Add is_entangled method to Statevector class (#14188)

[default741]

Summary

This PR addresses issue #14188 by introducing a new method is_entangled to the Statevector class. This method efficiently determines if a given quantum state is entangled based on the Von Neumann entropy criterion, evaluating individual qubit subsystems.

Details and comments

The added method simplifies the entanglement analysis process for users by providing a clear, built-in functionality directly within the Statevector class.

Implementation:

    def is_entangled(self, epsilon: float = 1e-10) -> bool:
        """Returns True is a Statevector is Entangled else False."""
        from qiskit.quantum_info import DensityMatrix, partial_trace, entropy

        density_matrix = DensityMatrix(self)

        for qubit in range(self.num_qubits):
            trace_out = [i for i in range(self.num_qubits) if i != qubit]
            subsystem_entropy = entropy(partial_trace(density_matrix, trace_out))

            if subsystem_entropy > epsilon:
                return True
        return False

Tests:
Added unit tests covering both entangled (Bell, GHZ) and separable (product) states.

def test_is_entangled(self):
        """Test is_entangled method."""
        qc_2 = QuantumCircuit(2)
        qc_3 = QuantumCircuit(3)
        qc_2.h(0)
        qc_2.cx(0, 1)
        qc_3.h(0)
        qc_3.cx(0, 1)
        qc_3.cx(0, 2)
        ne_state_2 = Statevector.from_label("++")
        ne_state_3 = Statevector.from_label("+++")
        self.assertFalse(ne_state_2.is_entangled())
        self.assertFalse(ne_state_3.is_entangled())
        e_state_2 = Statevector.from_instruction(qc_2)
        e_state_3 = Statevector.from_instruction(qc_3)
        self.assertTrue(e_state_2.is_entangled())
        self.assertTrue(e_state_3.is_entangled())

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[ShellyGarion]

Thank you for your contribution to qiskit, this seems as a potentially interesting and useful method to add. Do you have a reference for it?

Could you please better explain what you mean by "entangled" vs. "separable".
For example, take this state which is a tensor product of two entangled states:

        qc_4 = QuantumCircuit(4)
        qc_4.h(0)
        qc_4.cx(0, 1)
        qc_4.h(2)
        qc_4.cx(2, 3)
        state_4 = Statevector(qc_4)
        print(state_4.is_entangled()) # outputs True

What is the relation between the is_entangled method and other functions like
https://docs.quantum.ibm.com/api/qiskit/quantum_info#schmidt_decomposition and
https://docs.quantum.ibm.com/api/qiskit/quantum_info#entanglement_of_formation

I also have a few minor comments. In addition, could you add some release notes?

[ShellyGarion]

it is better to use the existing atol/rtol parameters than epsilon (see the is_valid function above)

[ShellyGarion]

it is better to use the existing atol/rtol parameters than epsilon (see the is_valid function above)

[ShellyGarion]

these imports yield lint errors due to cyclic imports.
can you try importing each one from the relevant file to avoid the lint errors?

[ShellyGarion]

can subsystem_entropy be negative?

[ShellyGarion]

you can use:

        e_state_2 = Statevector(qc_2)
        e_state_3 = Statevector(qc_3)

directly.