So, I was studying Quantum Information Theory, and I came up with Bell's inequality.

My notes then point out that there is no version of the Bell's inequality when we are in $d > 2$.

Is that still true? Is there any reference for this, or some new theories/attempt about?

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    $\begingroup$ $\phantom{xxx}$ What is $d$? $\endgroup$ – WillO Aug 19 '16 at 16:35
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    $\begingroup$ @WillO According to the notes, $d$ is the number of (nonzero ?) eigenvalues of a given state. For example when you use one qubit you have $d = 2$. $\endgroup$ – Les Adieux Aug 19 '16 at 16:49
  • $\begingroup$ just add an eigenvalue to the superposition sum and adjust the coefficients. d=2 is the value giving the greatest deviation ( from classical statistics ) and then it is the most interesting. $\endgroup$ – user46925 Aug 19 '16 at 19:29

There are Bell inequalities for basically all scenarios you can think of.

To be a bit more precise, I'll refer to a "Bell inequality" as a mathematical inequality describing the correlations of measurements in a local hidden variable model. A violation of a Bell inequality in an experiment then shows that this experiment cannot be described by local hidden variable theories.

This implies that we have two main parameters we can play with:

  • The number of space-like separated observers,
  • the number of outcomes for an observable,
  • and the number of measurement settings for every observer.

Note that the number of outcomes per observable is at most the number of local dimensions of the state so that's not a new parameter. If I understand you correctly, you are most interested in the second parameter, however since the numbers cannot really be considered completely separatly, I'll discuss all three of them.

If you have a state, that's just a probability distribution on the possible measurement outcomes and (at least for finite dimensions) the set of local hidden variable models turns out to be a convex polyhedron in that space. A Bell inequality is nothing but a hyperplane, where the local hidden variable models all lie to one side of the hyperplane (or on it). Given enough of those hyperplanes, they will characterise the set of local hidden variable models completely. The fact that this is possible at least in finite dimensions for any of the settings you can think of follows directly from convex analysis.

Okay, but what has been done? A lot! A short overview about what can be done for different setups can be found on the Hannover QIT page: Beware, it seems to only go up to 2010 (and a lot has happened since then). Let's get more specific:

What about Bell inequalities for observables with many outcomes? One of the first examples for such Bell inequalities (bipartite system, two measurement settings each, arbitrarily many outcomes for each measurement) seems to be the 2001 paper by Collins et al. For a recent paper discussing Bell inequalities for measurements with three outcomes and three observers, see for instance the paper arXiv:1606.01991.

What about Bell inequalities for many measurement settings? One of the early references considering this question is a paper by Bacon and Toner, which gives a complete classification for bipartite systems and three dichotomic (i.e. two-outcome) measurement settings.

What about Bell inequalities for more than two observers? Also here, tripartite Bell inequalities have been known for a long time. There is a much more interesting aspect I want to highlight here: You can also ask about the amount of violation of a Bell inequality by quantum mechanics. In other words: What is the ratio between the largest amount of correlations in quantum theory and the largest amount of correlations for local hidden variable models? In the famous CHSH inequality, that ratio is $\sqrt{2}$ and for arbitrary bipartite systems, it is finite (regardless of the number of measurement settings or the number of outcomes per measurement. The ratio is always upper bounded by some finite Grothendieck constants). However, for tripartite systems, it can already be infinite as shown in the seminal work by Perez-Garcia et al.


Martin's answer is very comprehensive, but here's a specific example of a game for $d=4$: the Mermin-Peres Magic Square Game.

Classical strategies win it at most 8/9'ths of the time in expectation, but entangled strategies can win with certainty.

Basically, the game amounts to two players each placing an even number of tokens on a 3x3 board. One has to play vertically on a column chosen at random by a referee, while the other plays horizontally on a row chosen similarly. The goal is for the cell at the intersection of the chosen column and row to end up with exactly one token:

Example outcomes

During the game each player doesn't know which row (col) the other player was told, obviously.


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