The four Bell states
Separate matching versus opposite outcomes from the relative sign of a Bell state.
Intuition
Φ states have matching computational-basis outcomes; Ψ states have opposite ones. Within each pair, the plus and minus states differ in relative phase, not in those measurement probabilities.
Build Φ+ with H(q0), CX(0,1). X on q1 changes it to Ψ+. Starting instead from Φ+, Z on either qubit gives Φ−. From Ψ+, Z on q0 gives Ψ− up to a global minus sign. Inspect amplitudes at each step; all four states are entangled, and none enables controlled faster-than-light messages.
The mathematics
Look a little deeper
These four normalized states are mutually orthogonal. Computational-basis counts distinguish the Φ family from the Ψ family, but not their signs. Accessing a sign difference requires a suitable change of measurement basis or circuit interference.
Use it
Prepare Ψ+ using H, CX, and X in at most three gates. Explore an additional Z, then remove it before verifying the Ψ+ target.
This exercise uses the same server-side state and circuit checks as its linked practice problem.
Build your circuit
Click a gate to append it, or drag it onto a wire. CX uses the selected control; SWAP uses it as the second operand. Basis order: |qₙ … q₀⟩.
Returned logical circuit · before transpilation · q0 on top
Connect to code
qc.h(0) qc.cx(0, 1) qc.x(1)
These operations go inside a circuit-building program. Practice problems provide a complete solve() template.
Check your understanding
First complete the circuit exercise successfully.