"""Tile state."""
import numpy as np
from toqito.states import basis
[docs]
def tile(idx: int) -> np.ndarray:
r"""
Produce a Tile state [UPBTile99]_.
The Tile states constitute five states on 3-by-3 dimensional space that form a UPB
(unextendible product basis).
Returns one of the following five tile states depending on the value of :code:`idx`:
.. math::
\begin{equation}
\begin{aligned}
|\psi_0 \rangle = \frac{1}{\sqrt{2}} |0 \rangle
\left(|0\rangle - |1\rangle \right),
\qquad &
|\psi_1\rangle = \frac{1}{\sqrt{2}}
\left(|0\rangle - |1\rangle \right) |2\rangle, \\
|\psi_2\rangle = \frac{1}{\sqrt{2}} |2\rangle
\left(|1\rangle - |2\rangle \right),
\qquad &
|\psi_3\rangle = \frac{1}{\sqrt{2}}
\left(|1\rangle - |2\rangle \right) |0\rangle, \\
\qquad &
|\psi_4\rangle = \frac{1}{3}
\left(|0\rangle + |1\rangle + |2\rangle)\right)
\left(|0\rangle + |1\rangle + |2\rangle \right).
\end{aligned}
\end{equation}
Examples
==========
When :code:`idx = 0`, this produces the following tile state
.. math::
\frac{1}{\sqrt{2}} |0\rangle \left( |0\rangle - |1\rangle \right).
Using :code:`toqito`, we can see that this yields the proper state.
>>> from toqito.states import tile
>>> import numpy as np
>>> tile(0)
[[ 0.70710678]
[-0. ]
[ 0. ]]
References
==========
.. [UPBTile99] 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]
:return: Tile state.
"""
e_0, e_1, e_2 = basis(3, 0), basis(3, 1), basis(3, 2)
if idx == 0:
return 1 / np.sqrt(2) * np.kron(e_0, (e_0 - e_1))
if idx == 1:
return 1 / np.sqrt(2) * np.kron((e_0 - e_1), e_2)
if idx == 2:
return 1 / np.sqrt(2) * np.kron(e_2, (e_1 - e_2))
if idx == 3:
return 1 / np.sqrt(2) * np.kron((e_1 - e_2), e_0)
if idx == 4:
return 1 / 3 * np.kron((e_0 + e_1 + e_2), (e_0 + e_1 + e_2))
raise ValueError("Invalid integer value for Tile state.")