Bell Inequality

The flagship experiment of quantum physics

Code: Cirq

Bell's theorem proves that quantum entanglement cannot be explained by any local realist theory. Bell inequality has been tested experimentally for more than 50 years and has proven conclusively that our physical reality always violates the inequality, i.e. reality is quantum.

There are many ways to setup an experiment that tests Bell Inequality. We will look at one that is close to quantum computing.

Objective

Victor is the dealer. Every round, Victor sends 1 random qubit to Alice, x, and 1 random qubit to Bob, y. Alice then returns 1 qubit to Victor, a, as does Bob, b. Alice and Bob win the round if logical AND of x, y is the logical XOR of a, b.

$$x \land b =a \oplus b$$

Prerequisites

• Victor's bits x, y are picked at random.

• Alice and Bob can perform any operations they want on x and y. They could also just have two readymade qubits a and b to send back.

• Before the experiment starts, Alice and Bob can agree on their team's strategy. But they cannot communicate during the experiment, e.g. by measuring 1 qubit and then deciding what to do with the other qubit.

Classical protocol

It can be proven that the best strategy Alice and Bob can use is "do nothing, just return 0", which has a 75% win rate.

Quantum protocol

The best quantum strategy Alice and Bob can use involves the use of quantum entanglement.

1. Before the experiment, Alice and Bob create an entangled pair of qubits, a and b. Alice keeps a and Bob keeps b.

2. During the experiment, Alice entangled her received qubit x with a. Bob entangled his received qubit y with b.

3. Alice returns a, x and Bob returns b, y to Victor.

4. Victor measures the quantities $a \oplus b$ and $x \land y$​.

Quantum best case

In the quantum protocol, all 4 qubits x, y, a and b are entangled, therefore their measurement results are correlated. Therefore, we find that the best quantum strategy can get a 85.3% win rate.

Expectation vs Reality

• An individual round results in a binary result, win or loss. The win rate is the probability of winning and requires a large number of rounds to observed in practice.

• In practice, entangled states decohere due to noise. Therefore 85.3% which is the maximum win rate possible is hard to achieve in practice.

• Making entangled states in the lab is hard. Bell inequality violation, i.e. achieving an observed win rate > 75%, is used as proof that non-classical, entangled states are being created in the lab.