Violation of the CHSH inequality

Demonstrate that quantum mechanics is not compatible with local realism.

Objectives

Understand why the violation of the CHSH inequality is important for the foundations of quantum mechanics.
Experimentally verify the violation of the CHSH inequality with two qubits on the quantum computer.

Background

In 2022 the scientists Alain Aspect, John F. Clauser and Anton Zeilinger won the Nobel prize in physics "for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science".

Theory

The origin of this reward goes back a long way until 1935, when a paper called Can Quantum-Mechanical Description of Physical Reality be Considered Complete? presented a thought experiment that became known as the EPR paradox, named after the authors' initials, Einstein, Podolsky and Rosen. In the setting of this experiment one considers two distant parties sharing an entangled pair (e.g. photons or spins). The spin of each party is measured using operators that do not commute.

To understand this, let's consider an entangled state of two spins one-half $|\phi_+\rangle = (|\uparrow\uparrow\rangle+|\downarrow\downarrow\rangle)/\sqrt{2}$ and two orthogonal spin components (e.g. $\hat{x}$ and $\hat{z}$). Now, we give one party to Alice and one to Bob and let theme choose which component of the spin they want to measure. If Alice measures along $\hat{z}$ and obtains $\frac{1}{2}$, Bob will get $\frac{1}{2}$ with probability 1 from the expression of the state $|\phi_+\rangle$ given he makes the same choice.

But what if he chooses instead to measure along $\hat{x}$? Then he would have still gotten a definite result (either $\frac{1}{2}$ or $-\frac{1}{2}$). This means, both spin components can be assigned sharp values, contradicting the uncertainty principle of quantum mechanics. The authors concluded then that quantum mechanics theory could not be complete and some "hidden variables" must exist which determined these measurement results.

This was due to some a priori assumptions that the authors made, such as locality, i.e. it is impossible to exchange information instantaneously due to the limit given by the finite speed of light, and realism, or in their own words "If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity".

This remained an open problem for many years, until Bell came up with a quantitative solution to show that quantum mechanics theory does not comply with these conditions. He derived an inequality for measurement correlations which is bounded for any local realist theory, and showed that for some experimental settings, the predictions of quantum mechanics violated the inequality.

The most famous version of Bell's inequality is the CHSH inequality, which is given by$$-2 \leq E(A_1 B_1)-E(A_1 B_2)+E(A_2 B_1)+E(A_2 B_2) \leq 2$$where $E(\cdot)$ indicates the expectation value and $A_i, B_i$ are the observables measured by Alice an Bob respectively. To see why this holds, let us consider that $A_i, B_i$ are variables taking values $+1$ or $-1$ randomly. Then if we consider the quantity $C=A_1(B_1-B_2)+A_2(B_1+B_2)$, we have that one of the two terms $(B_1\pm B_2)$ must be zero and the other $\pm 2$. The average of many measurements of $C$ is then between $-2$ and $+2$, giving the CHSH inequality above. It turns out that there are states for which the combination of expectation values in the inequality violates the bound, like the entagled state considered above. This demonstrates that quantum correlations can be non-local and more generally that quantum mechanics theory is not compatible with local realism, showing the importance of such result from a fundamental point of view, but also from a more practical one, since this also contributed to the use of entanglement as a resource in quantum information protocols.

Although straightforward, rigorously speaking, the experimental violation of the CHSH inequality conducted on a quantum computer leaves an open loophole, known as “locality loophole”. This means that we cannot rule out with complete certainty that no communication between the two parties (here represented by Alice and Bob measuring different observables on their qubits) happened before the actual measurement. This occurs because in our setting the measurements of the qubits are not space-like separated events. In state-of-the-art experiments for the violation of Bell inequalities, this loophole can be avoided by setting the distance between the measurement detectors and the time limit for the measurement in such a way that no information traveling at the speed of light can be communicated between the two parties in time. In this way the experiment demonstrates that there is a non-local correlation between the two parties and that entanglement cannot be used for superluminal communication, since there is no information propagating.

Methods

The expectation values in the CHSH inequality above ideally require an infinite number of circuit runs and measurements, so called shots. In reality, what we can have is only a finite number of shots and thus an approximation of the expectation values. To account for fluctuations of these values, we need to calculate error bars to associate them to our experiment results. One way to do this is via bootstrapping, a classical statistical technique that uses random sampling with replacements. In practice, we resample several times the probability distribution obtained from each circuit run and we use these reprocessed results, called resample or bootstrap, to calculate confidence intervals for our estimates. If the size of the original sample is big enough (i.e. the number of shots is sufficiently large), the probability of getting the same original sample is very low. The process of bootstrapping is then repeated a large number of times.

Moreover, due to noise affecting current quantum devices, the results of our experiments could be very far from the ideal case where all the quantum operations have no error and the coherence times of the qubits are very long. Error mitigation techniques have been devised to alleviate the effects of noise on the results and they act usually as postprocessing methods. Readout Error Mitigation (REM) deals with errors that occur in the measurement process of the qubits in a quantum computer; the basic implementation of this technique consists in generating a confusion matrix and in applying its Moore-Penrose inverse to the readout results. The confusion matrix encodes, for each pair of measurement basis states of two qubits $|ij\rangle, |kl\rangle$the probability that the device will return $|ij\rangle$ as outcome of the measurement when the true state being measured was $|kl\rangle$, i.e. $P\big(|ij\rangle\big| |kl\rangle\big)$. This matrix is usually calculated conducting a calibration experiment going through the repeated preparation and measurement of all basis states. Since the number of basis states scales exponentially with the number of qubits, computing the full confusion matrix is only feasible for small systems sizes.

Assignment

Prepare a circuit that creates a two-qubit maximally entangled state $|\phi_+\rangle$ (Bell states). Then, encode an angle in a parametrized $R_y$ gate to rotate the measurement basis of the first qubit with respect to the second qubit basis.
Evaluate the expectation values of the observables that appears in the CHSH inequality for different values of the rotation angle (here we will take $A_1=X$, $A_2=Z$, $B_1=X$ and $B_2=Z$).
Use bootstrapping to get error bars which take into account the finite number of shots used for the measurements (see the "Methods" section below for more details).
Analyze the results plotting them as a function of the rotation angle to understand if and when the CHSH inequality is violated.
Compare your experimentally obtained results with analytical results to see the effect of noise and eventually apply error mitigation techniques.

Questions for Preparation

Why are measurements of correlation functions in different bases necessary to prove entanglement? (For example, the ZZ correlation, i.e., the correlation of the measurement results when both qubits are measured in their respective Z basis, corresponds to the expectation value of the observable $\sigma_z \otimes \sigma_z$).
What paradox did the Bell's theorem solve in quantum mechanics?
What does a violation of the CHSH inequality mean?

Literature

How entanglement has become a powerful tool (The Nobel Prize in Physics 2022).

On-premises superconducting quantum computer for education and research (EPJ Quantum Technol. 11, 32, 2024).

Lab Course: Bell's Inequality and Quantum Tomography Course documents by Harald Weinfurter (LMU Munich).

Quantum Computation And Quantum Information (Nielsen and Chuang).

Lecture Notes: Quantum Information and Computation Chapter 4 (John Preskill).