Perfect-tensor code 

Also known as AME code.


Block quantum code encoding one subsystem into \(n\) subsystems whose encoding isometry is a perfect tensor. This code stems from an AME\((n,q)\) AME state, or equivalently, a \(((n+1,1,\lfloor (n+1)/2 \rfloor + 1))_{\mathbb{Z}_q}\) code.

Absolutely maximally entangled (AME) state: A state on \(n\) subsystems is \(d\)-uniform [1,2] (a.k.a. \(d\)-maximally mixed [3]) if all reduced density matrices on up to \(d\) subsystems are maximally mixed. A \(K\)-dimensional subspace of \(d-1\)-uniform states of \(q\)-dimensional subsystems is equivalent to a pure \(((n,K,d))_q\) code [4,5]. An AME state (a.k.a. maximally multi-partite entangled state [6,7]) is a \(\lfloor n/2 \rfloor\)-uniform state, corresponding to a pure \(((n,1,\lfloor n/2 \rfloor + 1))_{\mathbb{Z}_q}\) code. The rank-\(n\) tensor formed by the encoding isometry of such codes is a perfect tensor (a.k.a. multi-unitary tensor), meaning that it is proportional to an isometry for any bipartition of its indices into a set \(A\) and a complementary set \(A^{\perp}\) such that \(|A|\leq|A^{\perp}|\).

Stabilizer Galois-qudit perfect-tensor codes can be converted to AME states via established shortening/lengthening procedures [9][8; Table 1]. For example, an \([[n,0,d]]\) AME state can be converted into an \([[n-1,1,d-1]]\) perfect-tensor code by tracing over one qubit [10; Sec. 3.5].


See Ref. [11] and corresponding Table of AME states.





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“Perfect-tensor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.
@incollection{eczoo_ame, title={Perfect-tensor code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Perfect-tensor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.