toqito.states.tile

toqito.states.tile(idx)

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 idx:

\[\begin{split}\begin{equation} \begin{aligned} |\psi_0 \rangle = \frac{1}{\sqrt{2}} |0 \rangle \left(|0\rangle - |1\rangle \right), \qquad & |\psi_1\rangle = \frac{1}{\sqrt{2}} \left(|0\rangle - |1\rangle \right) |2\rangle, \\ |\psi_2\rangle = \frac{1}{\sqrt{2}} |2\rangle \left(|1\rangle - |2\rangle \right), \qquad & |\psi_3\rangle = \frac{1}{\sqrt{2}} \left(|1\rangle - |2\rangle \right) |0\rangle, \\ \qquad & |\psi_4\rangle = \frac{1}{3} \left(|0\rangle + |1\rangle + |2\rangle)\right) \left(|0\rangle + |1\rangle + |2\rangle \right). \end{aligned} \end{equation}\end{split}\]

Examples

When idx = 0, this produces the following tile state

\[\frac{1}{\sqrt{2}} |0\rangle \left( |0\rangle - |1\rangle \right).\]

Using 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 idx.

Parameters:

idx – A parameter in [0, 1, 2, 3, 4]

Returns:

Tile state.