Alternative names: Absolutely maximally entangled (AME) code.
Description
Block quantum code encoding one subsystem into an odd number \(n\) 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 [1–3] (a.k.a. \(d\)-undetermined [4] or \(d\)-maximally mixed [5]) 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 [6,7]. An AME state (a.k.a. maximally multi-partite entangled state [8,9]) 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 [11][10; 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 [12; Sec. 3.5].
Notes
See Ref. [13] and corresponding Table of AME states.\(d\)-uniform states are useful for masking quantum information [14].Quantum simulation of approximately \(d\)-uniform states is similar to that with random-state inputs in terms of Trotter error [15].Cousins
- Quantum maximum-distance-separable (MDS) code— AME states for even \(n\) are examples of quantum MDS codes with no logical qubits [2,16,17]. A family of conjectured perfect-tensor codes is quantum MDS [18].
- Combinatorial design— Combinatorial designs and \(d\)-uniform quantum states are related [19].
- Orthogonal array (OA)— Orthogonal arrays and \(d\)-uniform quantum states are related [3,20].
- Maximum distance separable (MDS) code— MDS codes can be used to obtain cluster states that are AME with minimal support [16,19,21–24].
- Modular-qudit cluster-state code— MDS codes can be used to obtain cluster states that are AME with minimal support [16,19,21–24].
- Galois-qudit RS code— AME states for even \(n\) are examples of quantum MDS codes with no logical qubits [2,16,17]. MDS RS codes can yield perfect tensors via the CSS and Hermitian constructions [18] (see also Refs. [25,26]).
- Approximate secret-sharing code— Perfect tensors are useful for quantum secret sharing and state teleportation [22,27].
- Qubit stabilizer code— The codespace of a qubit stabilizer code with pure distance \(d_{\textnormal{pure}}\) is a \((d_{\textnormal{pure}}-1)\)-uniform space.
- Neural network quantum code— Artificial intelligence can be used to find AME states [28].
- Hermitian qubit code— The sole codeword of some \([[n,0,d]]\) Hermitian codes is an AME state [29].
- Pastawski-Yoshida-Harlow-Preskill (HaPPY) code— The encoding of a HaPPy code is a holographic tensor network consisting of pentagon and hexagon perfect tensors.
Primary Hierarchy
Parents
Planar-perfect tensors are automatically perfect tensors.
Perfect-tensor code
Children
The \(((3,6,2))_{\mathbb{Z}_6}\) Euler code is an example of a non-stabilizer perfect-tensor code [30].
The \([[5,1,3]]_{\mathbb{Z}_q}\) code is a perfect-tensor code because it stems from the \([[6,0,4]]_{\mathbb{Z}_q}\) AME state [31; Thm. 13].
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Page edit log
- Victor V. Albert (2024-07-01) — most recent
Cite as:
“Perfect-tensor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/ame