"""Domino state."""
import numpy as np
from toqito.states import basis
[docs]
def domino(idx: int) -> np.ndarray:
r"""
Produce a domino state [CBDOM99]_, [UPB99]_.
The orthonormal product basis of domino states is given as
.. math::
\begin{equation}
\begin{aligned}
|\phi_0\rangle = |1\rangle |1 \rangle,
\qquad
|\phi_1\rangle = |0 \rangle \left(\frac{|0 \rangle + |1 \rangle}{\sqrt{2}} \right),
& \qquad
|\phi_2\rangle = |0\rangle \left(\frac{|0\rangle - |1\rangle}{\sqrt{2}}\right), \\
|\phi_3\rangle = |2\rangle \left(\frac{|0\rangle + |1\rangle}{\sqrt{2}}\right), \qquad
|\phi_4\rangle = |2\rangle \left(\frac{|0\rangle - |1\rangle}{\sqrt{2}}\right), & \qquad
|\phi_5\rangle = \left(\frac{|0\rangle + |1\rangle}{\sqrt{2}}\right) |0\rangle, \\
|\phi_6\rangle = \left(\frac{|0\rangle - |1\rangle}{\sqrt{2}}\right) |0\rangle, \qquad
|\phi_7\rangle = \left(\frac{|0\rangle + |1\rangle}{\sqrt{2}}\right) |2\rangle, & \qquad
|\phi_8\rangle = \left(\frac{|0\rangle - |1\rangle}{\sqrt{2}}\right) |2\rangle.
\end{aligned}
\end{equation}
Returns one of the following nine domino states depending on the value of :code:`idx`.
Examples
==========
When :code:`idx = 0`, this produces the following Domino state
.. math::
|\phi_0 \rangle = |11 \rangle |11 \rangle.
Using :code:`toqito`, we can see that this yields the proper state.
>>> from toqito.states import domino
>>> domino(0)
[[0],
[0],
[0],
[0],
[1],
[0],
[0],
[0],
[0]]
When :code:`idx = 3`, this produces the following Domino state
.. math::
|\phi_3\rangle = |2\rangle \left(\frac{|0\rangle + |1\rangle}
{\sqrt{2}}\right)
Using :code:`toqito`, we can see that this yields the proper state.
>>> from toqito.states import domino
>>> domino(3)
[[0. ],
[0. ],
[0. ],
[0. ],
[0. ],
[0. ],
[0. ],
[0.70710678],
[0.70710678]]
References
==========
.. [CBDOM99] Bennett, Charles H., et al.
Quantum nonlocality without entanglement.
Phys. Rev. A, 59:1070–1091, Feb 1999.
https://arxiv.org/abs/quant-ph/9804053
.. [UPB99] Bennett, Charles H., et al.
"Unextendible product bases and bound entanglement."
Physical Review Letters 82.26 (1999): 5385.
https://arxiv.org/abs/quant-ph/9808030
:raises ValueError: Invalid value for :code:`idx`.
:param idx: A parameter in [0, 1, 2, 3, 4, 5, 6, 7, 8]
:return: Domino state of index :code:`idx`.
"""
e_0, e_1, e_2 = basis(3, 0), basis(3, 1), basis(3, 2)
if idx == 0:
return np.kron(e_1, e_1)
if idx == 1:
return np.kron(e_0, 1 / np.sqrt(2) * (e_0 + e_1))
if idx == 2:
return np.kron(e_0, 1 / np.sqrt(2) * (e_0 - e_1))
if idx == 3:
return np.kron(e_2, 1 / np.sqrt(2) * (e_1 + e_2))
if idx == 4:
return np.kron(e_2, 1 / np.sqrt(2) * (e_1 - e_2))
if idx == 5:
return np.kron(1 / np.sqrt(2) * (e_1 + e_2), e_0)
if idx == 6:
return np.kron(1 / np.sqrt(2) * (e_1 - e_2), e_0)
if idx == 7:
return np.kron(1 / np.sqrt(2) * (e_0 + e_1), e_2)
if idx == 8:
return np.kron(1 / np.sqrt(2) * (e_0 - e_1), e_2)
raise ValueError("Invalid integer value for Domino state.")