Update documentation on bell_inequality.py example (#3281)

examples/bell_inequality.py

@@ -1,25 +1,56 @@
-"""Creates and simulates a circuit equivalent to a Bell inequality test.
+"""
+Bell's theorem or inequality proves that entanglement based
+observations can't be reproduced with any local realist theory [1].
+
+This example shows Bell's inequality in form of CHSH game where two
+players Alice and Bob receive an input bit x and y respectively and
+produce an output a and b based on the input bit.
+The goal is to maximize the probability to satisfy the condition [2]:
+    a XOR b = x AND y
+
+In the classical deterministic case, the highest probability
+achievable is 75%. While with quantum correlations, it can
+achieve higher success probability. In the quantum case, two players
+Alice and Bob start with a shared Bell-pair entangled state. The
+random input x and y is provided by referee for Alice and Bob. The
+success probability of satisfying the above condition will be
+cos(theta/2)^2 if Alice and Bob measure their entangled qubit in
+measurement basis V and W where angle between V and W is theta.
+Therefore, maximum success probability is cos(pi/8)^2 ~ 85.3%
+when theta = pi/4.
+
+In the usual implementation [2], Alice and Bob share the Bell state
+with the same value and opposite phase. If the input x (y) is 0, Alice (Bob)
+rotates in Y-basis by angle -pi/16 and if the input is 1, Alice (Bob)
+rotates by angle 3pi/16. Here, Alice and Bob start with the entangled
+Bell state with same value and phase. The same success probability is
+achieved by following procedure: Alice rotate in X-basis by angle
+-pi/4 followed by controlled-rotation by angle pi/2 in X-basis for
+Alice (Bob) based on input x (y).
+
+[1] https://en.wikipedia.org/wiki/Bell%27s_theorem
+[2] R. de Wolf. Quantum Computing: Lecture Notes
+(arXiv:1907.09415, Section 15.2)
 
 === EXAMPLE OUTPUT ===
-
 Circuit:
-(0, 0): ───H───@───X^-0.25───────X^0.5───M───
-               │                 │
-(0, 1): ───────┼─────────────H───@───────M───
+(0, 0): ───H───@───X^-0.25───X────────M('a')───
+               │             │
+(0, 1): ───H───┼─────────────@^0.5────M('x')───
                │
-(1, 0): ───────X─────────────────X^0.5───M───
-                                 │
-(1, 1): ─────────────────────H───@───────M───
+(1, 0): ───────X───X─────────M('b')────────────
+                   │
+(1, 1): ───H───────@^0.5─────M('y')────────────
 
 Simulating 75 repetitions...
 
 Results
-a: _1__11111___111111_1__1_11_11__111________1___1_111_111_____1111_1_111__1_1
-b: _1__1__11_1_1_1__1_1_1_1_11____111_111__11_____1_1__1111_1_11___111_11_1__1
-x: ____11______11111__1_1111111____1_111__1111___1111_1__11_1__11_11_11_1__11_
-y: 11_111_____1_1_111__11111_111_1____1____11_____11___11_1_1___1_111111_1_1_1
+a: _1___1_1_1111__11__11______11__1______111_______1_______11___11_11__1_1_1_1
+b: _1_______11_1_1__1_1111_11_11_1_1_____11___1__111__1_1_1_1__111_11_11_1_1_1
+x: 1_1____1______11_1_1_1_11111___111_1__1_1__11_111__1_11_11_11______1____1__
+y: ____1__111_______1___11_111__1_111______111___11_11_11__1_1_1111_1111__1_11
 (a XOR b) == (x AND y):
-   1111_1_111_11111111111111111_1111111__1111_111_111_11111111_11_11111111_111
+   11111_11111_11___11111_111_111_11_111111111_1111111_111_1111111111111111111
 Win rate: 84.0%
 """
 
@@ -71,7 +102,6 @@ def make_bell_test_circuit():
     circuit.append([
         cirq.H(alice),
         cirq.CNOT(alice, bob),
-        cirq.X(alice)**-0.25,
     ])
 
     # Referees flip coins.
@@ -82,6 +112,7 @@ def make_bell_test_circuit():
 
     # Players do a sqrt(X) based on their referee's coin.
     circuit.append([
+        cirq.X(alice)**-0.25,
         cirq.CNOT(alice_referee, alice)**0.5,
         cirq.CNOT(bob_referee, bob)**0.5,
     ])