Entanglement and Bell Tests

3.3.4 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 cannot be explained by any local hidden variable theory, and that there must be correlations that are beyond classical. 

In 1969 John Clauser, Michael Horne, Abner Shimony, and Richard Holt derived the following CHSH inequality |C|≤2 where (capital) C is

where <AB>, <AB’>, <A’B>, & <A’B’> are quantum correlations of the particle pairs, and the correlated expectation is given by 

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, (small) c. Hence the effect for a qubit to interact with or correlate to another qubit is delayed by a time=distance between 2 qubits/c.  So classical mechanics predicts that no instantaneous correlation can be built.  There is a hidden variable λ that defines all the correlations so that

and (capital) C becomes 

              Realism: All observables have a definite value independent of the measurement (+1 or -1). This implies that either |B(λ)+B′(λ)|=2 (or 0) while |B(λ)−B′(λ)|=0 (or 2) respectively. That is, |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 one must 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 have provided uses the entangled state (|00⟩+|11⟩) and the two measurements for system A are Z and X, while the two for B

which gives |C|=2√2. 

To run this experiment with our hardware we need the following quantum score for A and B (left diagram below), and 4 measurements Z, X, W, and V (right diagram below). 

After all the experiments below, you can see the final results. Here we got for you when we ran this experiment on the processor: 

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

Exercise 1. Bell State ZZ Measurement

Simulate the above circuits with ideal processor to get the following distribution:

Simulate the above circuits with realistic processor to get the following distribution:

Quantum Score file in both simulations are:

h q[1];
cx q[1], q[2];
measure q[1];
measure q[2];

This exercise has no corresponding items in the table of the text.  However, I think the author wants to initiate the interest to build the next several circuits, each with added complexity.  Meanwhile, it tells the purpose of building these circuits is to prove that because of quantum entanglement, there are correlations between qubits.  Famous CHSH inequality |C|≤2 is based on locality and realism, but Bell experiment can prove that |C|>2, as shown in the table of the text where |C|=2.56+-0.03.  We are here to figure out what kind of circuit can be built for such Bell experiment.

Exercise 2. Bell State ZW Measurement

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

Simulate the above circuits with ideal processor to get the following distribution:

Simulate the above circuits with realistic processor to get the following distribution:

Quantum Score file in both simulations are:

h q[1];
cx q[1], q[2];
s q[2];
h q[2];
t q[2];
h q[2];
measure q[1];
measure q[2];

The above distribution can be used to compare with the first row of table in the text.  I was kind of worrying because there is no single value with highest probability in the distribution, but the author offers the table of probabilities for each quantum value, to relieve my worry.  However, what if the author does not list the table?  In other words, I am asking a general question: if we are building circuits and trying to get a single (standard) answer that has highest probability, then we are destined to be disappointed if two or more answers come up with roughly the same probability.  Now, we are educated through the school tests all our youth life to figure out a single answer to match with instructor’s standard answer so that we can score higher in the competitive school environment.  However, in quantum computing, the probabilistic result discourages a standard answer from the deep roots of computing: the qubit.  In classical computing based on digital bits, ONE single answer is absolute and standard.  However, in quantum computing, you’d better not expect ONE standard answer.  Furthermore, if the theory of quantum brain is true, then the brain is random, and the world is random which is how we should teach our kids.  This definitely changes the current and will revolutionize the future education philosophy.

Exercise 3. Bell State ZV Measurement

In this experiment the two observables are Z and V=(Z-X)/√2 . To rotate to this basis we use the sequence of gates S -H -T† -H and then perform a standard measurement. The correlator ⟨ZV⟩ is found in a similar way as before and should be close to 1/√2.

Simulate the above circuits with ideal processor to get the following distribution:

Simulate the above circuits with realistic processor to get the following distribution:

Quantum Score file in both simulations are:

h q[1];
cx q[1], q[2];
s q[2];
h q[2];
tdg q[2];
h q[2];
measure q[1];
measure q[2];

The result seems not much different from the ZW measurement — is T replaced by T+ making no difference?  “+” means the transpose of the matrix.  Remember

So why the transpose does not make a difference?

Exercise 4. Bell State XW Measurement

In this experiment the correlators ⟨XW⟩ is measured and should be close to 1/√2. The W measurement is performed the same way as above and the X via a Hadamard gate before a standard measurement. 

Simulate the above circuits with ideal processor to get the following distribution:

Simulate the above circuits with realistic processor to get the following distribution:

Quantum Score file in both simulations are:

h q[1];
cx q[1], q[2];
h q[1];
s q[2];
h q[2];
t q[2];
h q[2];
measure q[1];
measure q[2];

Again the result is not much different from ZW & ZV measurements.

Exercise 5. Bell State XV Measurement

In this experiment the correlators ⟨XV⟩ is measured and should be close to −1/√2. The V measurement is performed the same way as above and the X via a Hadamard gate before a standard measurement. 

Simulate the above circuits with ideal processor to get the following distribution:

Simulate the above circuits with realistic processor to get the following distribution:

Quantum Score file in both simulations are:

h q[1];
cx q[1], q[2];
h q[1];
s q[2];
h q[2];
tdg q[2];
h q[2];
measure q[1];
measure q[2];

Now this result is very different from ZW, ZV & XW measurements – Pr(0,1) and Pr(1,0) is higher instead of Pr(0,0) and Pr(1,1).  Why is it? 

(1)   Is there some kind of design knowledge that teaches students to know how to do quantum circuit design (so that we can understand more the reason for running results)?  This question is partially answered by section 4.2 where circuit design is the major topic.

(2)   Can a circuit always be expressed in terms of general mathematical formulas (such as vector and matrix multiplications)?  This question may be partially answered by the Bell 1971 and Clauser/Horne 1974 derivations of CHSH inequality (see https://en.wikipedia.org/wiki/CHSH_inequality).

(3)   How the knowledge of this course unit have to do with quantum computer’s operating system?  This question is answered barely by section 4.3-5 where several famous quantum algorithms are the major topics.  From DWave’s taking Ising / graph algorithms () as its operating system, we realize quantum algorithms is the base of OS for a general-purpose quantum computer (GPQC).  Our brain-like computer, of course, is more complicated than a GPQC.  For instance, it should first resolve the constraints caused by Ising model because of the way DWave wires its qubits so that a qubit can only communicate with its neighbors, rather than with any qubits, especially with the non-local (i.e. faster than light) ones where entanglement may benefit the computing.  To resolve the constraints, there can be an entanglement unit (a networking resource) that helps the qubit (a computing resource) to have non-local entanglement so that a qubit no longer talks just locally and with neighbors only.  This entanglement unit may be made with ZV, ZW, XW, and XV circuits here for Bell state of dual qubits, and with YYX, YXY, XYY, & XXX circuits of next course unit for GHZ state of triple qubits!

This disproof against CHSH inequality using Bell experiment seems academic but also provides truth about entanglement: the Bell state cannot be explained by CHSH, and can only be proved by the circuits of Bell experiment.  These circuits gives probabilistic distributions to help calculating C, which is 2x1.414=2.828 > 2, a violation of CHSH inequality. However, I myself still feel there are many other good topics about quantum entanglement, like the OS of GPQC or brain-like QC as described above.