Examples for No-go theorems for photon state transformations in quantum linear optics¶

In this short notebook, we perform the calculations used in the article No-go theorems for photon state transformations in quantum linear optics using the library QOptCraft.

In [1]:

import numpy as np

from qoptcraft import (
    Fock,
    photon_invariant,
    forbidden_transition,
    get_algebra_basis,
    get_photon_basis,
)
from qoptcraft.math import gram_schmidt

import numpy as np from qoptcraft import ( Fock, photon_invariant, forbidden_transition, get_algebra_basis, get_photon_basis, ) from qoptcraft.math import gram_schmidt

Hilbert space and algebra bases¶

We begin by obtaining the basis of the Hilbert space with 2 modes and 2 photons:

In [2]:

modes = 2
photons = 2
get_photon_basis(modes, photons)

modes = 2 photons = 2 get_photon_basis(modes, photons)

Out[2]:

[(2, 0), (1, 1), (0, 2)]

QOptCraft returns the basis of the image algebra as sparse matrices to save space. The following cell returns a basis of the image of the Lie algebra of hermitian matrices of dimension 2 under the photonic homomorphism for 2 photons.

In [3]:

basis_algebra, basis_image_algebra = get_algebra_basis(modes, photons)
for i, matrix in enumerate(basis_image_algebra):
    print(f"b{i + 1} = {matrix.toarray()}\n")

basis_algebra, basis_image_algebra = get_algebra_basis(modes, photons) for i, matrix in enumerate(basis_image_algebra): print(f"b{i + 1} = {matrix.toarray()}\n")

b1 = [[0.+2.j 0.+0.j 0.+0.j]
 [0.+0.j 0.+1.j 0.+0.j]
 [0.+0.j 0.+0.j 0.+0.j]]

b2 = [[0.+0.j         0.+0.70710678j 0.+0.j        ]
 [0.+0.70710678j 0.+0.j         0.+0.70710678j]
 [0.+0.j         0.+0.70710678j 0.+0.j        ]]

b3 = [[0.+0.j 0.+0.j 0.+0.j]
 [0.+0.j 0.+1.j 0.+0.j]
 [0.+0.j 0.+0.j 0.+2.j]]

b4 = [[ 0.        +0.j -0.70710678+0.j  0.        +0.j]
 [ 0.70710678+0.j  0.        +0.j -0.70710678+0.j]
 [ 0.        +0.j  0.70710678+0.j  0.        +0.j]]

We can obtain an orthonormal basis using the gram-schmidt algorithm

In [4]:

ortho_basis_image = gram_schmidt(basis_image_algebra)
for i, matrix in enumerate(ortho_basis_image):
    print(f"c{i + 1} = {matrix.toarray()}\n")

ortho_basis_image = gram_schmidt(basis_image_algebra) for i, matrix in enumerate(ortho_basis_image): print(f"c{i + 1} = {matrix.toarray()}\n")

c1 = [[0.+0.89442719j 0.+0.j         0.+0.j        ]
 [0.+0.j         0.+0.4472136j  0.+0.j        ]
 [0.+0.j         0.+0.j         0.+0.j        ]]

c2 = [[0.+0.j  0.+0.5j 0.+0.j ]
 [0.+0.5j 0.+0.j  0.+0.5j]
 [0.+0.j  0.+0.5j 0.+0.j ]]

c3 = [[0.-0.18257419j 0.+0.j         0.+0.j        ]
 [0.+0.j         0.+0.36514837j 0.+0.j        ]
 [0.+0.j         0.+0.j         0.+0.91287093j]]

c4 = [[ 0. +0.j -0.5+0.j  0. +0.j]
 [ 0.5+0.j  0. +0.j -0.5+0.j]
 [ 0. +0.j  0.5+0.j  0. +0.j]]

Computing invariants¶

Fock state¶

The function photon_invariant computes the tangent and perpendicular invariants (in this order).

In [5]:

state = Fock(2, 0)
print(f"tangent_invariant = {photon_invariant(state)}")

state = Fock(2, 0) print(f"tangent_invariant = {photon_invariant(state)}")

tangent_invariant = 0.8333333333333333

In [6]:

state = Fock(1, 1)
print(f"tangent_invariant = {photon_invariant(state)}")

state = Fock(1, 1) print(f"tangent_invariant = {photon_invariant(state)}")

tangent_invariant = 0.3333333333333332

In [7]:

state = Fock(0, 2)
print(f"tangent_invariant = {photon_invariant(state)}")

state = Fock(0, 2) print(f"tangent_invariant = {photon_invariant(state)}")

tangent_invariant = 0.8333333333333336

Hong-Ou-Mandel¶

We also compute the invariant of the a 50:50 beamsplitter (Hong-Ou-Mandel experiment output)

In [8]:

state = (Fock(2, 0) - Fock(0, 2)) / np.sqrt(2)
print(f"tangent_invariant = {photon_invariant(state)}")

state = (Fock(2, 0) - Fock(0, 2)) / np.sqrt(2) print(f"tangent_invariant = {photon_invariant(state)}")

tangent_invariant = 0.3333333333333336

NOON states¶

In [9]:

state_in = Fock(1, 1)
state_out = Fock(2, 0) + Fock(0, 2)
print(f"\nForbidden transition? {forbidden_transition(state_in, state_out)}")

state_in = Fock(1, 1) state_out = Fock(2, 0) + Fock(0, 2) print(f"\nForbidden transition? {forbidden_transition(state_in, state_out)}")

In invariant = 0.3333333 	 Out invariant = 0.3333333

Forbidden transition? False

In [10]:

state_in = Fock(2, 2)
state_out = Fock(4, 0) + Fock(0, 4)
print(f"\nForbidden transition? {forbidden_transition(state_in, state_out)}")

state_in = Fock(2, 2) state_out = Fock(4, 0) + Fock(0, 4) print(f"\nForbidden transition? {forbidden_transition(state_in, state_out)}")

In invariant = 0.2000000 	 Out invariant = 0.2000000

Forbidden transition? False

In [11]:

state_in = Fock(3, 3)
state_out = Fock(6, 0) + Fock(0, 6)
print(f"\nForbidden transition? {forbidden_transition(state_in, state_out)}")

state_in = Fock(3, 3) state_out = Fock(6, 0) + Fock(0, 6) print(f"\nForbidden transition? {forbidden_transition(state_in, state_out)}")

In invariant = 0.1428571 	 Out invariant = 0.1428571

Forbidden transition? False

In [12]:

state_in = Fock(4, 4)
state_out = Fock(8, 0) + Fock(0, 8)
print(f"\nForbidden transition? {forbidden_transition(state_in, state_out)}")

state_in = Fock(4, 4) state_out = Fock(8, 0) + Fock(0, 8) print(f"\nForbidden transition? {forbidden_transition(state_in, state_out)}")

In invariant = 0.1111111 	 Out invariant = 0.1111111

Forbidden transition? False

In [13]:

state_in = Fock(5, 5)
state_out = Fock(10, 0) + Fock(0, 10)
print(f"\nForbidden transition? {forbidden_transition(state_in, state_out)}")

state_in = Fock(5, 5) state_out = Fock(10, 0) + Fock(0, 10) print(f"\nForbidden transition? {forbidden_transition(state_in, state_out)}")

In invariant = 0.0909091 	 Out invariant = 0.0909091

Forbidden transition? False

Reduced invariants¶

We can call the reduced invariant using the method='reduced' option in the functions can_transition and invariant.

In [14]:

state = Fock(0, 2)
print(f"tangent_invariant = {photon_invariant(state, method='reduced')}")

state = Fock(0, 2) print(f"tangent_invariant = {photon_invariant(state, method='reduced')}")

tangent_invariant = 0.0

For pure states, we can recover the full invariant using theorem 2 without using the algebra basis:

In [15]:

state = Fock(0, 2)
print(f"tangent_invariant = {photon_invariant(state, method='no basis')}")

state = Fock(0, 2) print(f"tangent_invariant = {photon_invariant(state, method='no basis')}")

tangent_invariant = 0.8333333333333334

The reduced invariant allows us to apply the invariant criterion

In [16]:

state_in = Fock(4, 4)
state_out = Fock(8, 0) + Fock(0, 8)
print(f"\nForbidden transition? {forbidden_transition(state_in, state_out, method='reduced')}")

state_in = Fock(4, 4) state_out = Fock(8, 0) + Fock(0, 8) print(f"\nForbidden transition? {forbidden_transition(state_in, state_out, method='reduced')}")

In reduced invariant = -16.0000000	 Out reduced invariant = -16.0000000

Forbidden transition? False

Most transitions between fock states are forbidden¶

In [17]:

not_forbidden = []
random_draws = 1000
max_photons_per_mode = 14
rng = np.random.default_rng()
min_modes = 2
max_modes = 25
for modes in range(min_modes, max_modes):
    for _ in range(random_draws):
        while True:
            photons_in = rng.integers(low=0, high=max_photons_per_mode, size=modes, endpoint=True)
            photons_out = rng.integers(low=0, high=max_photons_per_mode, size=modes - 1, endpoint=True)
            if sum(photons_in) >= sum(photons_out):
                photons_out = np.r_[photons_out, sum(photons_in) - sum(photons_out)]
                if (np.sort(photons_in) != np.sort(photons_out)).all():
                    state_in = Fock(*photons_in)
                    state_out = Fock(*photons_out)
                    break
        if not forbidden_transition(state_in, state_out, method="reduced", print_invariant=False):
            not_forbidden.append((state_in, state_out))

print(f"\nPercentage of not forbidden transitions = {len(not_forbidden) * 100 / (random_draws * (max_modes - min_modes))} %")

not_forbidden = [] random_draws = 1000 max_photons_per_mode = 14 rng = np.random.default_rng() min_modes = 2 max_modes = 25 for modes in range(min_modes, max_modes): for _ in range(random_draws): while True: photons_in = rng.integers(low=0, high=max_photons_per_mode, size=modes, endpoint=True) photons_out = rng.integers(low=0, high=max_photons_per_mode, size=modes - 1, endpoint=True) if sum(photons_in) >= sum(photons_out): photons_out = np.r_[photons_out, sum(photons_in) - sum(photons_out)] if (np.sort(photons_in) != np.sort(photons_out)).all(): state_in = Fock(*photons_in) state_out = Fock(*photons_out) break if not forbidden_transition(state_in, state_out, method="reduced", print_invariant=False): not_forbidden.append((state_in, state_out)) print(f"\nPercentage of not forbidden transitions = {len(not_forbidden) * 100 / (random_draws * (max_modes - min_modes))} %")

Percentage of not forbidden transitions = 0.16956521739130434 %

Forbidden (n, 0) to (n-k, k) transition¶

In [18]:

for n in range(5):
    for k in range(1, n):
        state_in = Fock(n, 0)
        state_out = Fock(n - k, k)
        print(
            f"Forbidden transition? {forbidden_transition(state_in, state_out, method='reduced')}\n"
        )

for n in range(5): for k in range(1, n): state_in = Fock(n, 0) state_out = Fock(n - k, k) print( f"Forbidden transition? {forbidden_transition(state_in, state_out, method='reduced')}\n" )

In reduced invariant = 0.0000000	 Out reduced invariant = -1.0000000
Forbidden transition? True

In reduced invariant = 0.0000000	 Out reduced invariant = -2.0000000
Forbidden transition? True

In reduced invariant = 0.0000000	 Out reduced invariant = -2.0000000
Forbidden transition? True

In reduced invariant = 0.0000000	 Out reduced invariant = -3.0000000
Forbidden transition? True

In reduced invariant = 0.0000000	 Out reduced invariant = -4.0000000
Forbidden transition? True

In reduced invariant = 0.0000000	 Out reduced invariant = -3.0000000
Forbidden transition? True

Impossibility of Bell state with fock ancilla generation from a fock state¶

In [19]:

rng = np.random.default_rng()

bell_state = (Fock(1, 0, 1, 0) + Fock(0, 1, 0, 1)) / np.sqrt(2)
for modes in range(5, 8):
    for max_photons_per_mode in range(1, 4):
        while True:
            photons_in = rng.integers(low=0, high=max_photons_per_mode, size=modes, endpoint=True)
            photons_out = rng.integers(
                low=0, high=max_photons_per_mode, size=modes - 4, endpoint=True
            )
            if sum(photons_in) - 2 == sum(photons_out) and sum(photons_in) != 0:
                state_in = Fock(*photons_in)
                state_out = bell_state * Fock(*photons_out)
                break
        print(
            f"Forbidden transition? {forbidden_transition(state_in, state_out, method='reduced')}\n"
        )

rng = np.random.default_rng() bell_state = (Fock(1, 0, 1, 0) + Fock(0, 1, 0, 1)) / np.sqrt(2) for modes in range(5, 8): for max_photons_per_mode in range(1, 4): while True: photons_in = rng.integers(low=0, high=max_photons_per_mode, size=modes, endpoint=True) photons_out = rng.integers( low=0, high=max_photons_per_mode, size=modes - 4, endpoint=True ) if sum(photons_in) - 2 == sum(photons_out) and sum(photons_in) != 0: state_in = Fock(*photons_in) state_out = bell_state * Fock(*photons_out) break print( f"Forbidden transition? {forbidden_transition(state_in, state_out, method='reduced')}\n" )

In reduced invariant = -1.0000000	 Out reduced invariant = -1.5000000
Forbidden transition? True

In reduced invariant = -3.0000000	 Out reduced invariant = -3.5000000
Forbidden transition? True

In reduced invariant = -8.0000000	 Out reduced invariant = -7.5000000
Forbidden transition? True

In reduced invariant = -3.0000000	 Out reduced invariant = -3.5000000
Forbidden transition? True

In reduced invariant = -13.0000000	 Out reduced invariant = -13.5000000
Forbidden transition? True

In reduced invariant = -11.0000000	 Out reduced invariant = -12.5000000
Forbidden transition? True

In reduced invariant = -6.0000000	 Out reduced invariant = -6.5000000
Forbidden transition? True

In reduced invariant = -18.0000000	 Out reduced invariant = -19.5000000
Forbidden transition? True

In reduced invariant = -29.0000000	 Out reduced invariant = -30.5000000
Forbidden transition? True

Impossibility of exact transformation of GHZ into W (with fock ancillas)¶

In [20]:

rng = np.random.default_rng()

ghz_state = (Fock(1, 0, 1, 0, 1, 0) + Fock(0, 1, 0, 1, 0, 1)) / np.sqrt(2)
w_state = (Fock(1, 0, 0, 1, 0, 1) + Fock(0, 1, 1, 0, 0, 1) + Fock(0, 1, 0, 1, 1, 0)) / np.sqrt(3)

for modes in range(1, 5):
    for max_photons_per_mode in range(1, 4):
        while True:
            photons_in = rng.integers(low=0, high=max_photons_per_mode, size=modes, endpoint=True)
            photons_out = rng.integers(low=0, high=max_photons_per_mode, size=modes, endpoint=True)
            if sum(photons_in) == sum(photons_out) and sum(photons_in) != 0:
                state_in = ghz_state * Fock(*photons_in)
                state_out = w_state * Fock(*photons_out)
                break
        print(
            f"Forbidden transition? {forbidden_transition(state_in, state_out, method='reduced')}\n"
        )

rng = np.random.default_rng() ghz_state = (Fock(1, 0, 1, 0, 1, 0) + Fock(0, 1, 0, 1, 0, 1)) / np.sqrt(2) w_state = (Fock(1, 0, 0, 1, 0, 1) + Fock(0, 1, 1, 0, 0, 1) + Fock(0, 1, 0, 1, 1, 0)) / np.sqrt(3) for modes in range(1, 5): for max_photons_per_mode in range(1, 4): while True: photons_in = rng.integers(low=0, high=max_photons_per_mode, size=modes, endpoint=True) photons_out = rng.integers(low=0, high=max_photons_per_mode, size=modes, endpoint=True) if sum(photons_in) == sum(photons_out) and sum(photons_in) != 0: state_in = ghz_state * Fock(*photons_in) state_out = w_state * Fock(*photons_out) break print( f"Forbidden transition? {forbidden_transition(state_in, state_out, method='reduced')}\n" )

In reduced invariant = -6.7500000	 Out reduced invariant = -6.6666667
Forbidden transition? True

In reduced invariant = -9.7500000	 Out reduced invariant = -9.6666667
Forbidden transition? True

In reduced invariant = -6.7500000	 Out reduced invariant = -6.6666667
Forbidden transition? True

In reduced invariant = -6.7500000	 Out reduced invariant = -6.6666667
Forbidden transition? True

In reduced invariant = -6.7500000	 Out reduced invariant = -6.6666667
Forbidden transition? True

In reduced invariant = -30.7500000	 Out reduced invariant = -30.6666667
Forbidden transition? True

In reduced invariant = -6.7500000	 Out reduced invariant = -6.6666667
Forbidden transition? True

In reduced invariant = -9.7500000	 Out reduced invariant = -10.6666667
Forbidden transition? True

In reduced invariant = -19.7500000	 Out reduced invariant = -20.6666667
Forbidden transition? True

In reduced invariant = -10.7500000	 Out reduced invariant = -10.6666667
Forbidden transition? True

In reduced invariant = -20.7500000	 Out reduced invariant = -20.6666667
Forbidden transition? True

In reduced invariant = -70.7500000	 Out reduced invariant = -70.6666667
Forbidden transition? True