qml.math.max_entropy¶
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max_entropy(state, indices, base=None, check_state=False, c_dtype='complex128')[source]¶ Compute the maximum entropy of a density matrix on a given subsystem. It supports all interfaces (NumPy, Autograd, Torch, TensorFlow and Jax).
\[S_{\text{max}}( \rho ) = \log( \text{rank} ( \rho ))\]- 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 maximum entropy of the considered subsystem.
- Return type
float
Example
The maximum 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) >>> max_entropy(x, indices=[0]) 0.6931472
The logarithm base can be changed. For example:
>>> max_entropy(x, indices=[0], base=2) 1.0
The maximum 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]] >>> max_entropy(y, indices=[0]) 0.6931472
The maximum entropy is always greater or equal to the Von Neumann entropy. In this maximally entangled example, they are equal:
>>> vn_entropy(x, indices=[0]) 0.6931472
However, in general, the Von Neumann entropy is lower:
>>> 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 >>> max_entropy(x, indices=[1]) 0.6931472