code: # This file is part of QuTiP: Quantum Toolbox in Python.
  #
  #    Copyright (c) 2011 and later, Paul D. Nation and Robert J. Johansson.
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  ###############################################################################
  """The Quantum Object (Qobj) class, for representing quantum states and
  operators, and related functions.
  """
  
  __all__ = ['Qobj', 'qobj_list_evaluate', 'ptrace', 'dag', 'isequal',
             'issuper', 'isoper', 'isoperket', 'isoperbra', 'isket', 'isbra',
             'isherm', 'shape', 'dims']
  
  import warnings
  import types
  
  try:
      import builtins
  except:
      import __builtin__ as builtins
  
  # import math functions from numpy.math: required for td string evaluation
  from numpy import (arccos, arccosh, arcsin, arcsinh, arctan, arctan2, arctanh,
                     ceil, copysign, cos, cosh, degrees, e, exp, expm1, fabs,
                     floor, fmod, frexp, hypot, isinf, isnan, ldexp, log, log10,
                     log1p, modf, pi, radians, sin, sinh, sqrt, tan, tanh, trunc)
  
  import numpy as np
  import scipy.sparse as sp
  import scipy.linalg as la
  import qutip.settings as settings
  from qutip import __version__
  from qutip.fastsparse import fast_csr_matrix, fast_identity
  from qutip.cy.ptrace import _ptrace
  from qutip.permute import _permute
  from qutip.sparse import (sp_eigs, sp_expm, sp_fro_norm, sp_max_norm,
                            sp_one_norm, sp_L2_norm)
  from qutip.dimensions import type_from_dims, enumerate_flat, collapse_dims_super
  from qutip.cy.spmath import (zcsr_transpose, zcsr_adjoint, zcsr_isherm,
                              zcsr_trace, zcsr_proj, zcsr_inner)
  from qutip.cy.spmatfuncs import zcsr_mat_elem
  from qutip.cy.sparse_utils import cy_tidyup
  import sys
  if sys.version_info.major >= 3:
      from itertools import zip_longest
  elif sys.version_info.major < 3:
      from itertools import izip_longest
      zip_longest = izip_longest
  
  #OPENMP stuff
  from qutip.cy.openmp.utilities import use_openmp
  if settings.has_openmp:
      from qutip.cy.openmp.omp_sparse_utils import omp_tidyup
  
  
  class Qobj(object):
      """A class for representing quantum objects, such as quantum operators
      and states.
  
      The Qobj class is the QuTiP representation of quantum operators and state
      vectors. This class also implements math operations +,-,* between Qobj
      instances (and / by a C-number), as well as a collection of common
      operator/state operations.  The Qobj constructor optionally takes a
      dimension ``list`` and/or shape ``list`` as arguments.
  
      Parameters
      ----------
      inpt : array_like
          Data for vector/matrix representation of the quantum object.
      dims : list
          Dimensions of object used for tensor products.
      shape : list
          Shape of underlying data structure (matrix shape).
      copy : bool
          Flag specifying whether Qobj should get a copy of the
          input data, or use the original.
      fast : bool
          Flag for fast qobj creation when running ode solvers.
          This parameter is used internally only.
  
  
      Attributes
      ----------
      data : array_like
          Sparse matrix characterizing the quantum object.
      dims : list
          List of dimensions keeping track of the tensor structure.
      shape : list
          Shape of the underlying `data` array.
      type : str
          Type of quantum object: 'bra', 'ket', 'oper', 'operator-ket',
          'operator-bra', or 'super'.
      superrep : str
          Representation used if `type` is 'super'. One of 'super'
          (Liouville form) or 'choi' (Choi matrix with tr = dimension).
      isherm : bool
          Indicates if quantum object represents Hermitian operator.
      isunitary : bool
          Indictaes if quantum object represents unitary operator.
      iscp : bool
          Indicates if the quantum object represents a map, and if that map is
          completely positive (CP).
      ishp : bool
          Indicates if the quantum object represents a map, and if that map is
          hermicity preserving (HP).
      istp : bool
          Indicates if the quantum object represents a map, and if that map is
          trace preserving (TP).
      iscptp : bool
          Indicates if the quantum object represents a map that is completely
          positive and trace preserving (CPTP).
      isket : bool
          Indicates if the quantum object represents a ket.
      isbra : bool
          Indicates if the quantum object represents a bra.
      isoper : bool
          Indicates if the quantum object represents an operator.
      issuper : bool
          Indicates if the quantum object represents a superoperator.
      isoperket : bool
          Indicates if the quantum object represents an operator in column vector
          form.
      isoperbra : bool
          Indicates if the quantum object represents an operator in row vector
          form.
  
      Methods
      -------
      copy()
          Create copy of Qobj
      conj()
          Conjugate of quantum object.
      cosm()
          Cosine of quantum object.
      dag()
          Adjoint (dagger) of quantum object.
      dnorm()
          Diamond norm of quantum operator.
      dual_chan()
          Dual channel of quantum object representing a CP map.
      eigenenergies(sparse=False, sort='low', eigvals=0, tol=0, maxiter=100000)
          Returns eigenenergies (eigenvalues) of a quantum object.
      eigenstates(sparse=False, sort='low', eigvals=0, tol=0, maxiter=100000)
          Returns eigenenergies and eigenstates of quantum object.
      expm()
          Matrix exponential of quantum object.
      full(order='C')
          Returns dense array of quantum object `data` attribute.
      groundstate(sparse=False, tol=0, maxiter=100000)
          Returns eigenvalue and eigenket for the groundstate of a quantum
          object.
      matrix_element(bra, ket)
          Returns the matrix element of operator between `bra` and `ket` vectors.
      norm(norm='tr', sparse=False, tol=0, maxiter=100000)
          Returns norm of a ket or an operator.
      permute(order)
          Returns composite qobj with indices reordered.
      proj()
          Computes the projector for a ket or bra vector.
      ptrace(sel)
          Returns quantum object for selected dimensions after performing
          partial trace.
      sinm()
          Sine of quantum object.
      sqrtm()
          Matrix square root of quantum object.
      tidyup(atol=1e-12)
          Removes small elements from quantum object.
      tr()
          Trace of quantum object.
      trans()
          Transpose of quantum object.
      transform(inpt, inverse=False)
          Performs a basis transformation defined by `inpt` matrix.
      trunc_neg(method='clip')
          Removes negative eigenvalues and returns a new Qobj that is
          a valid density operator.
      unit(norm='tr', sparse=False, tol=0, maxiter=100000)
          Returns normalized quantum object.
  
      """
      __array_priority__ = 100  # sets Qobj priority above numpy arrays
  
      def __init__(self, inpt=None, dims=[[], []], shape=[],
                   type=None, isherm=None, copy=True,
                   fast=False, superrep=None, isunitary=None):
          """
          Qobj constructor.
          """
          self._isherm = isherm
          self._type = type
          self.superrep = superrep
          self._isunitary = isunitary
  
          if fast == 'mc':
              # fast Qobj construction for use in mcsolve with ket output
              self._data = inpt
              self.dims = dims
              self._isherm = False
              return
  
          if fast == 'mc-dm':
              # fast Qobj construction for use in mcsolve with dm output
              self._data = inpt
              self.dims = dims
              self._isherm = True
              return
  
          if isinstance(inpt, Qobj):
              # if input is already Qobj then return identical copy
  
              self._data = fast_csr_matrix((inpt.data.data, inpt.data.indices,
                                            inpt.data.indptr),
                                            shape=inpt.shape, copy=copy)
  
              if not np.any(dims):
                  # Dimensions of quantum object used for keeping track of tensor
                  # components
                  self.dims = inpt.dims
              else:
                  self.dims = dims
  
              self.superrep = inpt.superrep
              self._isunitary = inpt._isunitary
  
          elif inpt is None:
              # initialize an empty Qobj with correct dimensions and shape
  
              if any(dims):
                  N, M = np.prod(dims[0]), np.prod(dims[1])
                  self.dims = dims
  
              elif shape:
                  N, M = shape
                  self.dims = [[N], [M]]
  
              else:
                  N, M = 1, 1
                  self.dims = [[N], [M]]
  
              self._data = fast_csr_matrix(shape=(N, M))
  
          elif isinstance(inpt, list) or isinstance(inpt, tuple):
              # case where input is a list
              data = np.array(inpt)
              if len(data.shape) == 1:
                  # if list has only one dimension (i.e [5,4])
                  data = data.transpose()
  
              _tmp = sp.csr_matrix(data, dtype=complex)
              self._data = fast_csr_matrix((_tmp.data, _tmp.indices, _tmp.indptr),
                                           shape=_tmp.shape)
              if not np.any(dims):
                  self.dims = [[int(data.shape[0])], [int(data.shape[1])]]
              else:
                  self.dims = dims
  
          elif isinstance(inpt, np.ndarray) or sp.issparse(inpt):
              # case where input is array or sparse
              if inpt.ndim == 1:
                  inpt = inpt[:, np.newaxis]
  
              do_copy = copy
              if not isinstance(inpt, fast_csr_matrix):
                  _tmp = sp.csr_matrix(inpt, dtype=complex, copy=do_copy)
                  _tmp.sort_indices() #Make sure indices are sorted.
                  do_copy = 0
              else:
                  _tmp = inpt
              self._data = fast_csr_matrix((_tmp.data, _tmp.indices, _tmp.indptr),
                                          shape=_tmp.shape, copy=do_copy)
  
              if not np.any(dims):
                  self.dims = [[int(inpt.shape[0])], [int(inpt.shape[1])]]
              else:
                  self.dims = dims
  
          elif isinstance(inpt, (int, float, complex,
                                 np.integer, np.floating, np.complexfloating)):
              # if input is int, float, or complex then convert to array
              _tmp = sp.csr_matrix([[inpt]], dtype=complex)
              self._data = fast_csr_matrix((_tmp.data, _tmp.indices, _tmp.indptr),
                                          shape=_tmp.shape)
              if not np.any(dims):
                  self.dims = [[1], [1]]
              else:
                  self.dims = dims
  
          else:
              warnings.warn("Initializing Qobj from unsupported type: %s" %
                            builtins.type(inpt))
              inpt = np.array([[0]])
              _tmp = sp.csr_matrix(inpt, dtype=complex, copy=copy)
              self._data = fast_csr_matrix((_tmp.data, _tmp.indices, _tmp.indptr),
                                          shape = _tmp.shape)
              self.dims = [[int(inpt.shape[0])], [int(inpt.shape[1])]]
  
          if type == 'super':
              # Type is not super, i.e. dims not explicitly passed, but oper shape
              if dims== [[], []] and self.shape[0] == self.shape[1]:
                  sub_shape = np.sqrt(self.shape[0])
                  # check if root of shape is int
                  if (sub_shape % 1) != 0:
                      raise Exception('Invalid shape for a super operator.')
                  else:
                      sub_shape = int(sub_shape)
                      self.dims = [[[sub_shape], [sub_shape]]]*2
  
  
          if superrep:
              self.superrep = superrep
          else:
              if self.type == 'super' and self.superrep is None:
                  self.superrep = 'super'
  
          # clear type cache
          self._type = None
  
      def copy(self):
          """Create identical copy"""
          return Qobj(inpt=self)
  
      def get_data(self):
          return self._data
      #Here we perfrom a check of the csr matrix type during setting of Q.data
      def set_data(self, data):
          if not isinstance(data, fast_csr_matrix):
              raise TypeError('Qobj data must be in fast_csr format.')
          else:
              self._data = data
      data = property(get_data, set_data)
  
      def __add__(self, other):
          """
          ADDITION with Qobj on LEFT [ ex. Qobj+4 ]
          """
          self._isunitary = None
  
          if isinstance(other, eseries):
              return other.__radd__(self)
  
          if not isinstance(other, Qobj):
              if isinstance(other, (int, float, complex, np.integer, np.floating,
                            np.complexfloating, np.ndarray, list, tuple)) \
                            or sp.issparse(other):
                  other = Qobj(other)
              else:
                  return NotImplemented
  
          if np.prod(other.shape) == 1 and np.prod(self.shape) != 1:
              # case for scalar quantum object
              dat = other.data[0, 0]
              if dat == 0:
                  return self
  
              out = Qobj()
  
              if self.type in ['oper', 'super']:
                  out.data = self.data + dat * fast_identity(
                      self.shape[0])
              else:
                  out.data = self.data
                  out.data.data = out.data.data + dat
  
              out.dims = self.dims
  
              if settings.auto_tidyup: out.tidyup()
  
              if isinstance(dat, (int, float)):
                  out._isherm = self._isherm
              else:
                  # We use _isherm here to prevent recalculating on self and
                  # other, relying on that bool(None) == False.
                  out._isherm = (True if self._isherm and other._isherm
                                      else out.isherm)
  
              out.superrep = self.superrep
  
              return out
  
          elif np.prod(self.shape) == 1 and np.prod(other.shape) != 1:
              # case for scalar quantum object
              dat = self.data[0, 0]
              if dat == 0:
                  return other
  
              out = Qobj()
              if other.type in ['oper', 'super']:
                  out.data = dat * fast_identity(other.shape[0]) + other.data
              else:
                  out.data = other.data
                  out.data.data = out.data.data + dat
              out.dims = other.dims
  
              if settings.auto_tidyup: out.tidyup()
  
              if isinstance(dat, complex):
                  out._isherm = out.isherm
              else:
                  out._isherm = self._isherm
  
              out.superrep = self.superrep
  
              return out
  
          elif self.dims != other.dims:
              raise TypeError('Incompatible quantum object dimensions')
  
          elif self.shape != other.shape:
              raise TypeError('Matrix shapes do not match')
  
          else:  # case for matching quantum objects
              out = Qobj()
              out.data = self.data + other.data
              out.dims = self.dims
              if settings.auto_tidyup: out.tidyup()
  
              if self.type in ['ket', 'bra', 'operator-ket', 'operator-bra']:
                  out._isherm = False
              elif self._isherm is None or other._isherm is None:
                  out._isherm = out.isherm
              elif not self._isherm and not other._isherm:
                  out._isherm = out.isherm
              else:
                  out._isherm = self._isherm and other._isherm
  
              if self.superrep and other.superrep:
                  if self.superrep != other.superrep:
                      msg = ("Adding superoperators with different " +
                             "representations")
                      warnings.warn(msg)
  
                  out.superrep = self.superrep
  
              return out
  
      def __radd__(self, other):
          """
          ADDITION with Qobj on RIGHT [ ex. 4+Qobj ]
          """
          return self + other
  
      def __sub__(self, other):
          """
          SUBTRACTION with Qobj on LEFT [ ex. Qobj-4 ]
          """
          return self + (-other)
  
      def __rsub__(self, other):
          """
          SUBTRACTION with Qobj on RIGHT [ ex. 4-Qobj ]
          """
          return (-self) + other
  
      def __mul__(self, other):
          """
          MULTIPLICATION with Qobj on LEFT [ ex. Qobj*4 ]
          """
          self._isunitary = None
  
          if isinstance(other, Qobj):
              if self.dims[1] == other.dims[0]:
                  out = Qobj()
                  out.data = self.data * other.data
                  dims = [self.dims[0], other.dims[1]]
                  out.dims = dims
                  if settings.auto_tidyup: out.tidyup()
                  if (settings.auto_tidyup_dims
                          and not isinstance(dims[0][0], list)
                          and not isinstance(dims[1][0], list)):
                      # If neither left or right is a superoperator,
                      # we should implicitly partial trace over
                      # matching dimensions of 1.
                      # Using izip_longest allows for the left and right dims
                      # to have uneven length (non-square Qobjs).
                      # We use None as padding so that it doesn't match anything,
                      # and will never cause a partial trace on the other side.
                      mask = [l == r == 1 for l, r in zip_longest(dims[0], dims[1],
                                                                  fillvalue=None)]
                      # To ensure that there are still any dimensions left, we
                      # use max() to add a dimensions list of [1] if all matching dims
                      # are traced out of that side.
                      out.dims = [max([1],
                                      [dim for dim, m in zip(dims[0], mask)
                                      if not m]),
                                  max([1],
                                      [dim for dim, m in zip(dims[1], mask)
                                      if not m])]
  
                  else:
                      out.dims = dims
  
                  out._isherm = None
  
                  if self.superrep and other.superrep:
                      if self.superrep != other.superrep:
                          msg = ("Multiplying superoperators with different " +
                                 "representations")
                          warnings.warn(msg)
  
                      out.superrep = self.superrep
  
                  return out
  
              elif np.prod(self.shape) == 1:
                  out = Qobj(other)
                  out.data *= self.data[0, 0]
                  out.superrep = other.superrep
                  return out.tidyup() if settings.auto_tidyup else out
  
              elif np.prod(other.shape) == 1:
                  out = Qobj(self)
                  out.data *= other.data[0, 0]
                  out.superrep = self.superrep
                  return out.tidyup() if settings.auto_tidyup else out
  
              else:
                  raise TypeError("Incompatible Qobj shapes")
  
          elif isinstance(other, np.ndarray):
              if other.dtype=='object':
                  return np.array([self * item for item in other],
                                  dtype=object)
              else:
                  return self.data * other
  
  
          elif isinstance(other, list):
              # if other is a list, do element-wise multiplication
              return np.array([self * item for item in other],
                              dtype=object)
  
          elif isinstance(other, eseries):
              return other.__rmul__(self)
  
          elif isinstance(other, (int, float, complex,
                                  np.integer, np.floating, np.complexfloating)):
              out = Qobj()
              out.data = self.data * other
              out.dims = self.dims
              out.superrep = self.superrep
              if settings.auto_tidyup: out.tidyup()
              if isinstance(other, complex):
                  out._isherm = out.isherm
              else:
                  out._isherm = self._isherm
  
              return out
  
          else:
              return NotImplemented
  
      def __rmul__(self, other):
          """
          MULTIPLICATION with Qobj on RIGHT [ ex. 4*Qobj ]
          """
          if isinstance(other, np.ndarray):
              if other.dtype=='object':
                  return np.array([item * self for item in other],
                                              dtype=object)
              else:
                  return other * self.data
  
          elif isinstance(other, list):
              # if other is a list, do element-wise multiplication
              return np.array([item * self for item in other],
                              dtype=object)
  
          elif isinstance(other, eseries):
              return other.__mul__(self)
  
          elif isinstance(other, (int, float, complex,
                                np.integer, np.floating, np.complexfloating)):
              out = Qobj()
              out.data = other * self.data
              out.dims = self.dims
              out.superrep = self.superrep
              if settings.auto_tidyup: out.tidyup()
              if isinstance(other, complex):
                  out._isherm = out.isherm
              else:
                  out._isherm = self._isherm
  
              return out
  
          else:
              raise TypeError("Incompatible object for multiplication")
  
      def __truediv__(self, other):
          return self.__div__(other)
  
      def __div__(self, other):
          """
          DIVISION (by numbers only)
          """
          if isinstance(other, Qobj):  # if both are quantum objects
              raise TypeError("Incompatible Qobj shapes " +
                              "[division with Qobj not implemented]")
  
          if isinstance(other, (int, float, complex,
                                np.integer, np.floating, np.complexfloating)):
              out = Qobj()
              out.data = self.data / other
              out.dims = self.dims
              if settings.auto_tidyup: out.tidyup()
              if isinstance(other, complex):
                  out._isherm = out.isherm
              else:
                  out._isherm = self._isherm
  
              out.superrep = self.superrep
  
              return out
  
          else:
              raise TypeError("Incompatible object for division")
  
      def __neg__(self):
          """
          NEGATION operation.
          """
          out = Qobj()
          out.data = -self.data
          out.dims = self.dims
          out.superrep = self.superrep
          if settings.auto_tidyup: out.tidyup()
          out._isherm = self._isherm
          out._isunitary = self._isunitary
          return out
  
      def __getitem__(self, ind):
          """
          GET qobj elements.
          """
          out = self.data[ind]
          if sp.issparse(out):
              return np.asarray(out.todense())
          else:
              return out
  
      def __eq__(self, other):
          """
          EQUALITY operator.
          """
          if (isinstance(other, Qobj) and
                  self.dims == other.dims and
                  not np.any(np.abs((self.data - other.data).data) >
                             settings.atol)):
              return True
          else:
              return False
  
      def __ne__(self, other):
          """
          INEQUALITY operator.
          """
          return not (self == other)
  
      def __pow__(self, n, m=None):  # calculates powers of Qobj
          """
          POWER operation.
          """
          if self.type not in ['oper', 'super']:
              raise Exception("Raising a qobj to some power works only for " +
                              "operators and super-operators (square matrices).")
  
          if m is not None:
              raise NotImplementedError("modulo is not implemented for Qobj")
  
          try:
              data = self.data ** n
              out = Qobj(data, dims=self.dims)
              out.superrep = self.superrep
              return out.tidyup() if settings.auto_tidyup else out
  
          except:
              raise ValueError('Invalid choice of exponent.')
  
      def __abs__(self):
          return abs(self.data)
  
      def __str__(self):
          s = ""
          t = self.type
          shape = self.shape
          if self.type in ['oper', 'super']:
              s += ("Quantum object: " +
                    "dims = " + str(self.dims) +
                    ", shape = " + str(shape) +
                    ", type = " + t +
                    ", isherm = " + str(self.isherm) +
                    (
                        ", superrep = {0.superrep}".format(self)
                        if t == "super" and self.superrep != "super"
                        else ""
                    ) + "\n")
          else:
              s += ("Quantum object: " +
                    "dims = " + str(self.dims) +
                    ", shape = " + str(shape) +
                    ", type = " + t + "\n")
          s += "Qobj data =\n"
  
          if shape[0] > 10000 or shape[1] > 10000:
              # if the system is huge, don't attempt to convert to a
              # dense matrix and then to string, because it is pointless
              # and is likely going to produce memory errors. Instead print the
              # sparse data string representation
              s += str(self.data)
  
          elif all(np.imag(self.data.data) == 0):
              s += str(np.real(self.full()))
  
          else:
              s += str(self.full())
  
          return s
  
      def __repr__(self):
          # give complete information on Qobj without print statement in
          # command-line we cant realistically serialize a Qobj into a string,
          # so we simply return the informal __str__ representation instead.)
          return self.__str__()
  
      def __call__(self, other):
          """
          Acts this Qobj on another Qobj either by left-multiplication,
          or by vectorization and devectorization, as
          appropriate.
          """
          if not isinstance(other, Qobj):
              raise TypeError("Only defined for quantum objects.")
  
          if self.type == "super":
              if other.type == "ket":
                  other = qutip.states.ket2dm(other)
  
              if other.type == "oper":
                  return qutip.superoperator.vector_to_operator(
                      self * qutip.superoperator.operator_to_vector(other)
                  )
              else:
                  raise TypeError("Can only act super on oper or ket.")
  
          elif self.type == "oper":
              if other.type == "ket":
                  return self * other
              else:
                  raise TypeError("Can only act oper on ket.")
  
      def __getstate__(self):
          # defines what happens when Qobj object gets pickled
          self.__dict__.update({'qutip_version': __version__[:5]})
          return self.__dict__
  
      def __setstate__(self, state):
          # defines what happens when loading a pickled Qobj
          if 'qutip_version' in state.keys():
              del state['qutip_version']
          (self.__dict__).update(state)
  
      def _repr_latex_(self):
          """
          Generate a LaTeX representation of the Qobj instance. Can be used for
          formatted output in ipython notebook.
          """
          t = self.type
          shape = self.shape
          s = r''
          if self.type in ['oper', 'super']:
              s += ("Quantum object: " +
                    "dims = " + str(self.dims) +
                    ", shape = " + str(shape) +
                    ", type = " + t +
                    ", isherm = " + str(self.isherm) +
                    (
                        ", superrep = {0.superrep}".format(self)
                        if t == "super" and self.superrep != "super"
                        else ""
                    ))
          else:
              s += ("Quantum object: " +
                    "dims = " + str(self.dims) +
                    ", shape = " + str(shape) +
                    ", type = " + t)
  
          M, N = self.data.shape
  
          s += r'\begin{equation*}\left(\begin{array}{*{11}c}'
  
          def _format_float(value):
              if value == 0.0:
                  return "0.0"
              elif abs(value) > 1000.0 or abs(value) < 0.001:
                  return ("%.3e" % value).replace("e", r"\times10^{") + "}"
              elif abs(value - int(value)) < 0.001:
                  return "%.1f" % value
              else:
                  return "%.3f" % value
  
          def _format_element(m, n, d):
              s = " & " if n > 0 else ""
              if type(d) == str:
                  return s + d
              else:
                  if abs(np.imag(d)) < settings.atol:
                      return s + _format_float(np.real(d))
                  elif abs(np.real(d)) < settings.atol:
                      return s + _format_float(np.imag(d)) + "j"
                  else:
                      s_re = _format_float(np.real(d))
                      s_im = _format_float(np.imag(d))
                      if np.imag(d) > 0.0:
                          return (s + "(" + s_re + "+" + s_im + "j)")
                      else:
                          return (s + "(" + s_re + s_im + "j)")
  
          if M > 10 and N > 10:
              # truncated matrix output
              for m in range(5):
                  for n in range(5):
                      s += _format_element(m, n, self.data[m, n])
                  s += r' & \cdots'
                  for n in range(N - 5, N):
                      s += _format_element(m, n, self.data[m, n])
                  s += r'\\'
  
              for n in range(5):
                  s += _format_element(m, n, r'\vdots')
              s += r' & \ddots'
              for n in range(N - 5, N):
                  s += _format_element(m, n, r'\vdots')
              s += r'\\'
  
              for m in range(M - 5, M):
                  for n in range(5):
                      s += _format_element(m, n, self.data[m, n])
                  s += r' & \cdots'
                  for n in range(N - 5, N):
                      s += _format_element(m, n, self.data[m, n])
                  s += r'\\'
  
          elif M > 10 and N <= 10:
              # truncated vertically elongated matrix output
              for m in range(5):
                  for n in range(N):
                      s += _format_element(m, n, self.data[m, n])
                  s += r'\\'
  
              for n in range(N):
                  s += _format_element(m, n, r'\vdots')
              s += r'\\'
  
              for m in range(M - 5, M):
                  for n in range(N):
                      s += _format_element(m, n, self.data[m, n])
                  s += r'\\'
  
          elif M <= 10 and N > 10:
              # truncated horizontally elongated matrix output
              for m in range(M):
                  for n in range(5):
                      s += _format_element(m, n, self.data[m, n])
                  s += r' & \cdots'
                  for n in range(N - 5, N):
                      s += _format_element(m, n, self.data[m, n])
                  s += r'\\'
  
          else:
              # full output
              for m in range(M):
                  for n in range(N):
                      s += _format_element(m, n, self.data[m, n])
                  s += r'\\'
  
          s += r'\end{array}\right)\end{equation*}'
          return s
  
      def dag(self):
          """Adjoint operator of quantum object.
          """
          out = Qobj()
          out.data = zcsr_adjoint(self.data)
          out.dims = [self.dims[1], self.dims[0]]
          out._isherm = self._isherm
          out.superrep = self.superrep
          return out
  
      def dual_chan(self):
          """Dual channel of quantum object representing a completely positive
          map.
          """
          # Uses the technique of Johnston and Kribs (arXiv:1102.0948), which
          # is only valid for completely positive maps.
          if not self.iscp:
              raise ValueError("Dual channels are only implemented for CP maps.")
          J = sr.to_choi(self)
          tensor_idxs = enumerate_flat(J.dims)
          J_dual = tensor.tensor_swap(J, *(
                  list(zip(tensor_idxs[0][1], tensor_idxs[0][0])) +
                  list(zip(tensor_idxs[1][1], tensor_idxs[1][0]))
          )).trans()
          J_dual.superrep = 'choi'
          return J_dual
  
  
      def conj(self):
          """Conjugate operator of quantum object.
          """
          out = Qobj()
          out.data = self.data.conj()
          out.dims = [self.dims[0], self.dims[1]]
          return out
  
      def norm(self, norm=None, sparse=False, tol=0, maxiter=100000):
          """Norm of a quantum object.
  
          Default norm is L2-norm for kets and trace-norm for operators.
          Other ket and operator norms may be specified using the `norm` and
          argument.
  
          Parameters
          ----------
          norm : str
              Which norm to use for ket/bra vectors: L2 'l2', max norm 'max',
              or for operators: trace 'tr', Frobius 'fro', one 'one', or max
              'max'.
  
          sparse : bool
              Use sparse eigenvalue solver for trace norm.  Other norms are not
              affected by this parameter.
  
          tol : float
              Tolerance for sparse solver (if used) for trace norm. The sparse
              solver may not converge if the tolerance is set too low.
  
          maxiter : int
              Maximum number of iterations performed by sparse solver (if used)
              for trace norm.
  
          Returns
          -------
          norm : float
              The requested norm of the operator or state quantum object.
  
  
          Notes
          -----
          The sparse eigensolver is much slower than the dense version.
          Use sparse only if memory requirements demand it.
  
          """
          if self.type in ['oper', 'super']:
              if norm is None or norm == 'tr':
                  _op = self*self.dag()
                  vals = sp_eigs(_op.data, _op.isherm, vecs=False,
                                 sparse=sparse, tol=tol, maxiter=maxiter)
                  return np.sum(np.sqrt(np.abs(vals)))
              elif norm == 'fro':
                  return sp_fro_norm(self.data)
              elif norm == 'one':
                  return sp_one_norm(self.data)
              elif norm == 'max':
                  return sp_max_norm(self.data)
              else:
                  raise ValueError(
                      "For matrices, norm must be 'tr', 'fro', 'one', or 'max'.")
          else:
              if norm is None or norm == 'l2':
                  return sp_L2_norm(self.data)
              elif norm == 'max':
                  return sp_max_norm(self.data)
              else:
                  raise ValueError("For vectors, norm must be 'l2', or 'max'.")
  
      def proj(self):
          """Form the projector from a given ket or bra vector.
  
          Parameters
          ----------
          Q : :class:`qutip.Qobj`
              Input bra or ket vector
  
          Returns
          -------
          P : :class:`qutip.Qobj`
              Projection operator.
          """
          if self.isket:
              _out = zcsr_proj(self.data,1)
              _dims = [self.dims[0],self.dims[0]]
          elif self.isbra:
              _out = zcsr_proj(self.data,0)
              _dims = [self.dims[1],self.dims[1]]
          else:
              raise TypeError('Projector can only be formed from a bra or ket.')
  
          return Qobj(_out,dims=_dims)
  
  
      def tr(self):
          """Trace of a quantum object.
  
          Returns
          -------
          trace : float
              Returns ``real`` if operator is Hermitian, returns ``complex``
              otherwise.
  
          """
          return zcsr_trace(self.data, self.isherm)
  
      def full(self, order='C', squeeze=False):
          """Dense array from quantum object.
  
          Parameters
          ----------
          order : str {'C', 'F'}
              Return array in C (default) or Fortran ordering.
          squeeze : bool {False, True}
              Squeeze output array.
  
          Returns
          -------
          data : array
              Array of complex data from quantum objects `data` attribute.
          """
          if squeeze:
              return self.data.toarray(order=order).squeeze()
          else:
              return self.data.toarray(order=order)
  
      def __array__(self, *arg, **kwarg):
          """Numpy array from Qobj
          For compatibility with np.array
          """
          return self.full()
  
      def diag(self):
          """Diagonal elements of quantum object.
  
          Returns
          -------
          diags : array
              Returns array of ``real`` values if operators is Hermitian,
              otherwise ``complex`` values are returned.
  
          """
          out = self.data.diagonal()
          if np.any(np.imag(out) > settings.atol) or not self.isherm:
              return out
          else:
              return np.real(out)
  
      def expm(self, method='dense'):
          """Matrix exponential of quantum operator.
  
          Input operator must be square.
  
          Parameters
          ----------
          method : str {'dense', 'sparse'}
              Use set method to use to calculate the matrix exponentiation. The
              available choices includes 'dense' and 'sparse'.  Since the
              exponential of a matrix is nearly always dense, method='dense'
              is set as default.s
  
          Returns
          -------
          oper : :class:`qutip.Qobj`
              Exponentiated quantum operator.
  
          Raises
          ------
          TypeError
              Quantum operator is not square.
  
          """
          if self.dims[0][0] != self.dims[1][0]:
              raise TypeError('Invalid operand for matrix exponential')
  
          if method == 'dense':
              F = sp_expm(self.data, sparse=False)
  
          elif method == 'sparse':
              F = sp_expm(self.data, sparse=True)
  
          else:
              raise ValueError("method must be 'dense' or 'sparse'.")
  
          out = Qobj(F, dims=self.dims)
          return out.tidyup() if settings.auto_tidyup else out
  
      def check_herm(self):
          """Check if the quantum object is hermitian.
  
          Returns
          -------
          isherm : bool
              Returns the new value of isherm property.
          """
          self._isherm = None
          return self.isherm
  
      def sqrtm(self, sparse=False, tol=0, maxiter=100000):
          """Sqrt of a quantum operator.
  
          Operator must be square.
  
          Parameters
          ----------
          sparse : bool
              Use sparse eigenvalue/vector solver.
          tol : float
              Tolerance used by sparse solver (0 = machine precision).
          maxiter : int
              Maximum number of iterations used by sparse solver.
  
          Returns
          -------
          oper : :class:`qutip.Qobj`
              Matrix square root of operator.
  
          Raises
          ------
          TypeError
              Quantum object is not square.
  
          Notes
          -----
          The sparse eigensolver is much slower than the dense version.
          Use sparse only if memory requirements demand it.
  
          """
          if self.dims[0][0] == self.dims[1][0]:
              evals, evecs = sp_eigs(self.data, self.isherm, sparse=sparse,
                                     tol=tol, maxiter=maxiter)
              numevals = len(evals)
              dV = sp.spdiags(np.sqrt(evals, dtype=complex), 0, numevals,
                              numevals, format='csr')
              if self.isherm:
                  spDv = dV.dot(evecs.T.conj().T)
              else:
                  spDv = dV.dot(np.linalg.inv(evecs.T))
  
              out = Qobj(evecs.T.dot(spDv), dims=self.dims)
              return out.tidyup() if settings.auto_tidyup else out
  
          else:
              raise TypeError('Invalid operand for matrix square root')
  
  
      def cosm(self):
          """Cosine of a quantum operator.
  
          Operator must be square.
  
          Returns
          -------
          oper : :class:`qutip.Qobj`
              Matrix cosine of operator.
  
          Raises
          ------
          TypeError
              Quantum object is not square.
  
          Notes
          -----
          Uses the Q.expm() method.
  
          """
          if self.dims[0][0] == self.dims[1][0]:
               return 0.5 * ((1j * self).expm() + (-1j * self).expm())
          else:
              raise TypeError('Invalid operand for matrix square root')
  
  
      def sinm(self):
          """Sine of a quantum operator.
  
          Operator must be square.
  
          Returns
          -------
          oper : :class:`qutip.Qobj`
              Matrix sine of operator.
  
          Raises
          ------
          TypeError
              Quantum object is not square.
  
          Notes
          -----
          Uses the Q.expm() method.
  
          """
          if self.dims[0][0] == self.dims[1][0]:
               return -0.5j * ((1j * self).expm() - (-1j * self).expm())
          else:
              raise TypeError('Invalid operand for matrix square root')
  
  
  
      def unit(self, inplace=False,
              norm=None, sparse=False,
              tol=0, maxiter=100000):
          """Operator or state normalized to unity.
  
          Uses norm from Qobj.norm().
  
          Parameters
          ----------
          inplace : bool
              Do an in-place normalization
          norm : str
              Requested norm for states / operators.
          sparse : bool
              Use sparse eigensolver for trace norm. Does not affect other norms.
          tol : float
              Tolerance used by sparse eigensolver.
          maxiter : int
              Number of maximum iterations performed by sparse eigensolver.
  
          Returns
          -------
          oper : :class:`qutip.Qobj`
              Normalized quantum object if not in-place,
              else None.
  
          """
          if inplace:
              nrm = self.norm(norm=norm, sparse=sparse,
                             tol=tol, maxiter=maxiter)
  
              self.data /= nrm
          elif not inplace:
              out = self / self.norm(norm=norm, sparse=sparse,
                                     tol=tol, maxiter=maxiter)
              if settings.auto_tidyup:
                  return out.tidyup()
              else:
                  return out
          else:
              raise Exception('inplace kwarg must be bool.')
  
      def ptrace(self, sel):
          """Partial trace of the quantum object.
  
          Parameters
          ----------
          sel : int/list
              An ``int`` or ``list`` of components to keep after partial trace.
  
          Returns
          -------
          oper : :class:`qutip.Qobj`
              Quantum object representing partial trace with selected components
              remaining.
  
          Notes
          -----
          This function is identical to the :func:`qutip.qobj.ptrace` function
          that has been deprecated.
  
          """
          q = Qobj()
          q.data, q.dims, _ = _ptrace(self, sel)
          return q.tidyup() if settings.auto_tidyup else q
  
      def permute(self, order):
          """Permutes a composite quantum object.
  
          Parameters
          ----------
          order : list/array
              List specifying new tensor order.
  
          Returns
          -------
          P : :class:`qutip.Qobj`
              Permuted quantum object.
  
          """
          q = Qobj()
          q.data, q.dims = _permute(self, order)
          return q.tidyup() if settings.auto_tidyup else q
  
      def tidyup(self, atol=settings.auto_tidyup_atol):
          """Removes small elements from the quantum object.
  
          Parameters
          ----------
          atol : float
              Absolute tolerance used by tidyup. Default is set
              via qutip global settings parameters.
  
          Returns
          -------
          oper : :class:`qutip.Qobj`
              Quantum object with small elements removed.
  
          """
          if self.data.nnz:
              #This does the tidyup and returns True if
              #The sparse data needs to be shortened
              if use_openmp() and self.data.nnz > 500:
                  if omp_tidyup(self.data.data,atol,self.data.nnz,
                              settings.num_cpus):
                              self.data.eliminate_zeros()
              else:
                  if cy_tidyup(self.data.data,atol,self.data.nnz):
                      self.data.eliminate_zeros()
              return self
          else:
              return self
  
      def transform(self, inpt, inverse=False, sparse=True):
          """Basis transform defined by input array.
  
          Input array can be a ``matrix`` defining the transformation,
          or a ``list`` of kets that defines the new basis.
  
  
          Parameters
          ----------
          inpt : array_like
              A ``matrix`` or ``list`` of kets defining the transformation.
          inverse : bool
              Whether to return inverse transformation.
          sparse : bool
              Use sparse matrices when possible. Can be slower.
  
          Returns
          -------
          oper : :class:`qutip.Qobj`
              Operator in new basis.
  
          Notes
          -----
          This function is still in development.
  
  
          """
          if isinstance(inpt, list) or (isinstance(inpt, np.ndarray) and
                                        len(inpt.shape) == 1):
              if len(inpt) != max(self.shape):
                  raise TypeError(
                      'Invalid size of ket list for basis transformation')
              if sparse:
                  S = sp.hstack([psi.data for psi in inpt],
                              format='csr', dtype=complex).conj().T
              else:
                  S = np.hstack([psi.full() for psi in inpt],
                                 dtype=complex).conj().T
          elif isinstance(inpt, Qobj) and inpt.isoper:
              S = inpt.data
          elif isinstance(inpt, np.ndarray):
              S = inpt.conj()
              sparse = False
          else:
              raise TypeError('Invalid operand for basis transformation')
  
  
          # transform data
          if inverse:
              if self.isket:
                    data = (S.conj().T) * self.data
              elif self.isbra:
                  data = self.data.dot(S)
              else:
                  if sparse:
                      data = (S.conj().T) * self.data * S
                  else:
                      data = (S.conj().T).dot(self.data.dot(S))
          else:
              if self.isket:
                  data = S * self.data
              elif self.isbra:
                  data = self.data.dot(S.conj().T)
              else:
                  if sparse:
                      data = S * self.data * (S.conj().T)
                  else:
                      data = S.dot(self.data.dot(S.conj().T))
  
          out = Qobj(data, dims=self.dims)
          out._isherm = self._isherm
          out.superrep = self.superrep
  
          if settings.auto_tidyup:
              return out.tidyup()
          else:
              return out
  
  
  
      def trunc_neg(self, method="clip"):
          """Truncates negative eigenvalues and renormalizes.
  
          Returns a new Qobj by removing the negative eigenvalues
          of this instance, then renormalizing to obtain a valid density
          operator.
  
  
          Parameters
          ----------
          method : str
              Algorithm to use to remove negative eigenvalues. "clip"
              simply discards negative eigenvalues, then renormalizes.
              "sgs" uses the SGS algorithm (doi:10/bb76) to find the
              positive operator that is nearest in the Shatten 2-norm.
  
          Returns
          -------
          oper : :class:`qutip.Qobj`
              A valid density operator.
  
          """
          if not self.isherm:
              raise ValueError("Must be a Hermitian operator to remove negative "
                              "eigenvalues.")
  
          if method not in ('clip', 'sgs'):
              raise ValueError("Method {} not recognized.".format(method))
  
          eigvals, eigstates = self.eigenstates()
          if all([eigval >= 0 for eigval in eigvals]):
              # All positive, so just renormalize.
              return self.unit()
          idx_nonzero = eigvals != 0
          eigvals = eigvals[idx_nonzero]
          eigstates = eigstates[idx_nonzero]
  
          if method == 'clip':
              eigvals[eigvals < 0] = 0
          elif method == 'sgs':
              eigvals = eigvals[::-1]
              eigstates = eigstates[::-1]
  
              acc = 0.0
              dim = self.shape[0]
              n_eigs = len(eigvals)
  
              for idx in reversed(range(n_eigs)):
                  if eigvals[idx] + acc / (idx + 1) >= 0:
                      break
                  else:
                      acc += eigvals[idx]
                      eigvals[idx] = 0.0
  
              eigvals[:idx+1] += acc / (idx + 1)
  
          return sum([
                  val * qutip.states.ket2dm(state)
                  for val, state in zip(eigvals, eigstates)
              ], Qobj(np.zeros(self.shape), dims=self.dims)
          ).unit()
  
  
      def matrix_element(self, bra, ket):
          """Calculates a matrix element.
  
          Gives the matrix element for the quantum object sandwiched between a
          `bra` and `ket` vector.
  
          Parameters
          -----------
          bra : :class:`qutip.Qobj`
              Quantum object of type 'bra' or 'ket'
  
          ket : :class:`qutip.Qobj`
              Quantum object of type 'ket'.
  
          Returns
          -------
          elem : complex
              Complex valued matrix element.
  
          Note
          ----
          It is slightly more computationally efficient to use a ket
          vector for the 'bra' input.
  
          """
          if not self.isoper:
              raise TypeError("Can only get matrix elements for an operator.")
  
          else:
              if bra.isbra and ket.isket:
                  return zcsr_mat_elem(self.data,bra.data,ket.data,1)
  
              elif bra.isket and ket.isket:
                  return zcsr_mat_elem(self.data,bra.data,ket.data,0)
              else:
                  raise TypeError("Can only calculate matrix elements for bra and ket vectors.")
  
      def overlap(self, other):
          """Overlap between two state vectors or two operators.
  
          Gives the overlap (inner product) between the current bra or ket Qobj
          and and another bra or ket Qobj. It gives the Hilbert-Schmidt overlap
          when one of the Qobj is an operator/density matrix.
  
          Parameters
          -----------
          other : :class:`qutip.Qobj`
              Quantum object for a state vector of type 'ket', 'bra' or density
              matrix.
  
          Returns
          -------
          overlap : complex
              Complex valued overlap.
  
          Raises
          ------
          TypeError
              Can only calculate overlap between a bra, ket and density matrix
              quantum objects.
  
          Notes
          -----
          Since QuTiP mainly deals with ket vectors, the most efficient inner
          product call is the ket-ket version that computes the product
          <self|other> with both vectors expressed as kets.
          """
  
          if isinstance(other, Qobj):
  
              if self.isbra:
                  if other.isket:
                      return zcsr_inner(self.data, other.data, 1)
                  elif other.isbra:
                      #Since we deal mainly with ket vectors, the bra-bra combo
                      #is not common, and not optimized.
                      return zcsr_inner(self.data, other.dag().data, 1)
                  elif other.isoper:
                      return (qutip.states.ket2dm(self).dag() * other).tr()
                  else:
                      raise TypeError("Can only calculate overlap for state vector Qobjs")
  
              elif self.isket:
                  if other.isbra:
                      return zcsr_inner(other.data, self.data, 1)
                  elif other.isket:
                      return zcsr_inner(self.data, other.data, 0)
                  elif other.isoper:
                      return (qutip.states.ket2dm(self).dag() * other).tr()
                  else:
                      raise TypeError("Can only calculate overlap for state vector Qobjs")
  
              elif self.isoper:
                  if other.isket or other.isbra:
                      return (self.dag() * qutip.states.ket2dm(other)).tr()
                  elif other.isoper:
                      return (self.dag() * other).tr()
                  else:
                      raise TypeError("Can only calculate overlap for state vector Qobjs")
  
  
          raise TypeError("Can only calculate overlap for state vector Qobjs")
  
  
      def eigenstates(self, sparse=False, sort='low',
                      eigvals=0, tol=0, maxiter=100000):
          """Eigenstates and eigenenergies.
  
          Eigenstates and eigenenergies are defined for operators and
          superoperators only.
  
          Parameters
          ----------
          sparse : bool
              Use sparse Eigensolver
  
          sort : str
              Sort eigenvalues (and vectors) 'low' to high, or 'high' to low.
  
          eigvals : int
              Number of requested eigenvalues. Default is all eigenvalues.
  
          tol : float
              Tolerance used by sparse Eigensolver (0 = machine precision).
              The sparse solver may not converge if the tolerance is set too low.
  
          maxiter : int
              Maximum number of iterations performed by sparse solver (if used).
  
          Returns
          -------
          eigvals : array
              Array of eigenvalues for operator.
  
          eigvecs : array
              Array of quantum operators representing the oprator eigenkets.
              Order of eigenkets is determined by order of eigenvalues.
  
          Notes
          -----
          The sparse eigensolver is much slower than the dense version.
          Use sparse only if memory requirements demand it.
  
          """
          evals, evecs = sp_eigs(self.data, self.isherm, sparse=sparse,
                                 sort=sort, eigvals=eigvals, tol=tol,
                                 maxiter=maxiter)
          new_dims = [self.dims[0], [1] * len(self.dims[0])]
          ekets = np.array([Qobj(vec, dims=new_dims) for vec in evecs],
                           dtype=object)
          norms = np.array([ket.norm() for ket in ekets])
          return evals, ekets / norms
  
  
      def eigenenergies(self, sparse=False, sort='low',
                        eigvals=0, tol=0, maxiter=100000):
          """Eigenenergies of a quantum object.
  
          Eigenenergies (eigenvalues) are defined for operators or superoperators
          only.
  
          Parameters
          ----------
          sparse : bool
              Use sparse Eigensolver
          sort : str
              Sort eigenvalues 'low' to high, or 'high' to low.
          eigvals : int
              Number of requested eigenvalues. Default is all eigenvalues.
          tol : float
              Tolerance used by sparse Eigensolver (0=machine precision).
              The sparse solver may not converge if the tolerance is set too low.
          maxiter : int
              Maximum number of iterations performed by sparse solver (if used).
  
          Returns
          -------
          eigvals : array
              Array of eigenvalues for operator.
  
          Notes
          -----
          The sparse eigensolver is much slower than the dense version.
          Use sparse only if memory requirements demand it.
  
          """
          return sp_eigs(self.data, self.isherm, vecs=False, sparse=sparse,
                         sort=sort, eigvals=eigvals, tol=tol, maxiter=maxiter)
  
      def groundstate(self, sparse=False, tol=0, maxiter=100000, safe=True):
          """Ground state Eigenvalue and Eigenvector.
  
          Defined for quantum operators or superoperators only.
  
          Parameters
          ----------
          sparse : bool
              Use sparse Eigensolver
          tol : float
              Tolerance used by sparse Eigensolver (0 = machine precision).
              The sparse solver may not converge if the tolerance is set too low.
          maxiter : int
              Maximum number of iterations performed by sparse solver (if used).
          safe : bool (default=True)
              Check for degenerate ground state
  
          Returns
          -------
          eigval : float
              Eigenvalue for the ground state of quantum operator.
          eigvec : :class:`qutip.Qobj`
              Eigenket for the ground state of quantum operator.
  
          Notes
          -----
          The sparse eigensolver is much slower than the dense version.
          Use sparse only if memory requirements demand it.
  
          """
          if safe:
              evals = 2
          else:
              evals = 1
          grndval, grndvec = sp_eigs(self.data, self.isherm, sparse=sparse,
                                     eigvals=evals, tol=tol, maxiter=maxiter)
          if safe:
              if tol == 0: tol = 1e-15
              if (grndval[1]-grndval[0]) <= 10*tol:
                  print("WARNING: Ground state may be degenerate. "
                          "Use Q.eigenstates()")
          new_dims = [self.dims[0], [1] * len(self.dims[0])]
          grndvec = Qobj(grndvec[0], dims=new_dims)
          grndvec = grndvec / grndvec.norm()
          return grndval[0], grndvec
  
      def trans(self):
          """Transposed operator.
  
          Returns
          -------
          oper : :class:`qutip.Qobj`
              Transpose of input operator.
  
          """
          out = Qobj()
          out.data = zcsr_transpose(self.data)
          out.dims = [self.dims[1], self.dims[0]]
          return out
  
      def extract_states(self, states_inds, normalize=False):
          """Qobj with states in state_inds only.
  
          Parameters
          ----------
          states_inds : list of integer
              The states that should be kept.
  
          normalize : True / False
              Weather or not the new Qobj instance should be normalized (default
              is False). For Qobjs that represents density matrices or state
              vectors normalized should probably be set to True, but for Qobjs
              that represents operators in for example an Hamiltonian, normalize
              should be False.
  
          Returns
          -------
          q : :class:`qutip.Qobj`
              A new instance of :class:`qutip.Qobj` that contains only the states
              corresponding to the indices in `state_inds`.
  
          Notes
          -----
          Experimental.
  
          """
          if self.isoper:
              q = Qobj(self.data[states_inds, :][:, states_inds])
          elif self.isket:
              q = Qobj(self.data[states_inds, :])
          elif self.isbra:
              q = Qobj(self.data[:, states_inds])
          else:
              raise TypeError("Can only eliminate states from operators or " +
                              "state vectors")
  
          return q.unit() if normalize else q
  
      def eliminate_states(self, states_inds, normalize=False):
          """Creates a new quantum object with states in state_inds eliminated.
  
          Parameters
          ----------
          states_inds : list of integer
              The states that should be removed.
  
          normalize : True / False
              Weather or not the new Qobj instance should be normalized (default
              is False). For Qobjs that represents density matrices or state
              vectors normalized should probably be set to True, but for Qobjs
              that represents operators in for example an Hamiltonian, normalize
              should be False.
  
          Returns
          -------
          q : :class:`qutip.Qobj`
              A new instance of :class:`qutip.Qobj` that contains only the states
              corresponding to indices that are **not** in `state_inds`.
  
          Notes
          -----
          Experimental.
  
          """
          keep_indices = np.array([s not in states_inds
                                   for s in range(self.shape[0])]).nonzero()[0]
  
          return self.extract_states(keep_indices, normalize=normalize)
  
      def dnorm(self, B=None):
          """Calculates the diamond norm, or the diamond distance to another
          operator.
  
          Parameters
          ----------
          B : :class:`qutip.Qobj` or None
              If B is not None, the diamond distance d(A, B) = dnorm(A - B) between
              this operator and B is returned instead of the diamond norm.
  
          Returns
          -------
          d : float
              Either the diamond norm of this operator, or the diamond distance
              from this operator to B.
  
          """
          return mts.dnorm(self, B)
  
  
      @property
      def ishp(self):
          # FIXME: this needs to be cached in the same ways as isherm.
          if self.type in ["super", "oper"]:
              try:
                  J = sr.to_choi(self)
                  return J.isherm
              except:
                  return False
          else:
              return False
  
      @property
      def iscp(self):
          # FIXME: this needs to be cached in the same ways as isherm.
          if self.type in ["super", "oper"]:
              try:
                  J = (
                      self
                      # We can test with either Choi or chi, since the basis
                      # transformation between them is unitary and hence
                      # preserves the CP and TP conditions.
                      if self.superrep in ('choi', 'chi')
                      else sr.to_choi(self)
                  )
                  # If J isn't hermitian, then that could indicate either
                  # that J is not normal, or is normal, but has complex eigenvalues.
                  # In either case, it makes no sense to then demand that the
                  # eigenvalues be non-negative.
                  if not J.isherm:
                      return False
                  eigs = J.eigenenergies()
                  return all(eigs >= -settings.atol)
              except:
                  return False
          else:
              return False
  
      @property
      def istp(self):
          import qutip.superop_reps as sr
          if self.type in ["super", "oper"]:
              try:
                  # Normalize to a super of type choi or chi.
                  # We can test with either Choi or chi, since the basis
                  # transformation between them is unitary and hence
                  # preserves the CP and TP conditions.
                  if self.type == "super" and self.superrep in ('choi', 'chi'):
                      qobj = self
                  else:
                      qobj = sr.to_choi(self)
  
                  # Possibly collapse dims.
                  if any([len(index) > 1 for super_index in qobj.dims
                                              for index in super_index]):
                      qobj = Qobj(qobj, dims=collapse_dims_super(qobj.dims))
                  else:
                      qobj = qobj
  
                  # We use the condition from John Watrous' lecture notes,
                  # Tr_1(J(Phi)) = identity_2.
                  tr_oper = qobj.ptrace([0])
                  ident = ops.identity(tr_oper.shape[0])
                  return isequal(tr_oper, ident)
              except:
                  return False
          else:
              return False
  
      @property
      def iscptp(self):
          from qutip.superop_reps import to_choi
          if self.type == "super" or self.type == "oper":
              reps = ('choi', 'chi')
              q_oper = to_choi(self) if self.superrep not in reps else self
              return q_oper.iscp and q_oper.istp
          else:
              return False
  
      @property
      def isherm(self):
  
          if self._isherm is not None:
              # used previously computed value
              return self._isherm
  
          self._isherm = bool(zcsr_isherm(self.data))
  
          return self._isherm
  
      @isherm.setter
      def isherm(self, isherm):
          self._isherm = isherm
  
      def check_isunitary(self):
          """
          Checks whether qobj is a unitary matrix
          """
          if self.isoper:
              eye_data = fast_identity(self.shape[0])
              return not (np.any(np.abs((self.data*self.dag().data
                                         - eye_data).data)
                                  > settings.atol)
                          or
                          np.any(np.abs((self.dag().data*self.data
                                         - eye_data).data) >
                                settings.atol)
                          )
  
          else:
              return False
  
      @property
      def isunitary(self):
          if self._isunitary is not None:
              # used previously computed value
              return self._isunitary
  
          self._isunitary = self.check_isunitary()
  
          return self._isunitary
  
      @isunitary.setter
      def isunitary(self, isunitary):
          self._isunitary = isunitary
  
      @property
      def type(self):
          if not self._type:
              self._type = type_from_dims(self.dims)
  
          return self._type
  
      @property
      def shape(self):
          if self.data.shape == (1, 1):
              return tuple([np.prod(self.dims[0]), np.prod(self.dims[1])])
          else:
              return tuple(self.data.shape)
  
      @property
      def isbra(self):
          return self.type == 'bra'
  
      @property
      def isket(self):
          return self.type == 'ket'
  
      @property
      def isoperbra(self):
          return self.type == 'operator-bra'
  
      @property
      def isoperket(self):
          return self.type == 'operator-ket'
  
      @property
      def isoper(self):
          return self.type == 'oper'
  
      @property
      def issuper(self):
          return self.type == 'super'
  
      @staticmethod
      def evaluate(qobj_list, t, args):
          """Evaluate a time-dependent quantum object in list format. For
          example,
  
              qobj_list = [H0, [H1, func_t]]
  
          is evaluated to
  
              Qobj(t) = H0 + H1 * func_t(t, args)
  
          and
  
              qobj_list = [H0, [H1, 'sin(w * t)']]
  
          is evaluated to
  
              Qobj(t) = H0 + H1 * sin(args['w'] * t)
  
          Parameters
          ----------
          qobj_list : list
              A nested list of Qobj instances and corresponding time-dependent
              coefficients.
          t : float
              The time for which to evaluate the time-dependent Qobj instance.
          args : dictionary
              A dictionary with parameter values required to evaluate the
              time-dependent Qobj intance.
  
          Returns
          -------
          output : :class:`qutip.Qobj`
              A Qobj instance that represents the value of qobj_list at time t.
  
          """
  
          q_sum = 0
          if isinstance(qobj_list, Qobj):
              q_sum = qobj_list
          elif isinstance(qobj_list, list):
              for q in qobj_list:
                  if isinstance(q, Qobj):
                      q_sum += q
                  elif (isinstance(q, list) and len(q) == 2 and
                        isinstance(q[0], Qobj)):
                      if isinstance(q[1], types.FunctionType):
                          q_sum += q[0] * q[1](t, args)
                      elif isinstance(q[1], str):
                          args['t'] = t
                          q_sum += q[0] * float(eval(q[1], globals(), args))
                      else:
                          raise TypeError('Unrecognized format for ' +
                                          'specification of time-dependent Qobj')
                  else:
                      raise TypeError('Unrecognized format for specification ' +
                                      'of time-dependent Qobj')
          else:
              raise TypeError(
                  'Unrecongized format for specification of time-dependent Qobj')
  
          return q_sum
  
  
  # -----------------------------------------------------------------------------
  # This functions evaluates a time-dependent quantum object on the list-string
  # and list-function formats that are used by the time-dependent solvers.
  # Although not used directly in by those solvers, it can for test purposes be
  # conventient to be able to evaluate the expressions passed to the solver for
  # arbitrary value of time. This function provides this functionality.
  #
  def qobj_list_evaluate(qobj_list, t, args):
      """
      Depracated: See Qobj.evaluate
      """
      warnings.warn("Deprecated: Use Qobj.evaluate", DeprecationWarning)
      return Qobj.evaluate(qobj_list, t, args)
  
  
  # -----------------------------------------------------------------------------
  #
  # A collection of tests used to determine the type of quantum objects, and some
  # functions for increased compatibility with quantum optics toolbox.
  #
  
  def dag(A):
      """Adjont operator (dagger) of a quantum object.
  
      Parameters
      ----------
      A : :class:`qutip.Qobj`
          Input quantum object.
  
      Returns
      -------
      oper : :class:`qutip.Qobj`
          Adjoint of input operator
  
      Notes
      -----
      This function is for legacy compatibility only. It is recommended to use
      the ``dag()`` Qobj method.
  
      """
      if not isinstance(A, Qobj):
          raise TypeError("Input is not a quantum object")
  
      return A.dag()
  
  
  def ptrace(Q, sel):
      """Partial trace of the Qobj with selected components remaining.
  
      Parameters
      ----------
      Q : :class:`qutip.Qobj`
          Composite quantum object.
      sel : int/list
          An ``int`` or ``list`` of components to keep after partial trace.
  
      Returns
      -------
      oper : :class:`qutip.Qobj`
          Quantum object representing partial trace with selected components
          remaining.
  
      Notes
      -----
      This function is for legacy compatibility only. It is recommended to use
      the ``ptrace()`` Qobj method.
  
      """
      if not isinstance(Q, Qobj):
          raise TypeError("Input is not a quantum object")
  
      return Q.ptrace(sel)
  
  
  def dims(inpt):
      """Returns the dims attribute of a quantum object.
  
      Parameters
      ----------
      inpt : :class:`qutip.Qobj`
          Input quantum object.
  
      Returns
      -------
      dims : list
          A ``list`` of the quantum objects dimensions.
  
      Notes
      -----
      This function is for legacy compatibility only. Using the `Qobj.dims`
      attribute is recommended.
  
      """
      if isinstance(inpt, Qobj):
          return inpt.dims
      else:
          raise TypeError("Input is not a quantum object")
  
  
  def shape(inpt):
      """Returns the shape attribute of a quantum object.
  
      Parameters
      ----------
      inpt : :class:`qutip.Qobj`
          Input quantum object.
  
      Returns
      -------
      shape : list
          A ``list`` of the quantum objects shape.
  
      Notes
      -----
      This function is for legacy compatibility only. Using the `Qobj.shape`
      attribute is recommended.
  
      """
      if isinstance(inpt, Qobj):
          return Qobj.shape
      else:
          return np.shape(inpt)
  
  
  def isket(Q):
      """
      Determines if given quantum object is a ket-vector.
  
      Parameters
      ----------
      Q : :class:`qutip.Qobj`
          Quantum object
  
      Returns
      -------
      isket : bool
          True if qobj is ket-vector, False otherwise.
  
      Examples
      --------
      >>> psi = basis(5,2)
      >>> isket(psi)
      True
  
      Notes
      -----
      This function is for legacy compatibility only. Using the `Qobj.isket`
      attribute is recommended.
  
      """
      return True if isinstance(Q, Qobj) and Q.isket else False
  
  
  def isbra(Q):
      """Determines if given quantum object is a bra-vector.
  
      Parameters
      ----------
      Q : :class:`qutip.Qobj`
          Quantum object
  
      Returns
      -------
      isbra : bool
          True if Qobj is bra-vector, False otherwise.
  
      Examples
      --------
      >>> psi = basis(5,2)
      >>> isket(psi)
      False
  
      Notes
      -----
      This function is for legacy compatibility only. Using the `Qobj.isbra`
      attribute is recommended.
  
      """
      return True if isinstance(Q, Qobj) and Q.isbra else False
  
  
  def isoperket(Q):
      """Determines if given quantum object is an operator in column vector form
      (operator-ket).
  
      Parameters
      ----------
      Q : :class:`qutip.Qobj`
          Quantum object
  
      Returns
      -------
      isoperket : bool
          True if Qobj is operator-ket, False otherwise.
  
      Notes
      -----
      This function is for legacy compatibility only. Using the `Qobj.isoperket`
      attribute is recommended.
  
      """
      return True if isinstance(Q, Qobj) and Q.isoperket else False
  
  
  def isoperbra(Q):
      """Determines if given quantum object is an operator in row vector form
      (operator-bra).
  
      Parameters
      ----------
      Q : :class:`qutip.Qobj`
          Quantum object
  
      Returns
      -------
      isoperbra : bool
          True if Qobj is operator-bra, False otherwise.
  
      Notes
      -----
      This function is for legacy compatibility only. Using the `Qobj.isoperbra`
      attribute is recommended.
  
      """
      return True if isinstance(Q, Qobj) and Q.isoperbra else False
  
  
  def isoper(Q):
      """Determines if given quantum object is a operator.
  
      Parameters
      ----------
      Q : :class:`qutip.Qobj`
          Quantum object
  
      Returns
      -------
      isoper : bool
          True if Qobj is operator, False otherwise.
  
      Examples
      --------
      >>> a = destroy(5)
      >>> isoper(a)
      True
  
      Notes
      -----
      This function is for legacy compatibility only. Using the `Qobj.isoper`
      attribute is recommended.
  
      """
      return True if isinstance(Q, Qobj) and Q.isoper else False
  
  
  def issuper(Q):
      """Determines if given quantum object is a super-operator.
  
      Parameters
      ----------
      Q : :class:`qutip.Qobj`
          Quantum object
  
      Returns
      -------
      issuper  : bool
          True if Qobj is superoperator, False otherwise.
  
      Notes
      -----
      This function is for legacy compatibility only. Using the `Qobj.issuper`
      attribute is recommended.
  
      """
      return True if isinstance(Q, Qobj) and Q.issuper else False
  
  
  def isequal(A, B, tol=None):
      """Determines if two qobj objects are equal to within given tolerance.
  
      Parameters
      ----------
      A : :class:`qutip.Qobj`
          Qobj one
      B : :class:`qutip.Qobj`
          Qobj two
      tol : float
          Tolerence for equality to be valid
  
      Returns
      -------
      isequal : bool
          True if qobjs are equal, False otherwise.
  
      Notes
      -----
      This function is for legacy compatibility only. Instead, it is recommended
      to use the equality operator of Qobj instances instead: A == B.
  
      """
      if tol is None:
          tol = settings.atol
  
      if not isinstance(A, Qobj) or not isinstance(B, Qobj):
          return False
  
      if A.dims != B.dims:
          return False
  
      Adat = A.data
      Bdat = B.data
      elems = (Adat - Bdat).data
      if np.any(np.abs(elems) > tol):
          return False
  
      return True
  
  
  def isherm(Q):
      """Determines if given operator is Hermitian.
  
      Parameters
      ----------
      Q : :class:`qutip.Qobj`
          Quantum object
  
      Returns
      -------
      isherm : bool
          True if operator is Hermitian, False otherwise.
  
      Examples
      --------
      >>> a = destroy(4)
      >>> isherm(a)
      False
  
      Notes
      -----
      This function is for legacy compatibility only. Using the `Qobj.isherm`
      attribute is recommended.
  
      """
      return True if isinstance(Q, Qobj) and Q.isherm else False
  
  
  # TRAILING IMPORTS
  # We do a few imports here to avoid circular dependencies.
  from qutip.eseries import eseries
  import qutip.superop_reps as sr
  import qutip.tensor as tensor
  import qutip.operators as ops
  import qutip.metrics as mts
  import qutip.states
  import qutip.superoperator
  
apis: numpy.zeros scipy.sparse.issparse numpy.linalg.inv numpy.any numpy.abs scipy.sparse.csr_matrix numpy.shape numpy.prod numpy.sqrt scipy.sparse.hstack numpy.array numpy.real numpy.imag