QuTIpy 0.1.0a0

Description:

QuTIpy 0.1.0a0

QuTIpy

Quantum Theory of Information for Python
pronounced "cutie pie". A package for performing calculations with quantum states and channels. It is comparable to the QETLAB package for MATLAB / Octave.
Read more about the QuTIpy package at our GitBook.
Requirements
The code requires Python 3, and apart from the standard numpy and scipy packages, it requires cvxpy if you want to run SDPs (e.g., for the diamond norm). It requires sympy for symbolic computations.
Installation
A simple pip install from the github repository will install the package in your system.
$ pip install git+https://github.com/sumeetkhatri/QuTIpy

Examples
Here are some simple examples.
We start by importing the package:
>>> from qutipy import *
>>> from qutipy.general_functions import *

Creating basis vectors
To create the qubit state |0⟩, we execute the following line.
>>> ket(2,0)

The first argument specifies the dimension, in this case two, and the second argument is the index for the basis vector that we want. The output of the above line is the following numpy matrix object:
ndarray([[1.],
[0.]])

Similarly,
>>> ket(2,1)

gives the following output:
ndarray([[0.],
[1.]])

In general, ket(d,j), for j between 0 and d-1, generates a d-dimensional column vector (as a numpy matrix) in which the jth entry contains a one.
We can take tensor products of d-dimensional basis vectors using ket(). For example, the two-qubit state |0⟩|0⟩ can be created as follows:
>>> ket( 2, [0, 0] )

In general, ket(d, [j1, j2, ... , jn]) creates the n-fold tensor product |j1⟩|j2⟩...|jn⟩ of d-dimensional basis vectors.
Taking the partial trace
Given an operator RAB acting on a tensor product Hilbert space of the quantum systems A and B, the partial trace over B can be calculated as follows:
>>> partial_trace(R_AB, [2], [dimA, dimB])

Here, dimA is the dimension of system A and dimB is the dimension of system B. Similarly,
>>> partial_trace(R_AB, [1], [dimA, dimB])

takes the partial trace of R_AB over system A. In general, partial_trace(R,sys,dim) traces over the systems in the list sys, and dim is a list of the dimensions of all of the subsystems on which the operator R acts.
Quantum states
We can generate a random quantum state (i.e., density matrix) in d dimensions as follows:
>>> RandomDensityMatrix(d)

To generate a random pure state (i.e., state vector) in d dimensions:
>>> RandomPureState(d)

To generate an isotropic state in d dimensions:
>>> isotropic_state(p,d)

where p is the fidelity to the maximally entangled state.
Another special class of states is the Werner states:
>>> Werner_state(p,d)


The Isotropic State can be viewed as a probabilistic mixture of the Qudit Bell states, such that the state |ϕ⟩⟨ϕ| is prepared with probability p, and the states |ϕz,x⟩⟨ϕz,x|, with (z,x)≠(0,0), are prepared with probability 1−pd2−1. This implies that every isotropic state is a Bell-diagonal state, that it has full rank, and that its eigenvalues are p and 1−pd2−1 (the latter with multiplicity d2−1).


The Werner state WAB(p,d), for 2 quantum systems A and B, with dA=dB=d≥2, is a mixture of projectors onto the symmetric and antisymmetric subspaces, with the relative weight p∈[0,1] being the main parameter that defines the state, for ρAB=ρABW;p , such that ρABW;p:=pζAB+(1−p)ζAB⊥

where ζAB and ζAB⊥ are quantum states and are proportional to the projections onto the anti-symmetric and symmetric subspaces respectively.
Quantum channels
The package comes with functions for commonly-used channels such as the depolarizing channel and the amplitude damping channel. One can also create an arbitrary Qubit Pauli channel as follows:
>>> Pauli_channel(px, py, pz)

where px, py, pz are the probabilities of the individual Pauli Matrices. The output of this function contains the Kraus operators of the channel as well as an isometric extension of the channel.
In order to apply a quantum channel to a quantum state rho, we can use the function apply_channel. First, let us define the following amplitude damping channel :
>>> K = amplitude_damping_channel(0.2)

The variable K contains the Kraus operators of the channel. Then,
>>> rho_out = apply_channel(K, rho)

gives the state at the output of the channel when the input state is rho.
Other functions include:

Getting the Choi and natural representation of a channel from its Kraus representation
Converting between the Choi, natural, and Kraus representations of a channel

Summary of other features
The package also contains functions for:

Trace norm
Fidelity and entanglement fidelity
Random unitaries
Clifford unitaries
Generators of the su(d) Lie algebra(for d=2, this is the set of Pauli matrices)
Discrete Weyl operators
von Neumann entropy and relative entropy
Renyi entropies
Coherent information and Holevo information for states and channels

Acknowledgements
Thanks to Mark Wilde for suggesting the name for the package.