|toqito⟩: Theory of Quantum Information Toolkit

The |toqito⟩ package is an open-source Python library for studying various
objects in quantum information, namely, states, channels, and measurements.

<p align="center"> <a href="https://vprusso.github.io/toqito/"> <img width=30% src="https://img.shields.io/badge/documentation-blue?style=for-the-badge&logo=material%20for%20mkdocs" alt="Documentation" /> </a> </p>

|toqito⟩ focuses on providing numerical tools to study problems
about entanglement theory, nonlocal games, matrix analysis, and other
aspects of quantum information that are often associated with computer science.

|toqito⟩ aims to fill the needs of quantum information researchers who want
numerical and computational tools for manipulating quantum states,
measurements, and channels. It can also be used as a tool to enhance the
experience of students and instructors in classes about quantum
information.

Installing

|toqito⟩ is available via PyPi for Linux and macOS, with support for Python 3.11
to 3.14.

sh
pip install toqito

Examples

For the full documentation, please consult: Documentation | Examples

Example: Nonlocal games

Nonlocal games are a mathematical framework
that abstractly models a physical system. The CHSH game is a subtype of nonlocal game referred to as an XOR game that
characterizes the seminal CHSH inequality.

For XOR games, there exist optimization problems (that are provided via |toqito⟩) that one can compute to attain the
optimal values of such games when the players use either a classical or quantum strategy.

python
# Calculate the classical and quantum value of the CHSH game.
import numpy as np
from toqito.nonlocal_games.xor_game import XORGame

# The probability matrix.
prob_mat = np.array([[1/4, 1/4], [1/4, 1/4]])

# The predicate matrix.
pred_mat = np.array([[0, 0], [0, 1]])

# Define CHSH game from matrices.
chsh = XORGame(prob_mat, pred_mat)

chsh.classical_value()
# 0.75
chsh.quantum_value()
# 0.8535533

Indeed, using a quantum strategy for the CHSH game gives the known optimal result of $\frac{1}{4}\left(2 +
\sqrt{2}\right) \approx 0.8535...$

Example: Quantum state distinguishability

Quantum state distinguishability is a fundamental task in quantum information theory. Consider the set of four Bell
states:

$$
\begin{equation}
\begin{aligned}
|\psi_0\rangle = \frac{1}{\sqrt{2}} \left(|00\rangle + |11\rangle\right), \quad
|\psi_1\rangle = \frac{1}{\sqrt{2}} \left(|00\rangle - |11\rangle\right), \
|\psi_2\rangle = \frac{1}{\sqrt{2}} \left(|01\rangle + |10\rangle\right), \quad
|\psi_3\rangle = \frac{1}{\sqrt{2}} \left(|01\rangle - |10\rangle\right).
\end{aligned}
\end{equation}
$$

The optimal probability of globally distinguishing the four Bell states (assuming an equal weighting of probability) is
1 (i.e., it can be performed perfectly). However, under a more restrictive set of measurements (such as PPT measurement
operators), the optimal probability of distinguishing the four Bell states using PPT operators is 1/2.

|toqito⟩ offers a wide suite of functionality for computing the distinguishability of quantum states:

python
from toqito.states import bell
from toqito.state_opt import state_distinguishability, ppt_distinguishability

# Define the set of states as the four Bell states:
states = [bell(0), bell(1), bell(2), bell(3)]

# Distinguishing four Bell states (global measurements): 0.9999999999767388
pos_res, _ = state_distinguishability(states)
print(f"Distinguishing four Bell states (global measurements): {pos_res}")

# Distinguishing four Bell states (PPT measurements): 0.5000000000098367
ppt_res, _ = ppt_distinguishability(states, subsystems=[0], dimensions=[2, 2])
print(f"Distinguishing four Bell states (PPT measurements): {ppt_res}")

Consult the quantum state tutorials for
additional examples and information about quantum states within the context of |toqito⟩.

Testing

The pytest module is used for testing. To run the suite of tests for |toqito⟩,
run the following command in the root directory of this project.

pytest --cov-report term-missing --cov=toqito

Citing

You can cite |toqito⟩ using the following DOI:
10.5281/zenodo.4743211

If you are using the |toqito⟩ software package in research work, please include
an explicit mention of |toqito⟩ in your publication. Something along the lines
of:

To solve problem "X", we used |toqito⟩; a package for studying certain
aspects of quantum information.

A BibTeX entry that you can use to cite |toqito⟩ is provided here:

bib
@misc{toqito,
   author       = {Vincent Russo},
   title        = {toqito: A {P}ython toolkit for quantum information},
   howpublished = {\url{https://github.com/vprusso/toqito}},
   month        = May,
   year         = 2021,
   doi          = {10.5281/zenodo.4743211}
 }

Research

The |toqito⟩ project is, first and foremost, a quantum information theory research tool. Consult the following open
problems wiki
page
for a list of certain solved and unsolved problems in quantum information theory in which |toqito⟩ could be potentially
helpful in probing. Feel free to add to this list and/or contribute solutions!

The |toqito⟩ project has been used or referenced in the following works:

Johnston, Nathaniel and Russo, Vincent
"Distinguishability of locally diagonal orthogonally invariant quantum states", arXiv:2606.12808, (2026).
Gupta, Tathagata and Mohan, Ankith and Murshid, Shayeef and Russo, Vincent and Sikora, Jamie and Zeng, Alice
"Local strategies are pretty good at computing Boolean properties of quantum sequences", arXiv:2603.05452, (2026).
Arun, Venkat, Vijay Chidambaram, and Scott Aaronson
"Faster-than-light coordination for networked systems with quantum non-local games", Proceedings of the 24th ACM Workshop on Hot Topics in Networks, (2025).
Lovitz, Benjamin and Johnston, Nathaniel and Russo, Vincent and Sikora, Jamie
"The complexity of perfect quantum state classification", arXiv:2510.20789, (2025).
Johnston, Nathaniel and Russo, Vincent and Sikora, Jamie
"Tight bounds for antidistinguishability and circulant sets of pure quantum states", Quantum 9, 1622, (2025).
Philip, Aby
"On Multipartite Entanglement and Its Use", (2024).
Almasi, Ali
"Quantum Guessing Games", (2024).
Gupta, Tathagata and Mushid, Shayeef and Russo, Vincent and Bandyopadhyay, Somshubhro
"Optimal discrimination of quantum sequences", Physical Review A, 110, 062426, (2024).
Bandyopadhyay, Somshubhro and Russo, Vincent
"Distinguishing a maximally entangled basis using LOCC and shared entanglement", Physical Review A 110, 042406, (2024).
Tavakoli, Armin and Pozas-Kerstjens, Alejandro and Brown, Peter and Araújo, Mateus
"Semidefinite programming relaxations for quantum correlations", Reviews of Modern Physics, Volume 96, (2024).
Pelofske, Elijah and Bartschi, Andreas and Eidenbenz, Stephan and Garcia, Bryan and Kiefer, Boris
"Probing Quantum Telecloning on Superconducting Quantum Processors", IEEE Transactions on Quantum Engineering, (2024).
Philip, Aby and Rethinasamy, Soorya and Russo, Vincent and Wilde, Mark.
"Quantum Steering Algorithm for Estimating Fidelity of Separability", Quantum 8, 1366, (2023).
Miszczak, Jarosław Adam.
"Symbolic quantum programming for supporting applications of quantum computing technologies", Companion Proceedings of the 7th International Conference on the Art, Science, and Engineering of Programming, (2023).
Casalé, Balthazar and Di Molfetta, Giuseppe and Anthoine, Sandrine and Kadri, Hachem.
"Large-Scale Quantum Separability Through a Reproducible Machine Learning Lens", (2023).
Russo, Vincent and Sikora, Jamie "Inner products of pure states and their antidistinguishability", Physical Review A, Vol. 107, No. 3, (2023).