Four qubits are shared among four distant stations in the state
PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Quantum states, measurement and entanglement · Quantum states, measurement and entanglement
Quantum Information and Technology, Ch Nonlocality: EPR, Bell inequalities and the CHSH game
Problem
Four qubits are shared among four distant stations in the state ∣GHZ4⟩=(∣0000⟩+∣1111⟩)/2. Consider the four observables XXXX, XXYY, XYXY and XYYX, one Pauli measurement per station.
((a)) Using Y∣0⟩=i∣1⟩ and Y∣1⟩=−i∣0⟩, show that XXXX∣GHZ4⟩=+∣GHZ4⟩ and that each of XXYY, XYXY, XYYX gives −∣GHZ4⟩.
((b)) Suppose, as a local hidden-variable model would have it, each station's qubit carries predetermined answers xi,yi∈{±1} to the questions "X" and "Y". Using the three −1 results from part (a), show that the model is forced to predict x1x2x3x4=−1, contradicting the +1 that XXXX gives with certainty.
((c)) Confirm, as an operator identity independent of any state, that XXYY⋅XYXY⋅XYYX=−XXXX.