# How would a Quantum Computer (network) perform loophole-free Bell tests?

In a simple form, Bell's theorem states that:

No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.

Bell developed a series of inequalities to provide specific experimental examples to distinguish between the predictions of any theory relying on local hidden variables and those of quantum mechanics. As such, Bell test inequality experiments are of fundamental interest in quantum mechanics. However, if one wants to do things properly, one realizes that there are a number of loopholes that affects, in different degrees, all experiments trying to perform Bell tests.[1] Experiments trying to close these loopholes tend to be unique rather than routine. One of the results of having general-purpose quantum computers, or networks thereof, would be the ability to routinely perform sophisticated quantum experiments.

Question: What requirements would have to fulfill a general-purpose quantum computer (network) to be able to implement Bell tests that are at least as loophole-free as the best realization that has been done so far?

For clarity: ideally the best answer will take a quantum computing approach and contain close-to-engineering details, or at least close-to-architecture. For example, writing the experiment as a simple quantum circuit, one of the current architectures can be chosen and from that one would make some realistic order-of-magnitude estimates to the required times of the different quantum gates / measurements and of the required physical distance between the different qubits.

[1] As commented by @kludg, it has been argued that "..no experiment, as ideal as it is, can be said to be totally loophole-free.", see Viewpoint: Closing the Door on Einstein and Bohr’s Quantum Debate

• the problem is that the quantum circuit for a Bell test is rather trivial, and does not convey the critical features, particularly with regards to the locality loophole. May 18, 2018 at 7:15
• FWIW the circuit is the header image of my blog. Apr 12, 2021 at 12:13

When people talk about a loophole free Bell test, what they really mean is that the two loopholes that most concern the majority of people are closed simultaneously: the measurement loophole and the locality loophole.

Let us briefly review the protocol:

• A Bell state $(|00\rangle+|11\rangle)/\sqrt{2}$ is produced, and two parties, Alice and Bob, each take one qubit.

• Alice and Bob must be separated by a distance $d$.

• Alice and Bob each pick a random bit value.

• If Alice's random bit value, $x$, is 0, she measures her qubit in the $Z$ basis. If it's 1, she measures in the $X$ basis. Her measurement outcome is a bit, recorded in the variable $A_x$.

• If Bob's random bit value is $y\in\{0,1\}$, he measures in the basis $(Z+(-1)^yX)/\sqrt{2}$. His measurement outcome is a bit, recorded in the variable $B_y$.

• Alice and Bob repeat this many times, and evaluate the expected value of $$S=A_0(B_0+B_1)+A_1(B_0-B_1).$$

Closing the locality loophole requires that the two parties that are taking part are separated by a distance $d$ such that the time between Alice's measurement basis being chosen, and Bob's answer being given is less than $d/c$, where $c$ is the speed of light (so that there is no way an adversary choosing Bob's answer can know Alice's measurement basis). It also requires, symmetrically, that Alice's answer is given no later than a time $d/c$ after Bob's measurement choice is made.

Closing the measurement loophole requires the use of detectors that have a sufficiently high accuracy (otherwise, an adversary obeying a Local Hidden Variable model could replace your detectors with better detectors, and use the margin of error to throw away results that would betray the presence of eavesdropping/manipulation). The precise value of this threshold depends on your precise formulation of the Bell test. The commonly quoted value is a detector efficiency of about $83\%$ for the CHSH test.

Recently, there have been experiments that have closed both these loopholes simultaneously. See here, for example. Their results are good enough that they can quantify the likelihood of there being a local hidden variable model that describes their results ($P=0.039$). Ultimately, if you want to do better, you either need devices that perform better than theirs, or to perform more runs of the experiment. That is, perhaps, now the main experimental challenge; to improve the speed of such devices so that it doesn't take 18 days to generate 245 trials! These experiments also claim to remove the freedom of choice loophole, wherein one worries that the random number generators that are used for choosing the measurement bases of Alice and Bob are also governed by the same local hidden variable model, instead of generating perfect randomness that is uncorrelated with the rest of the experiment.

In terms of a quantum computing architecture for implementing this, that is not a particularly natural issue: for a quantum computer, one wants to be able to create as much connection and interaction between the qubits as possible, which is rather the opposite of needing to separate them over great distances. I suppose the sort of context which is starting to generate the right scenario are the designs for scalable ion trap quantum computers, where there are multiple separated traps, each of which only interacts occasionally. If each of these were far enough apart, you could think of a loophole-free Bell test. I believe the measurement efficiencies in these scenarios are high enough. The question then is, how far apart do these different locations have to be to close the locality loop hole? I haven't done any sort of calculation based on real data, but I think the answer is of the order of kilometers, i.e. completely unreasonable for a single computer. For me, those would be separate computers, working very hard to cooperatively compute using the minimum of shared resources (i.e. entanglement).

• While I agree with your answer, I was going for something in terms of a quantum computer, or possibly an architecture. Let me edit the question for clarity. May 18, 2018 at 7:01
• @agaitaarino That would be good - I didn't understand the connection to quantum computers per se. Bell violations are talked about in very different contexts to to quantum computers. May 18, 2018 at 7:04
• Actually, I'd say that there is another major loophole (maybe not as big as the detection and the locality ones, but still of main concern): the freedom-of-choice loophole (see here for example: arxiv.org/abs/0811.3129). The first three "loophole-free Bell tests" closed all the three main loopholes May 18, 2018 at 7:17
• @Fraccalo I agree it's an issue. It's not usually considered on the same level as the other two, and I haven't read enough detail to be convinced that it has been fully closed (to the extent that would be required for a device-independent test), so was avoiding saying too much! May 18, 2018 at 7:23
• I think the point is that the settings are chosen in a space-like separated framework, so that they can't influence each other (therefore, the freedom of choice). I should look into that into details, as the last time I spent some time thinking at the problem it's like 3 years ago, but I remember that their argument made sense to me at the time. Here is the original work were they claim to close just the freedom of choice loophole: pnas.org/content/early/2010/10/29/1002780107 May 18, 2018 at 11:48