Quantities#

This module provides functions and a mixin for calculating quantum quantities.

from qhronology.mechanics.quantities import trace, purity, distance, fidelity, entropy, mutual
from qhronology.mechanics.quantities import QuantitiesMixin

Functions#

trace(
matrix: mat | QuantumObject,
) → num | sym[source]#

Calculate the (complete) trace \(\trace[\op{\rho}]\) of matrix (\(\op{\rho}\)).

Parameters:

matrix (mat | QuantumObject) – The input matrix.

Returns:

num | sym – The trace of the input matrix.

Examples

>>> matrix = sp.Matrix([["a", "b"], ["c", "d"]])
>>> trace(matrix)
a + d

>>> matrix = sp.MatrixSymbol("U", 3, 3).as_mutable()
>>> trace(matrix)
U[0, 0] + U[1, 1] + U[2, 2]
purity(
state: mat | QuantumObject,
) → num | sym[source]#

Calculate the purity (\(\Purity\)) of state (\(\op{\rho}\)):

(314)#\[\Purity(\op{\rho}) = \trace[\op{\rho}^2].\]
Parameters:

state (mat | QuantumObject) – The matrix representation of the input state.

Returns:

num | sym – The purity of the input state.

Examples

>>> matrix = sp.Matrix([["a", "b"], ["c", "d"]])
>>> purity(matrix)
a**2 + 2*b*c + d**2

>>> matrix = sp.Matrix(
...     [["a*conjugate(a)", "a*conjugate(b)"], ["b*conjugate(a)", "b*conjugate(b)"]]
... )
>>> purity(matrix)
(a*conjugate(a) + b*conjugate(b))**2
distance(
state_A: mat | QuantumObject,
state_B: mat | QuantumObject,
) → num | sym[source]#

Calculate the trace distance (\(\TraceDistance\)) between two states state_A (\(\op{\rho}\)) and state_B (\(\op{\tau}\)):

(315)#\[\TraceDistance(\op{\rho}, \op{\tau}) = \frac{1}{2}\trace{\abs{\op{\rho} - \op{\tau}}}.\]
Parameters:

  • state_A (mat | QuantumObject) – The matrix representation of the first input state.

  • state_B (mat | QuantumObject) – The matrix representation of the second input state.

Returns:

num | sym – The trace distance between the inputs state_A and state_B.

Examples

>>> matrix_A = sp.Matrix([["p", 0], [0, "1 - p"]])
>>> matrix_B = sp.Matrix([["q", 0], [0, "1 - q"]])
>>> distance(matrix_A, matrix_B)
sqrt((p - q)*(conjugate(p) - conjugate(q)))

>>> matrix_A = sp.Matrix([["1/sqrt(2)"], ["1/sqrt(2)"]])
>>> matrix_B = sp.Matrix([["1/sqrt(2)"], ["-1/sqrt(2)"]])
>>> distance(matrix_A, matrix_B)
1

>>> matrix = sp.Matrix([["a", "b"], ["c", "d"]])
>>> distance(matrix, matrix)
0
fidelity(
state_A: mat | QuantumObject,
state_B: mat | QuantumObject,
) → num | sym[source]#

Calculate the fidelity (\(\Fidelity\)) between two states state_A (\(\op{\rho}\)) and state_B (\(\op{\tau}\)):

(316)#\[\Fidelity(\op{\rho}, \op{\tau}) = \left(\trace{\sqrt{\sqrt{\op{\rho}}\,\op{\tau}\sqrt{\op{\rho}}}}\right)^2.\]
Parameters:

  • state_A (mat | QuantumObject) – The matrix representation of the first input state.

  • state_B (mat | QuantumObject) – The matrix representation of the second input state.

Returns:

num | sym – The fidelity between the inputs state_A and state_B.

Examples

>>> matrix_A = sp.Matrix([["a"], ["b"]])
>>> matrix_B = sp.Matrix([["c"], ["d"]])
>>> fidelity(matrix_A, matrix_A)
(a*conjugate(a) + b*conjugate(b))**2
>>> fidelity(matrix_B, matrix_B)
(c*conjugate(c) + d*conjugate(d))**2
>>> fidelity(matrix_A, matrix_B)
(a*conjugate(c) + b*conjugate(d))*(c*conjugate(a) + d*conjugate(b))

>>> matrix_A = sp.Matrix([["p", 0], [0, "1 - p"]])
>>> matrix_B = sp.Matrix([["q", 0], [0, "1 - q"]])
>>> fidelity(matrix_A, matrix_B)
(sqrt(p*q) + sqrt((1 - p)*(1 - q)))**2

>>> matrix_A = sp.Matrix([["1/sqrt(2)"], ["1/sqrt(2)"]])
>>> matrix_B = sp.Matrix([["1/sqrt(2)"], ["-1/sqrt(2)"]])
>>> fidelity(matrix_A, matrix_B)
0
entropy(
state_A: mat | QuantumObject,
state_B: mat | QuantumObject | None = None,
base: num | None = None,
) → num | sym[source]#

Calculate the relative von Neumann entropy (\(\Entropy\)) between two states state_A (\(\op{\rho}\)) and state_B (\(\op{\tau}\)):

(317)#\[\Entropy(\op{\rho} \Vert \op{\tau}) = \trace\bigl[\op{\rho} (\log_\Base\op{\rho} - \log_\Base\op{\tau})\bigr].\]

If state_B is not specified (i.e., None), calculate the ordinary von Neumann entropy of state_A (\(\op{\rho}\)) instead:

(318)#\[\Entropy(\op{\rho}) = \trace[\op{\rho}\log_\Base\op{\rho}].\]

Here, \(\Base\) represents base, which is the dimensionality of the unit of information with which the entropy is measured.

Parameters:

  • state_A (mat | QuantumObject) – The matrix representation of the first input state.

  • state_B (mat | QuantumObject) – The matrix representation of the second input state.

  • base (num) – The dimensionality of the unit of information with which the entropy is measured. Defaults to 2.

Returns:

num | sym – The von Neumann entropy of the input state_A (if state_B is None) or the relative entropy between state_A and state_B (if state_B is not None).

Examples

>>> matrix = sp.Matrix([["a"], ["b"]])
>>> entropy(matrix, base='d')
-(a*conjugate(a) + b*conjugate(b))**2*log(a*conjugate(a) + b*conjugate(b))/(b*log(d)*conjugate(b))

>>> matrix_A = sp.Matrix([["p", 0], [0, "1 - p"]])
>>> matrix_B = sp.Matrix([["q", 0], [0, "1 - q"]])
>>> entropy(matrix_A, base="d")
(-p*log(p) + (p - 1)*log(1 - p))/log(d)
>>> entropy(matrix_B, base="d")
(-q*log(q) + (q - 1)*log(1 - q))/log(d)
>>> entropy(matrix_A, matrix_B, base="d")
(p*(log(p) - log(q)) - (p - 1)*(log(1 - p) - log(1 - q)))/log(d)

>>> matrix_A = sp.Matrix([["1/sqrt(2)"], ["1/sqrt(2)"]])
>>> matrix_B = sp.eye(2) / 2
>>> entropy(matrix_A)
0
>>> entropy(matrix_B)
1
>>> entropy(matrix_A, matrix_B)
1
>>> entropy(matrix_B, matrix_A)
-1
mutual(
state: mat | QuantumObject,
systems_A: int | list[int] | None = None,
systems_B: int | list[int] | None = None,
dim: int | None = None,
) → num | sym[source]#

Calculate the mutual information (\(\MutualInformation\)) between two subsystems systems_A (\(A\)) and systems_B (\(B\)) of a composite quantum system represented by state (\(\rho^{A,B}\)):

(319)#\[\MutualInformation(A : B) = \Entropy(\op{\rho}^A) + \Entropy(\op{\rho}^B) - \Entropy(\op{\rho}^{A,B})\]

where \(\Entropy(\op{\rho})\) is the von Neumann entropy of a state \(\op{\rho}\).

Parameters:

  • state (mat | QuantumObject) – The matrix representation of the composite input state.

  • systems_A (int | list[int]) – The indices of the first subsystem. Defaults to [0].

  • systems_B (int | list[int]) – The indices of the second subsystem. Defaults to the complement of systems_A with respect to the entire composition of subsystems of state.

  • dim (int) – The dimensionality of the composite quantum system (and its subsystems). Must be a non-negative integer. Defaults to 2.

Returns:

num | sym – The mutual information between the subsystems systems_A and systems_B of the composite input state.

Examples

>>> matrix = sp.Matrix([1, 0, 0, 1]) / sp.sqrt(2)
>>> mutual(matrix, [0], [1])
2

>>> matrix = sp.eye(4) / 4
>>> mutual(matrix, [0], [1])
0

Mixin#

class QuantitiesMixin[source]#

A mixin for endowing classes with the ability to calculate various quantum quantities.

Any inheriting class must possess a matrix representation that can be accessed by either an output() method or a matrix property.

Note

The QuantitiesMixin mixin is used exclusively by the QuantumState class—please see the corresponding section (Quantities) for documentation on its methods.