state_props.is_pure¶
Checks if a quantum state is a pure state.
Functions¶
Module Contents¶
- state_props.is_pure.is_pure(state)¶
Determine if a given state is pure or list of states are pure [1].
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:
\[\begin{split}\rho = u u^* = \frac{1}{2} \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \end{pmatrix} \in \text{D}(\mathcal{X}).\end{split}\]Calculating the rank of \(\rho\) yields that the \(\rho\) is a pure state. This can be confirmed in
toqito
as follows:>>> from toqito.states import bell >>> from toqito.state_props import is_pure >>> u = bell(0) >>> rho = u @ u.conj().T >>> is_pure(rho) True
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.>>> 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 >>> >>> is_pure([rho0, rho1, rho2, rho3]) True
References
[1] (1,2)Wikipedia. Quantum state - pure states. URL: https://en.wikipedia.org/wiki/Quantum_state#Pure_states_of_wave_functions.
- 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 andFalse
otherwise.- Return type:
bool