qml.math.purity

purity(state, indices, check_state=False, c_dtype='complex128')[source]

Computes the purity of a density matrix.

\[\gamma = \text{Tr}(\rho^2)\]

where \(\rho\) is the density matrix. The purity of a normalized quantum state satisfies \(\frac{1}{d} \leq \gamma \leq 1\), where \(d\) is the dimension of the Hilbert space. A pure state has a purity of 1.

It is possible to compute the purity of a sub-system from a given state. To find the purity of the overall state, include all wires in the indices argument.

Parameters
  • state (tensor_like) – Density matrix of shape (2**N, 2**N) or (batch_dim, 2**N, 2**N)

  • indices (list(int)) – List of indices in the considered subsystem.

  • check_state (bool) – If True, the function will check the state validity (shape and norm).

  • c_dtype (str) – Complex floating point precision type.

Returns

Purity of the considered subsystem.

Return type

float

Example

>>> x = [[1/2, 0, 0, 1/2], [0, 0, 0, 0], [0, 0, 0, 0], [1/2, 0, 0, 1/2]]
>>> purity(x, [0, 1])
1.0
>>> purity(x, [0])
0.5
>>> x = [[1/2, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 1/2]]
>>> purity(x, [0, 1])
0.5

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