Quantum Channels API

Quantum Channels

class graphcalc.quantum.channels.QuantumChannel(choi: Sequence[Sequence[complex]] | ndarray, *, input_dim: int, output_dim: int, validate: bool = True, tol: float = 1e-09)

Bases: object

Finite-dimensional quantum channel represented by its Choi operator.

The channel is stored as a triple (J, input_dim, output_dim) where J is the Choi matrix of shape (input_dim * output_dim, input_dim * output_dim).

Conventions

• the Choi matrix is defined by J(Phi) = sum_{i,j} |i><j| \otimes Phi(|i><j|)

• the input subsystem appears first and the output subsystem second

• complete positivity is equivalent to positive semidefiniteness of J

• trace preservation is equivalent to the partial trace over the output subsystem equaling the identity on the input space

• unitality is equivalent to the partial trace over the input subsystem equaling the identity on the output space

param choi:

Choi matrix of the channel.

type choi:

array-like

param input_dim:

Input Hilbert-space dimension.

type input_dim:

int

param output_dim:

Output Hilbert-space dimension.

type output_dim:

int

param validate:

Whether to validate the input as a completely positive trace-preserving map.

type validate:

bool, default=True

param tol:

Numerical tolerance used in validation.

type tol:

float, default=1e-9

Notes

The internal Choi matrix is stored privately as _choi. Public access is provided through a read-only property that returns a copy.

apply(state: QuantumState) → QuantumState

Apply the channel to a quantum state.

Parameters:

state (QuantumState) – Input state. Its total dimension must equal input_dim.

Returns:

Output state on a single subsystem of dimension output_dim.

Return type:

QuantumState

Notes

The action is computed from a Kraus decomposition: Phi(rho) = sum_a K_a rho K_a^dagger.

property choi: ndarray

Return a copy of the Choi matrix.

property choi_rank: int

Return the rank of the Choi matrix.

compose(other: QuantumChannel) → QuantumChannel

Return the composition self ∘ other.

Parameters:

other (QuantumChannel) – Channel applied first.

Notes

If other : A -> B and self : B -> C, then the result is a channel A -> C defined by rho |-> self(other(rho)).

copy() → QuantumChannel

Return a copy of the channel.

property dimension: int

Return the dimension of the Choi matrix.

classmethod from_choi(choi: Sequence[Sequence[complex]] | ndarray, *, input_dim: int, output_dim: int, validate: bool = True, tol: float = 1e-09) → QuantumChannel

Construct a quantum channel from a Choi matrix.

classmethod from_kraus(operators: Iterable[Sequence[Sequence[complex]] | ndarray], *, input_dim: int | None = None, output_dim: int | None = None, validate: bool = True, tol: float = 1e-09) → QuantumChannel

Construct a quantum channel from Kraus operators.

Parameters:

• operators (iterable of array-like) – Kraus operators K_a of shape (output_dim, input_dim).

• input_dim (int | None, default=None) – Optional input dimension. If omitted, inferred from the operators.

• output_dim (int | None, default=None) – Optional output dimension. If omitted, inferred from the operators.

• validate (bool, default=True) – Whether to validate the resulting channel as CPTP.

• tol (float, default=1e-9) – Numerical tolerance used in validation.

classmethod identity(dim: int, *, tol: float = 1e-09) → QuantumChannel

Return the identity channel on a dim-dimensional system.

is_completely_positive() → bool

Return whether the channel is completely positive.

Notes

With the Choi representation, complete positivity is equivalent to positive semidefiniteness of the Choi matrix.

is_trace_preserving() → bool

Return whether the channel is trace preserving.

Notes

For the adopted Choi convention, trace preservation is equivalent to Tr_output(J) = I_input.

is_unital() → bool

Return whether the channel is unital.

Notes

For the adopted Choi convention, unitality is equivalent to Tr_input(J) = I_output.

kraus_operators() → list[ndarray]

Return a Kraus representation of the channel.

Notes

If J = sum_a |K_a><K_a| is a spectral decomposition of the Choi matrix, then the Kraus operators are obtained by reshaping the eigenvectors into matrices using column-major order.

tensor(other: QuantumChannel) → QuantumChannel

Return the tensor product channel self ⊗ other.

validate() → None

Validate that the stored Choi matrix defines a CPTP channel.