Merge pull request #1456 from qiboteam/negativity

Add `negativity` to `quantum_info.entanglement`

Author: renatomello

doc/source/api-reference/qibo.rst

@@ -1704,6 +1704,12 @@ Entanglement of formation
 .. autofunction:: qibo.quantum_info.entanglement_of_formation
 
 
+Negativity
+""""""""""
+
+.. autofunction:: qibo.quantum_info.negativity
+
+
 Entanglement fidelity
 """""""""""""""""""""
 

src/qibo/quantum_info/entanglement.py

@@ -4,7 +4,11 @@
 
 from qibo.backends import _check_backend
 from qibo.config import PRECISION_TOL, raise_error
-from qibo.quantum_info.linalg_operations import partial_trace
+from qibo.quantum_info.linalg_operations import (
+    matrix_power,
+    partial_trace,
+    partial_transpose,
+)
 from qibo.quantum_info.metrics import fidelity, purity
 
 
@@ -116,6 +120,39 @@ def entanglement_of_formation(
     return ent_of_form
 
 
+def negativity(state, bipartition, backend=None):
+    """Calculates the negativity of a bipartite quantum state.
+
+    Given a bipartite state :math:`\\rho \\in \\mathcal{H}_{A} \\otimes \\mathcal{H}_{B}`,
+    the negativity :math:`\\operatorname{Neg}(\\rho)` is given by
+
+    .. math::
+        \\operatorname{Neg}(\\rho) = \\frac{1}{2} \\,
+            \\left( \\norm{\\rho_{B}}_{1} - 1 \\right) \\, ,
+
+    where :math:`\\rho_{B}` is the reduced density matrix after tracing out qubits in
+    partition :math:`A`, and :math:`\\norm{\\cdot}_{1}` is the Schatten :math:`1`-norm
+    (also known as nuclear norm or trace norm).
+
+    Args:
+        state (ndarray): statevector or density matrix.
+        bipartition (list or tuple or ndarray): qubits in the subsystem to be traced out.
+        backend (:class:`qibo.backends.abstract.Backend`, optional): backend to be used
+            in the execution. If ``None``, it uses :class:`qibo.backends.GlobalBackend`.
+            Defaults to ``None``.
+
+    Returns:
+        float: Negativity :math:`\\operatorname{Neg}(\\rho)` of state :math:`\\rho`.
+    """
+    backend = _check_backend(backend)
+
+    reduced = partial_transpose(state, bipartition, backend)
+    reduced = backend.np.conj(reduced.T) @ reduced
+    norm = backend.np.trace(matrix_power(reduced, 1 / 2, backend))
+
+    return float(backend.np.real((norm - 1) / 2))
+
+
 def entanglement_fidelity(
     channel, nqubits: int, state=None, check_hermitian: bool = False, backend=None
 ):

tests/test_quantum_info_entanglement.py

@@ -9,6 +9,7 @@
     entanglement_of_formation,
     entangling_capability,
     meyer_wallach_entanglement,
+    negativity,
 )
 from qibo.quantum_info.random_ensembles import random_density_matrix, random_statevector
 
@@ -66,6 +67,27 @@ def test_concurrence_and_formation(backend, bipartition, base, check_purity):
     backend.assert_allclose(ent_form, 0.0, atol=PRECISION_TOL)
 
 
+@pytest.mark.parametrize("p", [1 / 5, 1 / 3 + 0.01, 1.0])
+def test_negativity(backend, p):
+    # werner state
+    zero, one = np.array([1, 0]), np.array([0, 1])
+    psi = (np.kron(zero, one) - np.kron(one, zero)) / np.sqrt(2)
+    psi = np.outer(psi, psi.T)
+    psi = backend.cast(psi)
+    state = p * psi + (1 - p) * backend.identity_density_matrix(2, normalize=True)
+
+    neg = negativity(state, [0], backend=backend)
+
+    if p == 1 / 5:
+        target = 0.0
+    elif p == 1.0:
+        target = 1 / 2
+    else:
+        target = 3 / 400
+
+    backend.assert_allclose(neg, target, atol=1e-10)
+
+
 @pytest.mark.parametrize("check_hermitian", [False, True])
 @pytest.mark.parametrize("nqubits", [4, 6])
 @pytest.mark.parametrize("channel", [gates.DepolarizingChannel])