Differences

This shows you the differences between two versions of the page.

--- quantum-state [June 12, 2026 at 23:31] – Ivan Janevski
+++ quantum-state [August 26, 2026 at 15:46] (current) – external edit 127.0.0.1
@@ Line 1: / Line 1: @@
 # Quantum state
-A **quantum state** is a complete mathematical description of a quantum system. For a single qubit, a quantum state is a unit vector in a two-dimensional complex Hilbert space; for $n$ qubits, the state lives in a $2^n$-dimensional space formed by the tensor product of the individual qubit spaces.
-Pure states are written as ket vectors $\lvert\psi\rangle = \alpha\lvert 0\rangle + \beta\lvert 1\rangle$, where $\alpha, \beta \in \CC$ and $|\alpha|^2 + |\beta|^2 = 1$. The squared magnitudes $|\alpha|^2$ and $|\beta|^2$ give the probabilities of measuring $\lvert 0\rangle$ and $\lvert 1\rangle$ respectively. When a system cannot be described by a single ket — because it is entangled with the environment or prepared as a statistical mixture — the state is instead represented by a [[density-matrix|density matrix]] $\rho$.
+**Quantum states** describe the complete information content of a quantum system. Like classical bits, qubits occupy definite states; unlike classical bits, qubits can exist in superposition—a linear combination of basis states. A quantum state of $n$ qubits is represented as a vector $|\psi\rangle$ in a complex Hilbert space $\mathbb{C}^{2^n}$, normalized so $\langle\psi|\psi\rangle = 1$.
-The set of single-qubit pure states maps one-to-one onto the surface of the [[bloch-sphere|Bloch sphere]], where the north and south poles are $\lvert 0\rangle$ and $\lvert 1\rangle$, the equatorial $x$-axis states are $\lvert +\rangle$ and $\lvert -\rangle$, and the equatorial $y$-axis states are $\lvert +i\rangle$ and $\lvert -i\rangle$. Multi-qubit states can be entangled, meaning they cannot be written as a product of individual qubit states; the [[bell-states|Bell states]] are the canonical two-qubit entangled states.
+A single-qubit state can be written $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ where $|\alpha|^2 + |\beta|^2 = 1$. Multi-qubit states live in tensor products of qubit spaces. Entangled states cannot be factored into independent qubits—they exhibit non-local correlations measured by entropy or purity.
+States can be pure (perfect knowledge, represented by a state vector) or mixed (statistical mixture, represented by a density matrix). Measurement collapses a superposition to a definite basis state with probability determined by the state's amplitudes.
+## Concepts
+. [[quantum-state-bloch-sphere|Bloch sphere representation]]
+. [[quantum-state-superposition|Superposition]]
+. [[quantum-state-entanglement|Entanglement]]
+. [[quantum-state-basis|Basis states and measurement]]
+. [[quantum-state-density-matrix|Density matrices and mixed states]]
+. [[quantum-state-stabilizer|Stabilizer states]]
+. [[quantum-state-tomography|State tomography]]
+. [[quantum-state-fidelity|Fidelity and purity]]
+## States
+- [[quantum-state-single-qubit|Single-qubit states]]
+  - [[quantum-state-computational|Computational basis]]
+    - [[quantum-state-0|Zero state (|0⟩)]]
+    - [[quantum-state-1|One state (|1⟩)]]
+  - [[quantum-state-pauli-eigenstates|Pauli eigenstates]]
+    - [[quantum-state-plus|Plus state (|+⟩)]]
+    - [[quantum-state-minus|Minus state (|-⟩)]]
+    - [[quantum-state-plus-i|Plus-i state (|+i⟩)]]
+    - [[quantum-state-minus-i|Minus-i state (|-i⟩)]]
+- [[quantum-state-two-qubit|Two-qubit states]]
+  - [[quantum-state-computational-two-qubit|Two-qubit computational basis]]
+    - [[quantum-state-computational-00|Computational 00 (|00⟩)]]
+    - [[quantum-state-computational-01|Computational 01 (|01⟩)]]
+    - [[quantum-state-computational-10|Computational 10 (|10⟩)]]
+    - [[quantum-state-computational-11|Computational 11 (|11⟩)]]
+  - [[quantum-state-bell|Bell states]]
+    - [[quantum-state-bell-00|Bell 00 (|Φ⁺⟩)]]
+    - [[quantum-state-bell-01|Bell 01 (|Ψ⁺⟩)]]
+    - [[quantum-state-bell-10|Bell 10 (|Ψ⁻⟩)]]
+    - [[quantum-state-bell-11|Bell 11 (|Φ⁻⟩)]]
+- [[quantum-state-three-qubit|Three-qubit entangled states]]
+  - [[quantum-state-ghz|GHZ state]]
+  - [[quantum-state-w|W state]]
+- [[quantum-state-multiqubit|Multi-qubit states (n-qubit general)]]
+  - [[quantum-state-dicke|Dicke state]]
+  - [[quantum-state-graph|Graph state]]
+  - [[quantum-state-cluster|Cluster state]]
+  - [[quantum-state-maximally-entangled|Maximally entangled state]]
+  - [[quantum-state-choi|Choi state]]