qml.math.partial_trace

partial_trace(matrix, indices, c_dtype='complex128')

Compute the reduced density matrix by tracing out the provided indices.

Parameters

• matrix (tensor_like) – 2D or 3D density matrix tensor. For a 2D tensor, the size is assumed to be (2**n, 2**n), for some integer number of wires n. For a 3D tensor, the first dimension is assumed to be the batch dimension, (batch_dim, 2**N, 2**N).

• indices (list(int)) – List of indices to be traced.

Returns

(reduced) Density matrix of size (2**len(wires), 2**len(wires))

Return type

tensor_like

See also

pennylane.math.reduce_dm(), and pennylane.math.reduce_statevector()

Example

We can compute the partial trace of the matrix x with respect to its 0th index.

>>> x = np.array([[1, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]])
>>> partial_trace(x, indices=[0])
array([[1.+0.j, 0.+0.j],
       [0.+0.j, 0.+0.j]])

We can also pass a batch of matrices x to the function and return the partial trace of each matrix with respect to each matrix’s 0th index.

>>> x = np.array([
... [[1, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]],
... [[0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]
... ])
>>> partial_trace(x, indices=[0])
array([[[1.+0.j, 0.+0.j],
        [0.+0.j, 0.+0.j]],
       [[0.+0.j, 0.+0.j],
        [0.+0.j, 1.+0.j]]])

The partial trace can also be computed with respect to multiple indices within different frameworks such as TensorFlow.

>>> x = tf.Variable([[[1, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]],
... [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 0]]], dtype=tf.complex128)
>>> partial_trace(x, indices=[1])
<tf.Tensor: shape=(2, 2, 2), dtype=complex128, numpy=
array([[[1.+0.j, 0.+0.j],
        [0.+0.j, 0.+0.j]],
       [[0.+0.j, 0.+0.j],
        [0.+0.j, 1.+0.j]]])>

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