Source code for qutip.entropy

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__all__ = ['entropy_vn', 'entropy_linear', 'entropy_mutual', 'negativity',
           'concurrence', 'entropy_conditional', 'entangling_power']

from numpy import e, real, sort, sqrt
from numpy.lib.scimath import log, log2
from qutip.qobj import ptrace
from qutip.states import ket2dm
from qutip.tensor import tensor
from qutip.operators import sigmay
from qutip.sparse import sp_eigs
from qutip.partial_transpose import partial_transpose


[docs]def entropy_vn(rho, base=e, sparse=False): """ Von-Neumann entropy of density matrix Parameters ---------- rho : qobj Density matrix. base : {e,2} Base of logarithm. sparse : {False,True} Use sparse eigensolver. Returns ------- entropy : float Von-Neumann entropy of `rho`. Examples -------- >>> rho=0.5*fock_dm(2,0)+0.5*fock_dm(2,1) >>> entropy_vn(rho,2) 1.0 """ if rho.type == 'ket' or rho.type == 'bra': rho = ket2dm(rho) vals = sp_eigs(rho.data, rho.isherm, vecs=False, sparse=sparse) nzvals = vals[vals != 0] if base == 2: logvals = log2(nzvals) elif base == e: logvals = log(nzvals) else: raise ValueError("Base must be 2 or e.") return float(real(-sum(nzvals * logvals)))
[docs]def entropy_linear(rho): """ Linear entropy of a density matrix. Parameters ---------- rho : qobj sensity matrix or ket/bra vector. Returns ------- entropy : float Linear entropy of rho. Examples -------- >>> rho=0.5*fock_dm(2,0)+0.5*fock_dm(2,1) >>> entropy_linear(rho) 0.5 """ if rho.type == 'ket' or rho.type == 'bra': rho = ket2dm(rho) return float(real(1.0 - (rho ** 2).tr()))
[docs]def concurrence(rho): """ Calculate the concurrence entanglement measure for a two-qubit state. Parameters ---------- state : qobj Ket, bra, or density matrix for a two-qubit state. Returns ------- concur : float Concurrence References ---------- .. [1] http://en.wikipedia.org/wiki/Concurrence_(quantum_computing) """ if rho.isket and rho.dims != [[2, 2], [1, 1]]: raise Exception("Ket must be tensor product of two qubits.") elif rho.isbra and rho.dims != [[1, 1], [2, 2]]: raise Exception("Bra must be tensor product of two qubits.") elif rho.isoper and rho.dims != [[2, 2], [2, 2]]: raise Exception("Density matrix must be tensor product of two qubits.") if rho.isket or rho.isbra: rho = ket2dm(rho) sysy = tensor(sigmay(), sigmay()) rho_tilde = (rho * sysy) * (rho.conj() * sysy) evals = rho_tilde.eigenenergies() # abs to avoid problems with sqrt for very small negative numbers evals = abs(sort(real(evals))) lsum = sqrt(evals[3]) - sqrt(evals[2]) - sqrt(evals[1]) - sqrt(evals[0]) return max(0, lsum)
def negativity(rho, subsys, method='tracenorm', logarithmic=False): """ Compute the negativity for a multipartite quantum system described by the density matrix rho. The subsys argument is an index that indicates which system to compute the negativity for. .. note:: Experimental. """ mask = [idx == subsys for idx, n in enumerate(rho.dims[0])] rho_pt = partial_transpose(rho, mask) if method == 'tracenorm': N = ((rho_pt.dag() * rho_pt).sqrtm().tr().real - 1)/2.0 elif method == 'eigenvalues': l = rho_pt.eigenenergies() N = ((abs(l)-l)/2).sum() else: raise ValueError("Unknown method %s" % method) if logarithmic: return log2(2 * N + 1) else: return N
[docs]def entropy_mutual(rho, selA, selB, base=e, sparse=False): """ Calculates the mutual information S(A:B) between selection components of a system density matrix. Parameters ---------- rho : qobj Density matrix for composite quantum systems selA : int/list `int` or `list` of first selected density matrix components. selB : int/list `int` or `list` of second selected density matrix components. base : {e,2} Base of logarithm. sparse : {False,True} Use sparse eigensolver. Returns ------- ent_mut : float Mutual information between selected components. """ if isinstance(selA, int): selA = [selA] if isinstance(selB, int): selB = [selB] if rho.type != 'oper': raise TypeError("Input must be a density matrix.") if (len(selA) + len(selB)) != len(rho.dims[0]): raise TypeError("Number of selected components must match " + "total number.") rhoA = ptrace(rho, selA) rhoB = ptrace(rho, selB) out = (entropy_vn(rhoA, base, sparse=sparse) + entropy_vn(rhoB, base, sparse=sparse) - entropy_vn(rho, base, sparse=sparse)) return out
def _entropy_relative(rho, sigma, base=e, sparse=False): """ ****NEEDS TO BE WORKED ON**** Calculates the relative entropy S(rho||sigma) between two density matrices. Parameters ---------- rho : qobj First density matrix. sigma : qobj Second density matrix. base : {e,2} Base of logarithm. Returns ------- rel_ent : float Value of relative entropy. """ if rho.type != 'oper' or sigma.type != 'oper': raise TypeError("Inputs must be density matrices..") # sigma terms svals = sp_eigs(sigma.data, sigma.isherm, vecs=False, sparse=sparse) snzvals = svals[svals != 0] if base == 2: slogvals = log2(snzvals) elif base == e: slogvals = log(snzvals) else: raise ValueError("Base must be 2 or e.") # rho terms rvals = sp_eigs(rho.data, rho.isherm, vecs=False, sparse=sparse) rnzvals = rvals[rvals != 0] # calculate tr(rho*log sigma) rel_trace = float(real(sum(rnzvals * slogvals))) return -entropy_vn(rho, base, sparse) - rel_trace
[docs]def entropy_conditional(rho, selB, base=e, sparse=False): """ Calculates the conditional entropy :math:`S(A|B)=S(A,B)-S(B)` of a selected density matrix component. Parameters ---------- rho : qobj Density matrix of composite object selB : int/list Selected components for density matrix B base : {e,2} Base of logarithm. sparse : {False,True} Use sparse eigensolver. Returns ------- ent_cond : float Value of conditional entropy """ if rho.type != 'oper': raise TypeError("Input must be density matrix.") if isinstance(selB, int): selB = [selB] B = ptrace(rho, selB) out = (entropy_vn(rho, base, sparse=sparse) - entropy_vn(B, base, sparse=sparse)) return out
def participation_ratio(rho): """ Returns the effective number of states for a density matrix. The participation is unity for pure states, and maximally N, where N is the Hilbert space dimensionality, for completely mixed states. Parameters ---------- rho : qobj Density matrix Returns ------- pr : float Effective number of states in the density matrix """ if rho.type == 'ket' or rho.type == 'bra': return 1.0 else: return 1.0 / (rho ** 2).tr() def entangling_power(U): """ Calculate the entangling power of a two-qubit gate U, which is zero of nonentangling gates and 1 and 2/9 for maximally entangling gates. Parameters ---------- U : qobj Qobj instance representing a two-qubit gate. Returns ------- ep : float The entanglement power of U (real number between 0 and 1) References: Explorations in Quantum Computing, Colin P. Williams (Springer, 2011) """ if not U.isoper: raise Exception("U must be an operator.") if U.dims != [[2, 2], [2, 2]]: raise Exception("U must be a two-qubit gate.") from qutip.qip.operations.gates import swap a = (tensor(U, U).dag() * swap(N=4, targets=[1, 3]) * tensor(U, U) * swap(N=4, targets=[1, 3])) b = (tensor(swap() * U, swap() * U).dag() * swap(N=4, targets=[1, 3]) * tensor(swap() * U, swap() * U) * swap(N=4, targets=[1, 3])) return 5.0/9 - 1.0/36 * (a.tr() + b.tr()).real