About

This is the Quantum Channel Zoo, a website collecting useful mathematical and information-theoretic facts about various quantum channels. It was created in an IGL undergraduate research project conducted in the 2023 Fall term at the University of Illinois Urbana-Champaign. The IGL team consisted of undergraduate students Ben Booker, Tianshun Gao, Anne Que, Yuxuan Wan and Lumi Xu, graduate student Sujeet Bhalerao and faculty advisor Felix Leditzky. The website is created using ZooDb.

How to contribute

The zookeepers need your help keeping this database up-to-date! For more information, please consult the `README.md` file in our Github repository.

How to use this website

The Homepage currently lists all channels in the database. Each channel webpage has a short description of the channel, the channel dimensions, and different channel representations (if available), including Kraus representation, isometry, and Choi state. We will soon add mathematical and information-theoretic properties, such as whether a channel is entanglement-breaking, or what we know about its quantum capacity.

Mathematical conventions

We use the following mathematical conventions on this website (see Mark Wilde's textbook for explanations of the notation used). A channel $\mathcal{N}\colon A\to B$ maps operators on an input Hilbert space $\mathcal{H}_A$ to operators on an output Hilbert space $\mathcal{H}_B$. Then there exists an environment space $\mathcal{H}_E$ of dimension $d_E = \dim\mathcal{H}_E$ such that the channel action can be written as follows: $$\mathcal{N}(X_A) = \sum_{i=0}^{d_E-1} K_i X_A K_i^\dagger = \text{tr}_E VX_AV^\dagger = \text{tr}_A\left[\tau_{AB} \left(X_A^T\otimes \mathbf{1}_B\right)\right].$$ In the above, we have used the following objects:

• the Kraus operators $K_i\colon \mathcal{H}_A \to \mathcal{H}_B$ satisfy $$\sum_{i=0}^{d_E-1} K_i^\dagger K_i = \mathbf{1}_A,$$ where $\mathbf{1}_A$ denotes the identity operator on $\mathcal{H}_A$.
• The channel isometry $V\colon \mathcal{H}_A \to \mathcal{H}_B \otimes \mathcal{H}_E$ satisfies $V^\dagger V = \mathbf{1}_A$.
• The Choi operator $\tau_{AB}$ of $\mathcal{N}\colon A\to B$ is defined as $$\tau_{AB} = (\text{id}_A\otimes \mathcal{N})(|\gamma\rangle\langle\gamma|_{AA'}),$$ where $$|\gamma\rangle_{AA'} = \sum_{i=0}^{|A|-1} |i\rangle_A\otimes |i\rangle_{A'}$$ is an unnormalized maximally entangled state on $AA'$ defined in terms of an orthonormal basis $\lbrace |i\rangle_A\rbrace_{i=0}^{|A|-1}$ of $\mathcal{H}_A $. Here, we used the notation $|A| = \dim\mathcal{H}_A$. The Choi operator of a quantum channel is a positive semidefinite operator satisfying $\tau_A = \text{tr}_B\tau_{AB} = \mathbf{1}_A$.

For information on how to derive these three channel representations and how to switch between them, we refer to Mark Wilde's textbook or Felix Leditzky's lecture notes on quantum channels.

About ZooDB

The ZooDb database is used to store a database of people with parent/child and friend relationships. The source data providing the content of the database is a collection of YAML files. YAML is a common and useful markup language to store structured data. You can google "YAML tutorial" or check out the language's Wikipedia page.

The database object schemas are in schemas/. The schemas define what exactly is being stored in the database.

The data, provided in YAML files, is in the folder data/.

Minimal JS code to load the people DB in memory is in peopledbjs/. If you want to produce any different form of output, e.g., a handbook or print form of the database, then you can use this code to load the database in memory and use it to generate the required output.

To build the website: refer to the README file in the folder website/ folder. The website is powered by 11ty and parcel.

See also the documentation for the ZooDB package.