channels.partial_trace

Generates the partial trace of a matrix.

Functions

partial_trace(input_mat[, sys, dim])

Compute the partial trace of a matrix [1].

Module Contents

channels.partial_trace.partial_trace(input_mat, sys=None, dim=None)

Compute the partial trace of a matrix [1].

The partial trace is defined as

\[\left( \text{Tr} \otimes \mathbb{I}_{\mathcal{Y}} \right) \left(X \otimes Y \right) = \text{Tr}(X)Y\]

where \(X \in \text{L}(\mathcal{X})\) and \(Y \in \text{L}(\mathcal{Y})\) are linear operators over complex Euclidean spaces \(\mathcal{X}\) and \(\mathcal{Y}\).

Gives the partial trace of the matrix X, where the dimensions of the (possibly more than 2) subsystems are given by the vector dim and the subsystems to take the trace on are given by the scalar or vector sys.

Examples

Consider the following matrix

\[\begin{split}X = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{pmatrix}.\end{split}\]

Taking the partial trace over the second subsystem of \(X\) yields the following matrix

\[\begin{split}X_{pt, 2} = \begin{pmatrix} 7 & 11 \\ 23 & 27 \end{pmatrix}.\end{split}\]

By default, the partial trace function in toqito takes the trace of the second subsystem.

>>> from toqito.channels import partial_trace
>>> import numpy as np
>>> test_input_mat = np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]])
>>> partial_trace(test_input_mat)
array([[ 7, 11],
       [23, 27]])

By specifying the sys = [0] argument, we can perform the partial trace over the first subsystem (instead of the default second subsystem as done above). Performing the partial trace over the first subsystem yields the following matrix

\[\begin{split}X_{pt, 1} = \begin{pmatrix} 12 & 14 \\ 20 & 22 \end{pmatrix}\end{split}\]
>>> from toqito.channels import partial_trace
>>> import numpy as np
>>> test_input_mat = np.array(
...     [[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]]
... )
>>> partial_trace(test_input_mat, [0])
array([[12, 14],
       [20, 22]])

We can also specify both dimension and system size as list arguments. Consider the following \(16\)-by-\(16\) matrix.

>>> from toqito.channels import partial_trace
>>> import numpy as np
>>> test_input_mat = np.arange(1, 257).reshape(16, 16)
>>> test_input_mat
array([[  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,
         14,  15,  16],
       [ 17,  18,  19,  20,  21,  22,  23,  24,  25,  26,  27,  28,  29,
         30,  31,  32],
       [ 33,  34,  35,  36,  37,  38,  39,  40,  41,  42,  43,  44,  45,
         46,  47,  48],
       [ 49,  50,  51,  52,  53,  54,  55,  56,  57,  58,  59,  60,  61,
         62,  63,  64],
       [ 65,  66,  67,  68,  69,  70,  71,  72,  73,  74,  75,  76,  77,
         78,  79,  80],
       [ 81,  82,  83,  84,  85,  86,  87,  88,  89,  90,  91,  92,  93,
         94,  95,  96],
       [ 97,  98,  99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109,
        110, 111, 112],
       [113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125,
        126, 127, 128],
       [129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141,
        142, 143, 144],
       [145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157,
        158, 159, 160],
       [161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173,
        174, 175, 176],
       [177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189,
        190, 191, 192],
       [193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205,
        206, 207, 208],
       [209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221,
        222, 223, 224],
       [225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237,
        238, 239, 240],
       [241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253,
        254, 255, 256]])

We can take the partial trace on the first and third subsystems and assume that the size of each of the 4 systems is of dimension 2.

>>> from toqito.channels import partial_trace
>>> import numpy as np
>>> partial_trace(test_input_mat, [0, 2], [2, 2, 2, 2])
array([[344, 348, 360, 364],
       [408, 412, 424, 428],
       [600, 604, 616, 620],
       [664, 668, 680, 684]])

References

[1] (1,2)

Wikipedia. Partial trace. URL: https://en.wikipedia.org/wiki/Partial_trace.

Raises:

ValueError – If matrix dimension is not equal to the number of subsystems.

Parameters:

  • input_mat (numpy.ndarray | cvxpy.expressions.variable.Variable) – A square matrix.

  • sys (int | list[int]) – Scalar or vector specifying the size of the subsystems.

  • dim (int | list[int]) – Dimension of the subsystems. If None, all dimensions are assumed to be equal.

Returns:

The partial trace of matrix input_mat.

Return type:

numpy.ndarray | cvxpy.expressions.expression.Expression