quantum-state-computational-two-qubit
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| quantum-state-computational-two-qubit [August 22, 2026 at 22:09] – Ivan Janevski | quantum-state-computational-two-qubit [August 26, 2026 at 16:02] (current) – external edit 127.0.0.1 | ||
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| **Two-qubit computational basis states** are the four orthonormal product states |00⟩, |01⟩, |10⟩, |11⟩ that form the default measurement basis on quantum computers. They represent definite, unentangled configurations of two qubits and are eigenstates of the Z operator on both qubits. | **Two-qubit computational basis states** are the four orthonormal product states |00⟩, |01⟩, |10⟩, |11⟩ that form the default measurement basis on quantum computers. They represent definite, unentangled configurations of two qubits and are eigenstates of the Z operator on both qubits. | ||
| - | ## The Four Computational States | + | ## The four computational states |
| - | 1. **[[quantum-state-computational-00|Computational 00 (|00⟩)]]**: | + | - **[[quantum-state-computational-00|Computational 00 (|00⟩)]]**: |
| - Both qubits in ground state | - Both qubits in ground state | ||
| - Z eigenvalue: +1 on both qubits | - Z eigenvalue: +1 on both qubits | ||
| - Default initial state | - Default initial state | ||
| - | 2. **[[quantum-state-computational-01|Computational 01 (|01⟩)]]**: | + | - **[[quantum-state-computational-01|Computational 01 (|01⟩)]]**: |
| - First qubit ground, second excited | - First qubit ground, second excited | ||
| - Z eigenvalue: +1 on first, -1 on second | - Z eigenvalue: +1 on first, -1 on second | ||
| - | 3. **[[quantum-state-computational-10|Computational 10 (|10⟩)]]**: | + | - **[[quantum-state-computational-10|Computational 10 (|10⟩)]]**: |
| - First qubit excited, second ground | - First qubit excited, second ground | ||
| - Z eigenvalue: -1 on first, +1 on second | - Z eigenvalue: -1 on first, +1 on second | ||
| - | 4. **[[quantum-state-computational-11|Computational 11 (|11⟩)]]**: | + | - **[[quantum-state-computational-11|Computational 11 (|11⟩)]]**: |
| - Both qubits in excited state | - Both qubits in excited state | ||
| - Z eigenvalue: -1 on both qubits | - Z eigenvalue: -1 on both qubits | ||
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| Measuring any computational state in the Z basis yields a definite outcome (00, 01, 10, or 11) with probability 1. Measuring in other bases (X or Y) requires rotating both qubits before measurement. | Measuring any computational state in the Z basis yields a definite outcome (00, 01, 10, or 11) with probability 1. Measuring in other bases (X or Y) requires rotating both qubits before measurement. | ||
| - | ## Composition from Single-Qubit Basis | + | ## Composition from single-qubit basis |
| Each two-qubit computational state is a tensor product of single-qubit states: | Each two-qubit computational state is a tensor product of single-qubit states: | ||
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| See [[quantum-state-computational|single-qubit computational basis]] for the component states. | See [[quantum-state-computational|single-qubit computational basis]] for the component states. | ||
| - | ## Superpositions and Entanglement | + | ## Superpositions and entanglement |
| Superpositions of computational states create quantum phenomena: | Superpositions of computational states create quantum phenomena: | ||
| + | |||
| - Equal superposition of all four states: $\frac{1}{2}(|00\rangle + |01\rangle + |10\rangle + |11\rangle)$ | - Equal superposition of all four states: $\frac{1}{2}(|00\rangle + |01\rangle + |10\rangle + |11\rangle)$ | ||
| - [[quantum-state-bell|Bell states]] are maximally entangled superpositions of computational states | - [[quantum-state-bell|Bell states]] are maximally entangled superpositions of computational states | ||
| - Entanglement enables quantum speedup in algorithms | - Entanglement enables quantum speedup in algorithms | ||
| - | ## Role in Quantum | + | ## Role in quantum |
| - **Initial state**: qubits default to $|0\rangle$ (computational 00) at startup | - **Initial state**: qubits default to $|0\rangle$ (computational 00) at startup | ||
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| - **Error basis**: single bit-flip errors swap between computational states | - **Error basis**: single bit-flip errors swap between computational states | ||
| - | ## Relation to Bell States | + | ## Relation to Bell states |
| Bell states are entangled superpositions of computational states: | Bell states are entangled superpositions of computational states: | ||
| - | - $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ | + | - $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$: [[quantum-state-bell-00|Bell 00]] |
| - | - $|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)$ | + | - $|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)$: [[quantum-state-bell-11|Bell 11]] |
| - | - $|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$ | + | - $|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$: [[quantum-state-bell-01|Bell 01]] |
| - | - $|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)$ | + | - $|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)$: [[quantum-state-bell-10|Bell 10]] |
| ## Applications | ## Applications | ||
quantum-state-computational-two-qubit.1787436593.md.gz · Last modified: by Ivan Janevski