# qml.math.min_entropy¶

min_entropy(state, indices, base=None, check_state=False, c_dtype='complex128')[source]

Compute the minimum entropy from a density matrix.

$S_{\text{min}}( \rho ) = -\log( \max_{i} ( p_{i} ))$
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.

• base (float) – Base for the logarithm. If None, the natural logarithm is used.

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

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

Returns

The minimum entropy of the considered subsystem.

Return type

float

Example

The minimum entropy of a subsystem for any state vector can be obtained by first calling dm_from_state_vector() on the input. Here is an example for the maximally entangled state, where the subsystem entropy is maximal (default base for log is exponential).

>>> x = [1, 0, 0, 1] / np.sqrt(2)
>>> x = dm_from_state_vector(x)
>>> min_entropy(x, indices=[0])
0.6931472


The logarithm base can be changed. For example:

>>> min_entropy(x, indices=[0], base=2)
1.0


The minimum entropy can be obtained by providing a quantum state as a density matrix. For example:

>>> y = [[1/2, 0, 0, 1/2], [0, 0, 0, 0], [0, 0, 0, 0], [1/2, 0, 0, 1/2]]
>>> min_entropy(y, indices=[0])
0.6931472


The Von Neumann entropy is always greater than the minimum entropy.

>>> x = [np.cos(np.pi/8), 0, 0, -1j*np.sin(np.pi/8)]
>>> x = dm_from_state_vector(x)
>>> vn_entropy(x, indices=[1])
0.4164955
>>> min_entropy(x, indices=[1])
0.1583472