# QuTIpy

QuTIpy (Quantum Theory of Information for Python; pronounced /cutiɛ paɪ/)

Quantum Theory of Information for Python; pronounced "cutie pie". A package for performing calculations with quantum states, channels and protocols. It is comparable to the QETLAB package for MATLAB/Octave.

## 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.

## Examples

Here are some simple examples.

We start by importing the package:

### Creating basis vectors

To create the qubit state $|0\rangle$, we execute the following line.

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:

Similarly,

gives the following output:

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 $|0\rangle|0\rangle$ can be created as follows:

In general, `ket(d, [j1, j2, ... , jn])`

creates the n-fold tensor product $|j_1\rangle|j_2\rangle...|j_n\rangle$ of d-dimensional basis vectors.

### Taking the partial trace

Given an operator $R_{AB}$ acting on a tensor product Hilbert space of the quantum systems `A`

and `B`

, the partial trace over `B`

can be calculated as follows:

Here, `dimA`

is the dimension of system `A`

and `dimB`

is the dimension of system `B`

. Similarly,

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:

To generate a random pure state (i.e., state vector) in `d`

dimensions:

To generate an isotropic state in `d`

dimensions:

where `p`

is the fidelity to the maximally entangled state.

Another special class of states is the Werner states:

The **Isotropic State** can be viewed as a probabilistic mixture of the Qudit Bell states, such that the state $|\phi\rangle\langle\phi|$ is prepared with probability $p$, and the states $|\phi_{z,x}\rangle\langle\phi_{z,x}|$, with $(z, x) \neq (0, 0)$, are prepared with probability $\frac{1−p} {d^2−1}$. This implies that every isotropic state is a Bell-diagonal state, that it has full rank, and that its eigenvalues are $p$ and $\frac{1−p} {d^2−1}$ (the latter with multiplicity $d^2 − 1$).

The **Werner state** $W_{AB}^{(p,d)}$, for 2 quantum systems $A$ and $B$, with $d_A = d_B = d ≥ 2$, is a mixture of projectors onto the symmetric and antisymmetric subspaces, with the relative weight $p\in [0,1]$ being the main parameter that defines the state,

for $\rho_{AB}=\rho^{W;p}_{AB}$ ,

such that $\rho^{W;p}_{AB} := p\zeta_{AB} + (1 − p)\zeta^\bot_{AB}$

where $\zeta_{AB}$ and $\zeta^\bot_{AB}$ 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:

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 :

The variable `K`

contains the Kraus operators of the channel. Then,

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.

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