Skip to content

dq.thermal_dm 

thermal_dm(dim: int | tuple[int, ...], nth: ArrayLike) -> QArray

Returns the density matrix of a thermal state or a tensor product of thermal states.

For a single mode, it is defined for a thermal photon number \(n_{th}\) by:

\[ \rho = \sum_k \frac{(n_{th})^k}{(1+n_{th})^{1+k}} \ket{k}\bra{k}. \]

Parameters:

  • dim –

    Hilbert space dimension of each mode.

  • nth (array-like of shape (...) or (..., len(dim))) –

    Thermal photon number for each mode. If dim is a tuple, the last dimension of nth should match the length of dim.

Returns:

  • (qarray of shape (..., n, n)) –

    Density matrix of the thermal state or tensor product of thermal states, with n = prod(dims).

Examples:

Single-mode thermal state with thermal photon number \(n_{th}=0.1\):

>>> dq.thermal_dm(4, 0.1)
QArray: shape=(4, 4), dims=(4,), dtype=complex64, layout=dense
[[0.909+0.j 0.   +0.j 0.   +0.j 0.   +0.j]
 [0.   +0.j 0.083+0.j 0.   +0.j 0.   +0.j]
 [0.   +0.j 0.   +0.j 0.008+0.j 0.   +0.j]
 [0.   +0.j 0.   +0.j 0.   +0.j 0.001+0.j]]

Batched single-mode thermal states:

>>> dq.thermal_dm(4, [0.1, 0.2, 0.3]).shape
(3, 4, 4)

Multi-mode thermal state:

>>> dq.thermal_dm((4, 3), (0.1, 0.2)).shape
(12, 12)

Batched multi-mode thermal states:

>>> nth = [(0.1, 0.2), (0.2, 0.1), (0.2, 0.2)]
>>> dq.thermal_dm((4, 3), nth).shape
(3, 12, 12)