Choi State

Choi state is a canonical maximally entangled bipartite state used to represent a quantum channel. The Choi-Jamiolkowski correspondence maps quantum channels to density matrices, enabling process tomography, channel capacity analysis, and characterization of quantum operations via entangled state measurements.

For a quantum channel $\mathcal{E}: \mathcal{H}_A \to \mathcal{H}_B$ with input dimension $d_A$ and output dimension $d_B$, the Choi state is:

$$\rho_{\text{Choi}} = (\mathcal{I}_A \otimes \mathcal{E}_B)(|\Phi_d\rangle\langle\Phi_d|)$$

where $|\Phi_d\rangle = \frac{1}{\sqrt{d}} \sum_{i=0}^{d-1} |i\rangle_A \otimes |i\rangle_B$ is the maximally entangled state in system A (reference), and $\mathcal{E}$ acts on system B.

The Choi matrix lives in $\mathcal{H}_B \otimes \mathcal{H}_A$ with dimension $d_B \times d_A$.

The Choi state is entangled across two systems:

• Reference system A: carries information about input basis
• Output system B: records the channel's action

Subsystem A remains unentangled (identity operation applied); subsystem B receives the channel output. The Choi state contains complete information about the channel in its entanglement structure.

The correspondence between channels and Choi states is one-to-one:

• Channel → Choi state: apply channel to half of maximally entangled state
• Choi state → Channel: partial trace and density matrix manipulation recovers channel properties

Key properties translate:

• Completely positive channel ↔ positive semidefinite Choi state
• Trace-preserving channel ↔ partial trace of Choi state is identity
• Unitary channel ↔ Choi state is pure and maximally entangled

• Density matrix representation: encodes channel superoperator as explicit quantum state
• Maximally entangled structure: equal superposition of all input-output pairs
• Complete characterization: single measurement protocol in Bell basis determines full channel
• Trace normalization: $\text{Tr}(\rho_{\text{Choi}}) = 1$ for trace-preserving channels

Measure the Choi state to reconstruct the channel:

1. Prepare Choi state (apply channel to half of Bell pair)
2. Perform Bell measurement on output and reference systems
3. Collect statistics over many trials (basis measurements)
4. Reconstruct channel superoperator via classical post-processing

Requires $d^4$ measurement outcomes (for $d$-dimensional channel) and $O(d^4)$ trials for full tomography.

• Channel characterization: direct measurement reveals channel properties (depolarization rate, dephasing, amplitude damping)
• Error mitigation: Choi state analysis identifies dominant error channels in quantum processors
• Channel capacity: classical capacity and quantum capacity computed from Choi eigenvalues
• Approximate channels: comparing Choi states (via trace distance) quantifies channel similarity
• Fidelity benchmarking: Choi fidelity to ideal channel measures process fidelity

Unlike measuring individual qubit states, Choi-based process tomography:

• Requires only one entangled state preparation (not multiple input states)
• Directly reveals channel correlations and non-Markovian effects
• Enables simultaneous testing of all input-output pairs
• Provides single-measurement protocol (Bell basis) sufficient for full reconstruction

• Bell states: two-qubit Choi states for single-qubit channels
• Maximally entangled states: Choi construction starts from $|\Phi_d\rangle$
• Graph states: measurement-based characterization uses graph-structured entanglement similar to Choi structure

Prepare Choi state and measure:

1. Prepare reference qubit in standard state (e.g., $|0\rangle$)
2. Prepare output qubit in superposition (e.g., via Hadamard)
3. Entangle reference and output via controlled-unitary (creates $|\Phi_d\rangle$ entanglement)
4. Apply channel $\mathcal{E}$ to output subsystem
5. Perform Bell measurement (CNOT + Hadamard + measure in Z basis)

For multi-qubit channels ($n$-qubit input/output), Choi state is $2n$-qubit maximally entangled state; measurement requires $4^n$ outcomes.

• Exponential growth: Choi state dimension $d_A \times d_B$ grows exponentially with system size
• Measurement overhead: full process tomography requires $O(d^4)$ shots
• Practical limit: beyond 2–3 qubits, classical Choi matrix storage becomes prohibitive
• Classical simulation: for small systems (1–2 qubits), Choi state is practical; for larger, selective measurements target specific channel properties