Are there Bell-like violations that can be observed without collecting statistics?

Original paper: Going beyond Bell's theorem

Papers by Mermin:

• Quantum mysteries revisited
• Hidden variables and the two theorems of John Bell

---

We can give a non-probabilistic proof that local hidden variables theories are incompatible with Quantum Mechanics.

The proof does not explain how experimentally decide which theory is correct; it just shows more clearly than Bell or CHSH that Quantum Mechanics and local hidden variables theories are incompatible.

Proof:

Suppose we have 3-qubit GHZ state $$|GHZrangle=frac{1}{sqrt{2}}(|000rangle+|111rangle)$$ For visibility sake think of a qubit as of a spin $1/2$ particle.

We can measure spin in $X$ and $Y$ directions; $X$ direction means Hadamard basis ${|+rangle, |-rangle}$, $Y$ direction means ${frac{1}{sqrt{2}}(|0rangle+i|1rangle), frac{1}{sqrt{2}}(|0rangle-i|1rangle)}$ basis.

First, Quantum Mechanics predicts that if we measure 2 qubits in $Y$ direction and 1 qubit in $X$ direction, each qubit's measurement outcome is $50% / 50%$ random, but there is strong correlation: the outcomes of $Y$ measurements determine the outcome of $X$ measurement; namely, if $Y$ measurements are identical, $X$ measurement is $|-rangle$, otherwise $|+rangle$. Following Einstein's "elements of reality" concept, it means that each qubit has deterministic "instructions" how to behave if measured in $X$ and $Y$ direction, and the set of legal instructions for all 3 qubits in GHZ experiment is limited.

Following the 1st cited Mermin paper, the legal instructions are (first line $X$ measurement, second line $Y$ measurement):

 ---  -++  +-+  ++-  -++  ---  ++-  +-+
 ---  -++  +-+  ++-  +--  +++  --+  -+-

It now turns out that if we apply legal instructions approach to measuring all 3 qubits in $X$ direction, the only possible outcomes are $|---rangle$, $|-++rangle$, $|+-+rangle$ and $|++-rangle$.

On the other hand, quantum mechanics predicts that the only possible outcomes are $|+++rangle$, $|+--rangle$, $|-+-rangle$ and $|--+rangle$, because

$$|GHZrangle = frac12(|+++rangle+|+--rangle+|-+-rangle+|--+rangle)$$ and we have no overlap with legal instructions (hidden variables) approach.

I believe you're thinking of the all-versus-nothing proofs based on GHZ states. You start with a state such as $$ |Psirangle=frac{1}{sqrt{2}}(|000rangle+|111rangle) $$ and you select at random one of the four measurements to implement: $Xotimes Xotimes X$, $Xotimes Yotimes Y$, $Yotimes Xotimes Y$ or $Yotimes Yotimes X$. Assuming every one of these can be implemented perfectly every time, you must always get the answer -1 (just calculate $langlePsi|M|Psirangle$ where $M$ is one of the measurement operators) for any of the measurements with two $Y$s, and +1 for the $XXX$ measurement.

However, if the outcomes are associated with a local hidden variable model, this outcome is impossible. Let $x_iin{-1,1}$ and $y_iin{-1,1}$ be the hidden variables associated with the outcomes of measuring $X$ or $Y$ respectively on qubit $i$.

Now let us consider what would happen if we made each of the four measurements, and took the product of the outcomes. The quantum model says we get answer $-1$. The local hidden variable model, however, says that answer must be $$ (x_1x_2x_3)(x_1y_2y_3)(y_1x_2y_3)(y_1y_2x_3)=x_1^2x_2^2x_3^2y_1^2y_2^2y_3^2=1. $$ A local hidden variable model is incompatible with always getting the predicted set of quantum answers.

So, the idea is that whatever measurement you make, you know there is a fixed answer that you must get if the system is behaving properly quantumly (and assuming no errors whatsoever in the system). If you ever get a disagreement, the system cannot be behaving perfectly quantumly. So, that gives you the non-probabilstic answer you mention.

Of course, a more reliable way of running this is to repeat the experiment many times. If you ever get a false answer, that's a proof of non-quantumness. However, the conclusion of quantumness (i.e. always getting the expected answers) still has an associated probability because after $n$ runs, there's a probability of $1/4^n$ that the local hidden variable version gives all the expected quantum answer.

Yes, there are. I just wrote a paper about it, actually.

You need to define carefully what you mean by obtaining a Bell violation or ruling out local hidden variables. You can't demand to have a result which is impossible to explain with local hidden variables: the local bound of a Bell inequality is inherently probabilistic, so it is possible to obtain any statistic you observe merely by chance. What you can do is demand it to be unlikely, with $p$-value smaller than some threshold, or Bayes factor above some threshold. For example, in one of the loophole-free Bell tests they reported that the $p$-value of their data under the local hidden variables hypothesis was below $10^{-30}$.

In a single-shot scenario, the $p$-value of winning a single round of a nonlocal game is simple its local bound, the maximal probability of winning it with local hidden variables. If you can make the local bound smaller than this threshold, while keeping the quantum probability of victory close to 1, then you'll have a single-shot violation.

It turns out that it is possible to construct (families of) nonlocal games such that the local bound becomes arbitrarily close to 0, while the Tsirelson bound becomes arbitrarily close to 1, so yes, a single-shot violation is possible for any threshold you want.

The simplest way of constructing such a game is by doing parallel repetition of a pseudo-telepathy game, that is, playing $n$ times in parallel a nonlocal game with Tsirelson bound 1. The local bound goes down to zero exponentially in $n$ (this is highly nontrivial to prove), while the Tsirelson bound stays equal to 1, so there you have it.

This construction is physically meaningless though, because the local bound here is the probability of winning all parallel instances simultaneously, which you'll never be able to do in reality, so the fact that it is exponentially unlikely with local hidden variables doesn't help you. There are more sophisticated constructions that solve this problem: instead of considering the probability of winning all instances, you can also consider the probability of winning a fraction of instances higher than what you'd expect from the local bound: it turns out that this still goes down to zero exponentially with $n$ under the local hidden variables hypothesis, as proven by Rao's concentration bound, and goes up to one exponentially with $n$ under quantum mechanics. Now this is robust to experimental errors, so it is the way to do it in reality.

You might find unsatisfactory the use of parallel repetition; well there also exists a nonlocal game, the Khot-Vishnoi game, such that the local bound is arbitrarily close to zero and the Tsirelson bound arbitrarily close to one, and it is not based on parallel repetition. This game is very hard to implement experimentally, though.