Entanglement and Bell tests

One of the infamous counterintuitive ideas of quantum mechanics is that two systems that appear too far apart to influence each other can nevertheless behave in ways that, though individually random, are too strongly correlated to be described by any classical local theory.

To understand this, we have outlined a simple Bell test experiment here. Imagine you have two systems (see blue and red systems below). Within each there are two measurements performed: A , A' , B and B' , that have outcomes 1 (or -1 ). Bell showed that if these measurements are chosen correctly for a given entangled state, the statistics can not be explained by any local hidden variable theory, and that there must be correlations that are beyond classical.

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In 1969 John Clauser, Michael Horne, Abner Shimony, and Richard Holt derived the following CHSH inequality |\mathrm{C}|\leq 2  where

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and the correlated expectation is given by

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with 0 giving outcome +1 and 1 giving outcome -1 . A correlation of 1 means both observables have even parity, and a correlation of -1 means both observables have odd parity.

It is simple to show that this inequality must be true if the theory obeys the following two assumptions, locality and realism:

Locality: No information can travel faster than the speed of light. There is a hidden variable \lambda that defines all the correlations so that

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and C becomes

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Realism: All observables have a definite value independent of the measurement (+1 or -1). This implies that either |B(\lambda)+B'(\lambda)|=2 (or 0 ) while |B(\lambda)-B'(\lambda)| = 0 (or 2 ) respectively. That is, |\mathrm{C}|= 2 , and noise will only make this smaller.

Perfectly reasonable, right? However, as you see, |C|>2 . How is this possible? The above assumptions must not be valid, and this is one of those astonishing counterintuitive ideas necessary to accept in the quantum world. Before you launch the scores below, let’s try to understand what is happening and how each observable is measured and combined to give |C| .

The Bell experiment we provide uses the entangled state \frac{1}{\sqrt{2}}(|00\rangle+|11\rangle) , and the two measurements for system A are Z and X , while the two for B are W=\frac{1}{\sqrt{2}}(Z+X)  and V=\frac{1}{\sqrt{2}}(Z-X) .

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For an ideal implementation the four correlated expectation values give,

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which gives |C|=2\sqrt{2} .

To run this experiment with our hardware, we need the following quantum score and 4 measurements.

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In the first part of the experiment, the qubits are initially prepared in the ground state |00\rangle . The H takes the first qubit to the equal superposition \frac{|00\rangle+|10\rangle}{\sqrt{2}} , and the CNOT gate flips the second qubit if the first is excited, making the state \frac{|00\rangle+|11\rangle}{\sqrt{2}} . This is the entangled state (commonly called a Bell state) required for this test. In the first experiment, the measurements are of the observable Z and W= \frac{1}{\sqrt{2}}(X+Z).   To rotate the measurement basis to the W axis, use the sequence of gates S - H - T - H , then perform a standard measurement. The correlator  \langle ZW \rangle should be close to 1/\sqrt{2} , and is found using the above equation.

In the second experiment, the two observables are Z and V=\frac{1}{\sqrt{2}}(-X+Z) . To rotate to this basis, we use the sequence of gates S - H - T^\dagger - H , then perform a standard measurement. The correlator \langle ZV \rangle is found in a similar way as before; it should be close to 1/\sqrt{2} . Finally, in the third and fourth experiment, the correlators \langle XW\rangle and \langle XV\rangle and -1/\sqrt{2} respectively. The W and V measurement are performed the same way as above, and the X via a Hadamard gate before a standard measurement.

Here you can see the results of this experiment on the processor:

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Try it out for yourself! Compare what we got with the simulations (with both ideal and realistic parameters).

Bell State ZZ-Measurement image9

Open in composer

Bell State ZW-Measurement image10

Open in composer

Bell State ZV-Measurement image11

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Bell State XW-Measurement image12

Open in composer

Bell State XV-Measurement image13

Open in composer