toqito.state_props.is_pure

Checks if a quantum state is a pure state.

Module Contents

toqito.state_props.is_pure.is_pure(state)

Determine if a given state is pure or list of states are pure [@WikiPureSt].

A state is said to be pure if it is a density matrix with rank equal to 1. Equivalently, the state (rho) is pure if there exists a unit vector (u) such that:

[

rho = u u^*.

]

Examples

Consider the following Bell state:

[

u = frac{1}{sqrt{2}} left( |00 rangle + |11 rangle right) in mathcal{X}.

]

The corresponding density matrix of (u) may be calculated by:

[

rho = u u^* = frac{1}{2} begin{pmatrix}

1 & 0 & 0 & 1 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 1

end{pmatrix} in text{D}(mathcal{X}).

]

Calculating the rank of (rho) yields that the (rho) is a pure state. This can be confirmed in |toqito⟩ as follows:

`python exec="1" source="above" from toqito.states import bell from toqito.state_props import is_pure u = bell(0) rho = u @ u.conj().T print(is_pure(rho)) `

It is also possible to determine whether a set of density matrices are pure. For instance, we can see that the density matrices corresponding to the four Bell states yield a result of True indicating that all states provided to the function are pure.

`python exec="1" source="above" from toqito.states import bell from toqito.state_props import is_pure u0, u1, u2, u3 = bell(0), bell(1), bell(2), bell(3) rho0 = u0 @ u0.conj().T rho1 = u1 @ u1.conj().T rho2 = u2 @ u2.conj().T rho3 = u3 @ u3.conj().T print(is_pure([rho0, rho1, rho2, rho3])) `

Parameters:

• state (list[numpy.ndarray] | numpy.ndarray) – The density matrix representing the quantum state or a list of density matrices representing quantum

• states.

Returns:

True if state is pure and False otherwise.

Return type:

bool