state_props.is_antidistinguishable

Check if set of states are antidistinguishable.

Functions

is_antidistinguishable(states)

Check whether a collection of vectors are antidistinguishable or not.

Module Contents

state_props.is_antidistinguishable.is_antidistinguishable(states)

Check whether a collection of vectors are antidistinguishable or not.

For more information, see :cite: Heinosaari_2018_Antidistinguishability.

The ability to determine whether a set of quantum states are antidistinguishable can be obtained via the state exclusion SDP [2] such that we ignore the associated probabilities with which the states are chosen from the set of vectors.

Examples

The set of Bell states are an example of antidistinguishable states. Recall that the Bell states are defined as:

\[\begin{split}u_1 = \frac{1}{\sqrt{2}} \left(|00\rangle + |11\rangle\right), &\quad u_2 = \frac{1}{\sqrt{2}} \left(|00\rangle - |11\rangle\right), \\ u_3 = \frac{1}{\sqrt{2}} \left(|01\rangle + |10\rangle\right), &\quad u_4 = \frac{1}{\sqrt{2}} \left(|01\rangle - |10\rangle\right).\end{split}\]

It can be checked in :code`toqito` that the Bell states are antidistinguishable:

>>> from toqito.states import bell
>>> from toqito.state_props import is_antidistinguishable
>>>
>>> bell_states = [bell(0), bell(1), bell(2), bell(3)]
>>> is_antidistinguishable(bell_states)
np.True_

Consider the following measurement operators

\[M_i = \frac{1}{3}\left(\mathbb{I}_{\mathcal{X} - u_i u_i^*\right)\]

for all \(1 \leq i \leq 4\). It can be verified that these constitute a valid set of POVMs, that is \(\sum_{i=1}^4 M_i = \mathbb{I}_{\mathcal{X}}\) and \(M_i \in \text{Pos}(\mathcal{X})\) for all \(1 \leq i \leq 4\). It may also be verified that

\[\sum_{i=1}^4 \langle M_i, u_i u_i^* \rangle = 0,\]

and hence, the Bell states are antidistinguishable.

References

[1]

Somshubhro Bandyopadhyay, Rahul Jain, Jonathan Oppenheim, and Christopher Perry. Conclusive exclusion of quantum states. Physical Review A, Feb 2014. URL: http://dx.doi.org/10.1103/PhysRevA.89.022336, doi:10.1103/physreva.89.022336.

Parameters:

states (list[numpy.ndarray]) – A set of vectors consisting of quantum states to determine the antidistinguishability of.

Returns:

True if the vectors are antidistinguishable; False otherwise.

Return type:

bool