Defining states¶

There are various ways of defining states in QOptCraft. The main type (class) is State, which subclasses into PureState and MixedState. For convenience, we also have a Fock subclass of PureState, which allows us to create pure states in a handy way with sums and products.

In [1]:

import numpy as np
from qoptcraft.state import Fock, PureState, MixedState, State

import numpy as np from qoptcraft.state import Fock, PureState, MixedState, State

Pure states¶

Initialization¶

Let's define a state using the Fock subclass

In [2]:

state = (Fock(1, 0) + Fock(0, 1)) / np.sqrt(2)
state

state = (Fock(1, 0) + Fock(0, 1)) / np.sqrt(2) state

Out[2]:

0.71 * (1, 0) + 
0.71 * (0, 1)

By default, states are normalized conserving the proportions between the coefficients; thus, we can write the previous state as

In [3]:

state = Fock(1, 0) + Fock(0, 1)
state

state = Fock(1, 0) + Fock(0, 1) state

Out[3]:

0.71 * (1, 0) + 
0.71 * (0, 1)

This state is not the same as

In [4]:

state = Fock(1, 0) + 2 * Fock(0, 1)
state

state = Fock(1, 0) + 2 * Fock(0, 1) state

Out[4]:

0.45 * (1, 0) + 
0.89 * (0, 1)

The class keeps track of both the coefficients and the amplitudes:

In [5]:

print(f"coefficients = {state.coefs}")
print(f"amplitudes = {state.amplitudes}")

print(f"coefficients = {state.coefs}") print(f"amplitudes = {state.amplitudes}")

coefficients = [1 2]
amplitudes = [0.4472136  0.89442719]

We can also create pure states by initializing the PureState class. This is done providing a list of photon number (fock) states, as a list of tuples, and a list of coefficients.

In [6]:

fock_list = [(2, 0), (0, 2)]
coef_list = [1 / np.sqrt(2), -1 / np.sqrt(2)]
PureState(fock_list, coef_list)

fock_list = [(2, 0), (0, 2)] coef_list = [1 / np.sqrt(2), -1 / np.sqrt(2)] PureState(fock_list, coef_list)

Out[6]:

0.71 * (2, 0) + 
-0.71 * (0, 2)

Operations¶

We can do operations with pure states, like summing or computing tensor products:

In [7]:

state_1 = Fock(2, 0) + Fock(0, 2)
state_2 = Fock(1, 1)
state_1 - state_2

state_1 = Fock(2, 0) + Fock(0, 2) state_2 = Fock(1, 1) state_1 - state_2

Out[7]:

0.58 * (2, 0) + 
-0.58 * (1, 1) + 
0.58 * (0, 2)

In [8]:

state_1 * state_2

state_1 * state_2

Out[8]:

0.71 * (2, 0, 1, 1) + 
0.71 * (0, 2, 1, 1)

Attributes¶

There are a number of attributes we can access, including the number of photons

In [9]:

state_1.photons

state_1.photons

Out[9]:

2

the number of modes

In [10]:

state_1.modes

state_1.modes

Out[10]:

2

and the density matrix

In [11]:

state_1.density_matrix

state_1.density_matrix

Out[11]:

array([[0.5+0.j, 0. +0.j, 0.5+0.j],
       [0. +0.j, 0. +0.j, 0. +0.j],
       [0.5+0.j, 0. +0.j, 0.5+0.j]])

Methods¶

We can retrieve the state in the Fock basis:

In [12]:

state_1.state_in_basis()

state_1.state_in_basis()

Out[12]:

array([0.70710678+0.j, 0.        +0.j, 0.70710678+0.j])

We can apply creation and annihilation operators

In [13]:

state = Fock(1, 0) + Fock(0, 1)
state.creation(mode=1)

state = Fock(1, 0) + Fock(0, 1) state.creation(mode=1)

Out[13]:

0.58 * (1, 1) + 
0.82 * (0, 2)

In [14]:

state = Fock(2, 0) + Fock(0, 2)
state.annihilation(mode=0)

state = Fock(2, 0) + Fock(0, 2) state.annihilation(mode=0)

Out[14]:

(1, 0)

Mixed states¶

Mixed states can be created directly from the density matrix:

In [15]:

mixed_state = MixedState(density_matrix=np.eye(3) / 3, modes=2, photons=2)
mixed_state.density_matrix

mixed_state = MixedState(density_matrix=np.eye(3) / 3, modes=2, photons=2) mixed_state.density_matrix

Out[15]:

array([[0.33333333, 0.        , 0.        ],
       [0.        , 0.33333333, 0.        ],
       [0.        , 0.        , 0.33333333]])

or from a mixture of pure states:

In [16]:

mixed_state = MixedState.from_mixture(pure_states=[state_1, state_2], probs=[0.5, 0.5])
mixed_state.density_matrix

mixed_state = MixedState.from_mixture(pure_states=[state_1, state_2], probs=[0.5, 0.5]) mixed_state.density_matrix

Out[16]:

array([[0.25+0.j, 0.  +0.j, 0.25+0.j],
       [0.  +0.j, 0.5 +0.j, 0.  +0.j],
       [0.25+0.j, 0.  +0.j, 0.25+0.j]])