximal I3322 violation requires infinite local dimension
So it looks like I’m starting a blog. I don’t know why, honestly—maybe because I’m bored and have a big ego. Anyway, if you’re the target audience, you’ll have no idea what the next paragraph means. That’s not because I hate you, but because... well, you’ll see. However, if you keep reading, you’ll hopefully understand by the end of this post! If you don't like it (or even if you do!), you should let me know what you think (in order of preference: Signal > CMU email > Discord > violet_connor@proton.me > Insta).
I recently finished writing a paper that, hopefully, proves, the maximum violation of the Bell inequality for I3322 requires infinite local dimension (although one can get arbitrary close with finite dimensions). There's an ordinary normalizable infinite-dimension spatial strategy that reaches this limit. Here's a formal statement for those of you more mathematically inclined:
Basically, β* is the best possible quantum value. βd is the best value if Alice and Bob are allowed a quantum system with d dimensions. Every finite d fails to reach the maximum, but βd is nondecreasing for d and approaches B*, meaning we can get arbitrarily close.
What is I3322? / What's already known?
Bell inequalities are tests that separate quantum mechanics from classical explanations. We call something a violation if it achieves more correlation between particles than classical mechanics would allow, and these are one way we know classical mechanics is incomplete. The most famous and simplest Bell inequality is a test case called CHSH (Clauser–Horne–Shimony–Holt). In this, two people, Alice and Bob, receive random hidden instructions from a shared source; they measure paired particles at different angles. Without communicating, they try to produce outputs satisfying a prescribed relation. If particles act like pre-set objects, their maximum score is at most precisely 75% (They can reach this with some fancy game theory approaches but all that's by-the-by). For our purposes, note that, if they use some rules of quantum mechanics, they can break this limit, reaching up to approximately 85%.
I3322 is the simplest Bell inequality after CHSH. Instead of two measurements, Alice and Bob get three each, each of which has two possible answers. If Alice and Bob only use single qubits, they can reach a value of 1/4 (a qubit is a two-dimensional system; you may have noticed I use the word dimension a lot, and it's probably well past time I clear up confusion about that. In this area of physics, and many others, dimension doesn't typically refer to some grand concepts about the universe or whatever you learned from Marvel movies. Instead, it refers to the 'degrees of freedom', or however many possibilities there are. To elucidate, an example: if a switch has two options (on-off), it has two dimensions. If it has three options, it has three dimensions. If you have two switches each with two options, you have four options: off&off, off&on, on&off, on&on). However, larger quantum systems can do better than qubits. As the dimension increases, the limit appears to be around 0.250875 per Pál and Vértesi (2010); this paper found that as an infinite limit but did not prove finite dimensions fail to attain it. This suggested the possibility that there is no no finite-dimensional system that is actually best. Perhaps the bound at infinite dimensions could be approached but never reached.
My paper: getting rid of the dimension
The paper focuses on rewriting the problem so that the (Hilbert-space) dimension disappears. This is because a Bell strategy is typically a quantum state together with several measurement operators, which are matrices of arbitrarily large size. Instead of optimizing over all of those matrices, the problem is much easier if we reduce it to optimizing over a propability distribution η on the interval from −1 to 1
The function F is intimidating but, essentially, it contains an average, a term that pairs opposite parts of the distribution, and another term that compares the distribution with its reflection.
The important part is that the same function controls every possible quantum strategy, whether its dimension is finite or infinite.
Some manipulation reveals this also gives us analytic (analytic means proof-based; the opposite would be numerical, which is test-based) bounds on the answer:
This proves the true unrestricted value is larger than the qubit value of 1/4.
My paper: turning the answer back into a quantum state
Of course, now that everything is nice and simple, we have to make it complicated again. Because a probability distribution is not a full Bell experiment, we still need to show the best distribution corresponds to a quantum state and measurements.
The construction in the paper turns the distribution into a chain by the following process. Points in the distribution get paired with reflected points, and those pairings produces a sequence. That sequence determines a set of quantum amplitudes, which become the Schmidt coefficients of the state (Schmidt coefficients are basically how we define the state, by stating how tightly entangled particles are).The result is an infinite chain (as Yoda would put it: "Distribution becomes points. Points become paired points. Paired points become sequences. Sequences become amplitudes. Amplitudes become Schmidt coefficients. Schmidt coefficients become a chain.") with coefficients that get smaller quickly enough that their total squared size is finite. This means the state is perfectly normalizable (meaning that the probability of finding the particle across the whole space is exactly 1). This construction is based on an earlier numerical construction by Pál and Vértesi (the aforementioned 2010 paper). The difference is that here the chain comes from the conditions for reaching the unrestricted maximum, instead of being guessed at.
At least one of these infinite chains we've constructed (well, I have but you can read the paper and do it too. Try this at home kids!) reaches β* exactly
My paper: why finite dimension fails
Let us assume a finite-dimensional strategy reaches β*. If so, the probability distribution produced by the reduction above would only need finitely many points to maximize F.
If this is the case, the conditions for equality force the points and amplitudes in the chain to obey a strict recurrence. If the distribution had only finitely many points, the labels on the infinite chain would eventually have to repeat in a pattern and, ultimately, become constant. However, that pattern does not satisfy all of the equations simultaneously: the amplitudes would have to shrink (so that the quantum state remains normalizable), while the equality equations force the same ratios to keep propagating backward through the chain. Continuing this process eventually either hits an impossible endpoint or produces a state whose amplitudes do not have a finite squared sum.
This means a finite-support maximizing distribution does not exist, and therefore neither can a finite local dimension strategy that attains β*. There is also a slightly stronger version of this result the paper proves: every nonzero part of a maximizing normal state has infinite Schmidt rank. In other words, the infinite-dimensionality extends to every part of the construction.
My paper: getting arbitrarily close
However, like I've mentioned, none of the above means that finite-dimensional systems stay far below the maximum. You can cut off the infinite chain at an arbitrary point, with each cutoff giving a finite-dimensional quantum strategy. As you keep more terms, its I3322 value gets closer and closer to β*. To any precision, a sufficiently large finite system can imitate it.
Future work: open questions
What actually is the optimal strategy? How quickly can you approach β* with that?
Are there unique infinite-dimensional states that reach β*? For reference, CHSH has only one viable strategy (well technically there are local isometries, ancillary systems, and the usual symmetries but if you're quibbling over that, you really should just read the paper).
There are also several generalizations that suggest themselves, to larger systems, to apply the Jacobi chain elsewhere, and so on.
Future work: cryptographic applications
Cryptography is the study of how we can make communications and keep data secure. Particularly relevant is device-independent quantum cryptography, which creates security from the observed input-output behavior of devices rather than from trusting their internal construction. This is, naturally, based on Bell violations. Such violations also relate to device-independent randomness generation and related certification protocols.
This paper does not show that I3322 gives a better cryptographic protocol, or that anyone needs an infinite-dimensional computer to send a secure key. Real experiments always have noise and finite precision, and but so a sufficiently large finite-dimensional system can imitate the infinite-dimensional optimum as closely as the Bell experiment can resolve.
However, could the observed I3322 value be proven to form a lower bound on the amount of entanglement inside an untrusted device or the eavesdropper's information? Could one build a protocol in which approaching the I3322 maximum necessitates more and more quantum resources?
AFAICT, all of these would require a theorem for β* − ε.
Umm, I don't know how to conclude. This is more of a practice blog post than anything so ... yeah. Have a nice day!
--V. (I always think it's cool when people sign off like this and now I get the chance to do it mwahahaha)
Manuscripts
I wrote two versions that I'll upload here and to arxiv concurrently. I haven't done so yet because I'm a perfectionist and procrastinator.
- Letter: Maximal I3322 Violation Requires Infinite Local Dimension (PDF)
- Full article: Variational Characterization and Rigidity of I3322 Violation (PDF)
- LaTeX source package