dq.ptrace

ptrace(
    x: QArrayLike, keep: int | tuple[int, ...], dims: tuple[int, ...] | None = None
) -> QArray

Returns the partial trace of a ket, bra or density matrix.

Parameters

• x (qarray-like of shape (..., n, 1) or (..., 1, n) or (..., n, n)) –

  Ket, bra or density matrix of a composite system.

• keep (int or tuple of ints) –

  Dimensions to keep after partial trace.

• dims (tuple of ints or None) –

  Dimensions of each subsystem in the composite system Hilbert space tensor product. Defaults to None (x.dims if available, single Hilbert space dims=(n,) otherwise).

Returns

(qarray of shape (..., m, m)) Density matrix (with m <= n).

Note

The returned object is always a density matrix, even if the input is a ket or a bra.

Examples

>>> psi_abc = dq.fock((3, 4, 5), (0, 2, 1))
>>> psi_abc.dims
(3, 4, 5)
>>> psi_abc.shape
(60, 1)
>>> rho_a = dq.ptrace(psi_abc, 0)
>>> rho_a.dims
(3,)
>>> rho_a.shape
(3, 3)
>>> rho_bc = dq.ptrace(psi_abc, (1, 2))
>>> rho_bc.dims
(4, 5)
>>> rho_bc.shape
(20, 20)

If the input qarray-like x does not hold Hilbert space dimensions, you can specify them with the argument dims. For example, to trace out the second subsystem of the Bell state \((\ket{00}+\ket{11})/\sqrt2\):

>>> bell_state = np.array([1, 0, 0, 1])[:, None] / np.sqrt(2)
>>> bell_state.shape
(4, 1)
>>> dq.ptrace(bell_state, 0, dims=(2, 2))
QArray: shape=(2, 2), dims=(2,), dtype=float32, layout=dense
[[0.5 0. ]
 [0.  0.5]]