Modifying Method of Constructing Quantum Codes From Highly Entangled States

There is a connection between classical codes, highly entangled pure states (called
$k$
-uniform or absolutely maximally entangled (AME) states), and quantum error correcting codes (QECCs). This leads to a systematic method to construct stabilizer QECCs by starting from a
$k$
-uniform state or the corresponding classical code and tracing out one party at each step. We provide explicit constructions for codewords, encoding procedure and stabilizer formalism of the QECCs by describing the changes that partial traces cause on the corresponding generator matrix of the classical codes. We then modify the method to produce another set of stabilizer QECCs that encode a logical qudit into a subspace spanned by AME states. This construction produces quantum codes starting from an AME state without tracing out any party. Therefore, quantum stabilizer codes with larger codespace can be constructed.