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# Two-Qubit States

**Two-qubit states** are quantum states of two qubits, normalized vectors in a four-dimensional Hilbert space. Two-qubit states are fundamental to understanding entanglement and form the basis for many quantum algorithms and protocols.

## Overview

Two-qubit states span a Hilbert space of dimension $2 \times 2 = 4$. They are classified into two families:

### Product States

States that factor into independent single-qubit states:
$$|\psi\rangle = |\psi\rangle_A \otimes |\phi\rangle_B$$

These exhibit no entanglement; measuring one qubit does not constrain the other. The default product states are the [[quantum-state-computational-two-qubit|computational basis states]] $|00\rangle$, $|01\rangle$, $|10\rangle$, $|11\rangle$.

### Entangled States

States that cannot be factored into independent single-qubit states. The most important entangled two-qubit states are:

- **[[quantum-state-bell|Bell states]]**: maximally entangled basis 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⁻⟩)]]

## Two-Qubit Computational States

The four computational (Z) basis states form the default measurement basis for two-qubit systems:

- **[[quantum-state-computational-00|Computational 00 (|00⟩)]]**: both qubits in ground state
- **[[quantum-state-computational-01|Computational 01 (|01⟩)]]**: first qubit ground, second excited
- **[[quantum-state-computational-10|Computational 10 (|10⟩)]]**: first qubit excited, second ground
- **[[quantum-state-computational-11|Computational 11 (|11⟩)]]**: both qubits in excited state

These product states are separable (unentangled) and are the eigenstates of the Z operator on both qubits. They form a complete orthonormal basis for two-qubit Hilbert space.

## Bell States as Measurement Basis

The four Bell states form a complete orthonormal basis for two-qubit Hilbert space. Any two-qubit state can be decomposed in the Bell basis, enabling "Bell measurement" to distinguish all four states.

## Separability

A two-qubit state $\rho$ is **separable** if it can be written as:
$$\rho = \sum_i p_i \rho_i^A \otimes \rho_i^B$$

where $p_i \geq 0$ and $\sum_i p_i = 1$. Separable states (pure or mixed) exhibit no entanglement and can be prepared classically.

A state is **entangled** if it is not separable. Bell states are maximally entangled; general two-qubit entangled states have varying degrees of entanglement.

## Entanglement Measures

For two-qubit states:
- **Concurrence**: quantifies bipartite entanglement; ranges from 0 (separable) to 1 (maximally entangled)
- **Entanglement of formation**: minimum number of Bell pairs needed to create the state
- **Bell inequality violation**: entangled states violate CHSH inequality; maximally entangled states violate it maximally

## Two-Qubit Gates

Two-qubit gates couple qubits and create entanglement:
- **CNOT (controlled-NOT)**: Pauli X on target qubit, controlled by source qubit state
- **CZ (controlled-Z)**: Pauli Z on target, controlled by source
- **SWAP**: exchange qubit states
- **iSWAP**: SWAP with $i$ phase on exchange
- **Parametric gates** (XX, YY, ZZ, CPHASE): angle-tunable entangling gates

See [[quantum-gate-two-qubit|two-qubit gates]] for details on each gate.

## Creation of Bell States

The most common Bell state, $|\Phi^+\rangle$, is created via:
1. Prepare initial state $|00\rangle$
2. Apply Hadamard to first qubit: $\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) \otimes |0\rangle$
3. Apply CNOT with first as control: yields $|\Phi^+\rangle$

Other Bell states are obtained by applying single-qubit rotations (X or Z gates) before or after CNOT.

## Applications

### Quantum Communication
- **Quantum teleportation**: Bell pair enables sending unknown qubit state via two classical bits
- **Superdense coding**: Bell pair allows encoding two classical bits by manipulating one qubit
- **Quantum key distribution**: E91 protocol uses Bell states to establish shared secret key

### Quantum Networks
- **Entanglement swapping**: connecting Bell pairs extends entanglement across networks
- **Quantum repeaters**: swap Bell pairs to extend range beyond direct gate distance
- **Distributed quantum computing**: Bell pairs form links between distant quantum processors

### Quantum Algorithms
- **Deutsch-Jozsa algorithm**: uses Bell state phase patterns
- **QAOA**: two-qubit entanglement creates search space for optimization
- **VQE**: Bell state ansätze parameterize trial states

### Quantum Error Correction
- **Surface codes**: local two-qubit interactions detect errors via Bell-like measurements
- **Stabilizer codes**: measure two-qubit Pauli products to extract syndrome information

## Pure vs Mixed Two-Qubit States

**Pure states**: two-qubit density matrix $\rho = |\psi\rangle\langle\psi|$ with rank 1. Maximum purity.

**Mixed states**: density matrix with rank > 1. Represent statistical mixtures or decoherence-affected states. Purity $\text{Tr}(\rho^2) < 1$.

## Decoherence Mechanisms

Common two-qubit decoherence sources:
- **Dephasing**: loss of phase coherence in superposition; affects X and Y basis overlaps
- **Amplitude damping**: decay to lower energy state (usually $|00\rangle$)
- **Bit flip**: random flips $|0\rangle \leftrightarrow |1\rangle$
- **Depolarization**: random projection to random state

Bell states are particularly fragile under decoherence due to maximal entanglement.

## Relation to Multi-Qubit States

Two-qubit entanglement generalizes to multi-qubit systems:
- **Three qubits**: Bell states embed as subspaces; [[quantum-state-ghz|GHZ]] and [[quantum-state-w|W states]] have different three-qubit entanglement structures
- **$n$ qubits**: entanglement complexity grows; two-qubit correlations are building blocks for global correlations
- **Graph states**: two-qubit CZ gates create larger graph-structured entanglement

## Measurement and State Tomography

Characterizing a two-qubit state requires:
1. Measuring in multiple bases (Z, X, Y on each qubit independently)
2. Collecting statistics from many trials
3. Reconstructing density matrix via classical processing

Full state tomography of a two-qubit state requires measurements in all $3 \times 3 = 9$ two-qubit bases (e.g., ZZ, ZX, ZY, XZ, XX, XY, YZ, YX, YY).

## Scalability

Two-qubit systems are the simplest non-trivial quantum computers. Scaling to many qubits requires:
- Precise two-qubit gate control (errors accumulate with circuit depth)
- Minimal crosstalk between gate operations
- Long coherence times for many-qubit entanglement maintenance
- Classical processing for measurement feedback and error correction