Basic Operations on Quantum Objects

First things first

Warning

Do not run QuTiP from the installation directory.

To load the qutip modules, we must first call the import statement:

In [1]: from qutip import *

that will load all of the user available functions. Often, we also need to import the NumPy and Matplotlib libraries with:

In [2]: import numpy as np

In [3]: import matplotlib.pyplot as plt

Note that, in the rest of the documentation, functions are written using qutip.module.function() notation which links to the corresponding function in the QuTiP API: Functions. However, in calling import *, we have already loaded all of the QuTiP modules. Therefore, we will only need the function name and not the complete path when calling the function from the interpreter prompt, Python script, or Jupyter notebook.

The quantum object class

Introduction

The key difference between classical and quantum mechanics lies in the use of operators instead of numbers as variables. Moreover, we need to specify state vectors and their properties. Therefore, in computing the dynamics of quantum systems we need a data structure that is capable of encapsulating the properties of a quantum operator and ket/bra vectors. The quantum object class, qutip.Qobj, accomplishes this using matrix representation.

To begin, let us create a blank Qobj:

In [4]: Qobj()
Out[4]: 
Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra
Qobj data =
[[ 0.]]

where we see the blank Qobj object with dimensions, shape, and data. Here the data corresponds to a 1x1-dimensional matrix consisting of a single zero entry.

Hint

By convention, Class objects in Python such as Qobj() differ from functions in the use of a beginning capital letter.

We can create a Qobj with a user defined data set by passing a list or array of data into the Qobj:

In [5]: Qobj([[1],[2],[3],[4],[5]])
Out[5]: 
Quantum object: dims = [[5], [1]], shape = (5, 1), type = ket
Qobj data =
[[ 1.]
 [ 2.]
 [ 3.]
 [ 4.]
 [ 5.]]

Notice how both the dims and shape change according to the input data. Although dims and shape appear to have the same function, the difference will become quite clear in the section on tensor products and partial traces.

Note

If you are running QuTiP from a python script you must use the print function to view the Qobj attributes.

States and operators

Manually specifying the data for each quantum object is inefficient. Even more so when most objects correspond to commonly used types such as the ladder operators of a harmonic oscillator, the Pauli spin operators for a two-level system, or state vectors such as Fock states. Therefore, QuTiP includes predefined objects for a variety of states:

States Command (# means optional) Inputs
Fock state ket vector basis(N,#m)/fock(N,#m) N = number of levels in Hilbert space, m = level containing excitation (0 if no m given)
Fock density matrix (outer product of basis) fock_dm(N,#p) same as basis(N,m) / fock(N,m)
Coherent state coherent(N,alpha) alpha = complex number (eigenvalue) for requested coherent state
Coherent density matrix (outer product) coherent_dm(N,alpha) same as coherent(N,alpha)
Thermal density matrix (for n particles) thermal_dm(N,n) n = particle number expectation value

and operators:

Operators Command (# means optional) Inputs
Charge operator charge(N,M=-N) Diagonal operator with entries from M..0..N.
Commutator commutator(A, B, kind) Kind = ‘normal’ or ‘anti’.
Diagonals operator qdiags(N) Quantum object created from arrays of diagonals at given offsets.
Displacement operator (Single-mode) displace(N,alpha) N=number of levels in Hilbert space, alpha = complex displacement amplitude.
Higher spin operators jmat(j,#s) j = integer or half-integer representing spin, s = ‘x’, ‘y’, ‘z’, ‘+’, or ‘-‘
Identity qeye(N) N = number of levels in Hilbert space.
Lowering (destruction) operator destroy(N) same as above
Momentum operator momentum(N) same as above
Number operator num(N) same as above
Phase operator (Single-mode) phase(N, phi0) Single-mode Pegg-Barnett phase operator with ref phase phi0.
Position operator position(N) same as above
Raising (creation) operator create(N) same as above
Squeezing operator (Single-mode) squeeze(N, sp) N=number of levels in Hilbert space, sp = squeezing parameter.
Squeezing operator (Generalized) squeezing(q1, q2, sp) q1,q2 = Quantum operators (Qobj) sp = squeezing parameter.
Sigma-X sigmax()  
Sigma-Y sigmay()  
Sigma-Z sigmaz()  
Sigma plus sigmap()  
Sigma minus sigmam()  
Tunneling operator tunneling(N,m) Tunneling operator with elements of the form \(|N><N+m| + |N+m><N|\).

As an example, we give the output for a few of these functions:

In [6]: basis(5,3)
Out[6]: 
Quantum object: dims = [[5], [1]], shape = (5, 1), type = ket
Qobj data =
[[ 0.]
 [ 0.]
 [ 0.]
 [ 1.]
 [ 0.]]

In [7]: coherent(5,0.5-0.5j)
Out[7]: 
Quantum object: dims = [[5], [1]], shape = (5, 1), type = ket
Qobj data =
[[ 0.77880170+0.j        ]
 [ 0.38939142-0.38939142j]
 [ 0.00000000-0.27545895j]
 [-0.07898617-0.07898617j]
 [-0.04314271+0.j        ]]

In [8]: destroy(4)
Out[8]: 
Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = False
Qobj data =
[[ 0.          1.          0.          0.        ]
 [ 0.          0.          1.41421356  0.        ]
 [ 0.          0.          0.          1.73205081]
 [ 0.          0.          0.          0.        ]]

In [9]: sigmaz()
Out[9]: 
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[ 1.  0.]
 [ 0. -1.]]

In [10]: jmat(5/2.0,'+')
Out[10]: 
Quantum object: dims = [[6], [6]], shape = (6, 6), type = oper, isherm = False
Qobj data =
[[ 0.          2.23606798  0.          0.          0.          0.        ]
 [ 0.          0.          2.82842712  0.          0.          0.        ]
 [ 0.          0.          0.          3.          0.          0.        ]
 [ 0.          0.          0.          0.          2.82842712  0.        ]
 [ 0.          0.          0.          0.          0.          2.23606798]
 [ 0.          0.          0.          0.          0.          0.        ]]

Qobj attributes

We have seen that a quantum object has several internal attributes, such as data, dims, and shape. These can be accessed in the following way:

In [11]: q = destroy(4)

In [12]: q.dims
Out[12]: [[4], [4]]

In [13]: q.shape
Out[13]: (4, 4)

In general, the attributes (properties) of a Qobj object (or any Python class) can be retrieved using the Q.attribute notation. In addition to the attributes shown with the print function, the Qobj class also has the following:

Property Attribute Description
Data Q.data Matrix representing state or operator
Dimensions Q.dims List keeping track of shapes for individual components of a multipartite system (for tensor products and partial traces).
Shape Q.shape Dimensions of underlying data matrix.
is Hermitian? Q.isherm Is the operator Hermitian or not?
Type Q.type Is object of type ‘ket, ‘bra’, ‘oper’, or ‘super’?
../images/quide-basics-qobj-box.png

The Qobj Class viewed as a container for the properties need to characterize a quantum operator or state vector.

For the destruction operator above:

In [14]: q.type
Out[14]: 'oper'

In [15]: q.isherm
Out[15]: False

In [16]: q.data
Out[16]: 
<4x4 sparse matrix of type '<class 'numpy.complex128'>'
	with 3 stored elements in Compressed Sparse Row format>

The data attribute returns a message stating that the data is a sparse matrix. All Qobj instances store their data as a sparse matrix to save memory. To access the underlying dense matrix one needs to use the qutip.Qobj.full function as described below.

Qobj Math

The rules for mathematical operations on Qobj instances are similar to standard matrix arithmetic:

In [17]: q = destroy(4)

In [18]: x = sigmax()

In [19]: q + 5
Out[19]: 
Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = False
Qobj data =
[[ 5.          1.          0.          0.        ]
 [ 0.          5.          1.41421356  0.        ]
 [ 0.          0.          5.          1.73205081]
 [ 0.          0.          0.          5.        ]]

In [20]: x * x
Out[20]: 
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[ 1.  0.]
 [ 0.  1.]]

In [21]: q ** 3
Out[21]: 
Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = False
Qobj data =
[[ 0.          0.          0.          2.44948974]
 [ 0.          0.          0.          0.        ]
 [ 0.          0.          0.          0.        ]
 [ 0.          0.          0.          0.        ]]

In [22]: x / np.sqrt(2)
Out[22]: 
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[ 0.          0.70710678]
 [ 0.70710678  0.        ]]

Of course, like matrices, multiplying two objects of incompatible shape throws an error:

In [23]: q * x
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-23-c5138e004127> in <module>()
----> 1 q * x

~/anaconda/lib/python3.6/site-packages/qutip-4.3.0.dev0+0c3b404c-py3.6-macosx-10.9-x86_64.egg/qutip/qobj.py in __mul__(self, other)
    513 
    514             else:
--> 515                 raise TypeError("Incompatible Qobj shapes")
    516 
    517         elif isinstance(other, np.ndarray):

TypeError: Incompatible Qobj shapes

In addition, the logic operators is equal == and is not equal != are also supported.

Functions operating on Qobj class

Like attributes, the quantum object class has defined functions (methods) that operate on Qobj class instances. For a general quantum object Q:

Function Command Description
Check Hermicity Q.check_herm() Check if quantum object is Hermitian
Conjugate Q.conj() Conjugate of quantum object.
Cosine Q.cosm() Cosine of quantum object.
Dagger (adjoint) Q.dag() Returns adjoint (dagger) of object.
Diagonal Q.diag() Returns the diagonal elements.
Diamond Norm Q.dnorm() Returns the diamond norm.
Eigenenergies Q.eigenenergies() Eigenenergies (values) of operator.
Eigenstates Q.eigenstates() Returns eigenvalues and eigenvectors.
Eliminate States Q.eliminate_states(inds) Returns quantum object with states in list inds removed.
Exponential Q.expm() Matrix exponential of operator.
Extract States Q.extract_states(inds) Qobj with states listed in inds only.
Full Q.full() Returns full (not sparse) array of Q’s data.
Groundstate Q.groundstate() Eigenval & eigket of Qobj groundstate.
Matrix Element Q.matrix_element(bra,ket) Matrix element <bra|Q|ket>
Norm Q.norm() Returns L2 norm for states, trace norm for operators.
Overlap Q.overlap(state) Overlap between current Qobj and a given state.
Partial Trace Q.ptrace(sel) Partial trace returning components selected using ‘sel’ parameter.
Permute Q.permute(order) Permutes the tensor structure of a composite object in the given order.
Sine Q.sinm() Sine of quantum operator.
Sqrt Q.sqrtm() Matrix sqrt of operator.
Tidyup Q.tidyup() Removes small elements from Qobj.
Trace Q.tr() Returns trace of quantum object.
Transform Q.transform(inpt) A basis transformation defined by matrix or list of kets ‘inpt’ .
Transpose Q.trans() Transpose of quantum object.
Truncate Neg Q.trunc_neg() Truncates negative eigenvalues
Unit Q.unit() Returns normalized (unit) vector Q/Q.norm().
In [24]: basis(5, 3)
Out[24]: 
Quantum object: dims = [[5], [1]], shape = (5, 1), type = ket
Qobj data =
[[ 0.]
 [ 0.]
 [ 0.]
 [ 1.]
 [ 0.]]

In [25]: basis(5, 3).dag()
Out[25]: 
Quantum object: dims = [[1], [5]], shape = (1, 5), type = bra
Qobj data =
[[ 0.  0.  0.  1.  0.]]

In [26]: coherent_dm(5, 1)
Out[26]: 
Quantum object: dims = [[5], [5]], shape = (5, 5), type = oper, isherm = True
Qobj data =
[[ 0.36791117  0.36774407  0.26105441  0.14620658  0.08826704]
 [ 0.36774407  0.36757705  0.26093584  0.14614018  0.08822695]
 [ 0.26105441  0.26093584  0.18523331  0.10374209  0.06263061]
 [ 0.14620658  0.14614018  0.10374209  0.05810197  0.035077  ]
 [ 0.08826704  0.08822695  0.06263061  0.035077    0.0211765 ]]

In [27]: coherent_dm(5, 1).diag()
Out[27]: array([ 0.36791117,  0.36757705,  0.18523331,  0.05810197,  0.0211765 ])

In [28]: coherent_dm(5, 1).full()
Out[28]: 
array([[ 0.36791117+0.j,  0.36774407+0.j,  0.26105441+0.j,  0.14620658+0.j,
         0.08826704+0.j],
       [ 0.36774407+0.j,  0.36757705+0.j,  0.26093584+0.j,  0.14614018+0.j,
         0.08822695+0.j],
       [ 0.26105441+0.j,  0.26093584+0.j,  0.18523331+0.j,  0.10374209+0.j,
         0.06263061+0.j],
       [ 0.14620658+0.j,  0.14614018+0.j,  0.10374209+0.j,  0.05810197+0.j,
         0.03507700+0.j],
       [ 0.08826704+0.j,  0.08822695+0.j,  0.06263061+0.j,  0.03507700+0.j,
         0.02117650+0.j]])

In [29]: coherent_dm(5, 1).norm()
Out[29]: 1.0000000000000004

In [30]: coherent_dm(5, 1).sqrtm()
Out[30]: 
Quantum object: dims = [[5], [5]], shape = (5, 5), type = oper, isherm = False
Qobj data =
[[ 0.36791119 +0.00000000e+00j  0.36774406 +0.00000000e+00j
   0.26105440 +0.00000000e+00j  0.14620658 +0.00000000e+00j
   0.08826704 +0.00000000e+00j]
 [ 0.36774406 +0.00000000e+00j  0.36757705 +3.17312987e-11j
   0.26093584 +5.80360966e-11j  0.14614018 -4.24987846e-11j
   0.08822695 -2.33450385e-10j]
 [ 0.26105440 +0.00000000e+00j  0.26093584 +5.80360966e-11j
   0.18523332 +1.06147200e-10j  0.10374209 -7.77296761e-11j
   0.06263061 -4.26977453e-10j]
 [ 0.14620658 +0.00000000e+00j  0.14614018 -4.24987846e-11j
   0.10374209 -7.77296761e-11j  0.05810197 +5.69200369e-11j
   0.03507700 +3.12667872e-10j]
 [ 0.08826704 +0.00000000e+00j  0.08822695 -2.33450385e-10j
   0.06263061 -4.26977453e-10j  0.03507700 +3.12667872e-10j
   0.02117650 +1.71751818e-09j]]

In [31]: coherent_dm(5, 1).tr()
Out[31]: 1.0000000000000002

In [32]: (basis(4, 2) + basis(4, 1)).unit()
Out[32]: 
Quantum object: dims = [[4], [1]], shape = (4, 1), type = ket
Qobj data =
[[ 0.        ]
 [ 0.70710678]
 [ 0.70710678]
 [ 0.        ]]