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This code is part of Qiskit.","#","# (C) Copyright IBM 2017, 2020.","#","# This code is licensed under the Apache License, Version 2.0. You may","# obtain a copy of this license in the LICENSE.txt file in the root directory","# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.","#","# Any modifications or derivative works of this code must retain this","# copyright notice, and modified files need to carry a notice indicating","# that they have been altered from the originals.","","\"\"\"","Quantum information utility functions for states.","\"\"\"","","from __future__ import annotations","","import numpy as np","","from qiskit.exceptions import QiskitError","from qiskit.quantum_info.states.statevector import Statevector","from qiskit.quantum_info.states.densitymatrix import DensityMatrix","from qiskit.quantum_info.operators.channel import SuperOp","from qiskit.quantum_info.operators.predicates import ATOL_DEFAULT","","","def partial_trace(state: Statevector | DensityMatrix, qargs: list) -> DensityMatrix:"," \"\"\"Return reduced density matrix by tracing out part of quantum state.",""," If all subsystems are traced over this returns the"," :meth:`~qiskit.quantum_info.DensityMatrix.trace` of the"," input state.",""," Args:"," state (Statevector or DensityMatrix): the input state."," qargs (list): The subsystems to trace over.",""," Returns:"," DensityMatrix: The reduced density matrix.",""," Raises:"," QiskitError: if input state is invalid."," \"\"\""," state = _format_state(state, validate=False)",""," # Compute traced shape"," traced_shape = state._op_shape.remove(qargs=qargs)"," # Convert vector shape to matrix shape"," traced_shape._dims_r = traced_shape._dims_l"," traced_shape._num_qargs_r = traced_shape._num_qargs_l",""," # If we are tracing over all subsystems we return the trace"," if traced_shape.size == 0:"," return state.trace()",""," # Statevector case"," if isinstance(state, Statevector):"," trace_systems = len(state._op_shape.dims_l()) - 1 - np.array(qargs)"," arr = state._data.reshape(state._op_shape.tensor_shape)"," rho = np.tensordot(arr, arr.conj(), axes=(trace_systems, trace_systems))"," rho = np.reshape(rho, traced_shape.shape)"," return DensityMatrix(rho, dims=traced_shape._dims_l)",""," # Density matrix case"," # Empty partial trace case."," if not qargs:"," return state.copy()"," # Trace first subsystem to avoid coping whole density matrix"," dims = state.dims(qargs)"," tr_op = SuperOp(np.eye(dims[0]).reshape(1, dims[0] ** 2), input_dims=[dims[0]], output_dims=[1])"," ret = state.evolve(tr_op, [qargs[0]])"," # Trace over remaining subsystems"," for qarg, dim in zip(qargs[1:], dims[1:]):"," tr_op = SuperOp(np.eye(dim).reshape(1, dim**2), input_dims=[dim], output_dims=[1])"," ret = ret.evolve(tr_op, [qarg])"," # Remove traced over subsystems which are listed as dimension 1"," ret._op_shape = traced_shape"," return ret","","","def shannon_entropy(pvec: list | np.ndarray, base: int = 2) -> float:"," r\"\"\"Compute the Shannon entropy of a probability vector.",""," The shannon entropy of a probability vector"," :math:`\\vec{p} = [p_0, ..., p_{n-1}]` is defined as",""," .. math::",""," H(\\vec{p}) = \\sum_{i=0}^{n-1} p_i \\log_b(p_i)",""," where :math:`b` is the log base and (default 2), and"," :math:`0 \\log_b(0) \\equiv 0`.",""," Args:"," pvec (array_like): a probability vector."," base (int): the base of the logarithm [Default: 2].",""," Returns:"," float: The Shannon entropy H(pvec)."," \"\"\""," if base == 2:",""," def logfn(x):"," return -x * np.log2(x)",""," elif base == np.e:",""," def logfn(x):"," return -x * np.log(x)",""," else:",""," def logfn(x):"," return -x * np.log(x) / np.log(base)",""," h_val = 0.0"," for x in pvec:"," if 0 < x < 1:"," h_val += logfn(x)"," return h_val","","","def schmidt_decomposition(state, qargs):"," r\"\"\"Return the Schmidt Decomposition of a pure quantum state.",""," For an arbitrary bipartite state:",""," .. math::"," |\\psi\\rangle_{AB} = \\sum_{i,j} c_{ij}"," |x_i\\rangle_A \\otimes |y_j\\rangle_B,",""," its Schmidt Decomposition is given by the single-index sum over k:",""," .. math::"," |\\psi\\rangle_{AB} = \\sum_{k} \\lambda_{k}"," |u_k\\rangle_A \\otimes |v_k\\rangle_B",""," where :math:`|u_k\\rangle_A` and :math:`|v_k\\rangle_B` are an"," orthonormal set of vectors in their respective spaces :math:`A` and :math:`B`,"," and the Schmidt coefficients :math:`\\lambda_k` are positive real values.",""," Args:"," state (Statevector or DensityMatrix): the input state."," qargs (list): the list of Input state positions corresponding to subsystem :math:`B`.",""," Returns:"," list: list of tuples ``(s, u, v)``, where ``s`` (float) are the Schmidt coefficients"," :math:`\\lambda_k`, and ``u`` (Statevector), ``v`` (Statevector) are the Schmidt vectors"," :math:`|u_k\\rangle_A`, :math:`|u_k\\rangle_B`, respectively.",""," Raises:"," QiskitError: if Input qargs is not a list of positions of the Input state."," QiskitError: if Input qargs is not a proper subset of Input state.",""," .. note::"," In Qiskit, qubits are ordered using little-endian notation, with the least significant"," qubits having smaller indices. For example, a four-qubit system is represented as"," :math:`|q_3q_2q_1q_0\\rangle`. Using this convention, setting ``qargs=[0]`` will partition the"," state as :math:`|q_3q_2q_1\\rangle_A\\otimes|q_0\\rangle_B`. Furthermore, qubits will be organized"," in this notation regardless of the order they are passed. For instance, passing either"," ``qargs=[1,2]`` or ``qargs=[2,1]`` will result in partitioning the state as"," :math:`|q_3q_0\\rangle_A\\otimes|q_2q_1\\rangle_B`."," \"\"\""," state = _format_state(state, validate=False)",""," # convert to statevector if state is density matrix. Errors if state is mixed."," if isinstance(state, DensityMatrix):"," state = state.to_statevector()",""," # reshape statevector into state tensor"," dims = state.dims()"," state_tens = state._data.reshape(dims[::-1])"," ndim = state_tens.ndim"," qudits = list(range(ndim))",""," # check if qargs are valid"," if not isinstance(qargs, (list, np.ndarray)):"," raise QiskitError(\"Input qargs is not a list of positions of the Input state\")"," qargs = set(qargs)"," if qargs == set(qudits) or not qargs.issubset(qudits):"," raise QiskitError(\"Input qargs is not a proper subset of Input state\")",""," # define subsystem A and B qargs and dims"," qargs_b = list(qargs)"," qargs_a = [i for i in qudits if i not in qargs_b]"," dims_b = state.dims(qargs_b)"," dims_a = state.dims(qargs_a)"," ndim_b = np.prod(dims_b)"," ndim_a = np.prod(dims_a)",""," # permute state for desired qargs order"," qargs_axes = [qudits[::-1].index(i) for i in qargs_b + qargs_a][::-1]"," state_tens = state_tens.transpose(qargs_axes)",""," # convert state tensor to matrix of prob amplitudes and perform svd."," state_mat = state_tens.reshape([ndim_a, ndim_b])"," u_mat, s_arr, vh_mat = np.linalg.svd(state_mat, full_matrices=False)",""," schmidt_components = ["," (s, Statevector(u, dims=dims_a), Statevector(v, dims=dims_b))"," for s, u, v in zip(s_arr, u_mat.T, vh_mat)"," if s > ATOL_DEFAULT"," ]",""," return schmidt_components","","","def _format_state(state, validate=True):"," \"\"\"Format input state into class object\"\"\""," if isinstance(state, list):"," state = np.array(state, dtype=complex)"," if isinstance(state, np.ndarray):"," ndim = state.ndim"," if ndim == 1:"," state = Statevector(state)"," elif ndim == 2:"," dim1, dim2 = state.shape"," if dim2 == 1:"," state = Statevector(state)"," elif dim1 == dim2:"," state = DensityMatrix(state)"," if not isinstance(state, (Statevector, DensityMatrix)):"," raise QiskitError(\"Input is not a quantum state\")"," if validate and not 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