From 0f2c3f9a2c32f7078c810e92de9986f84d2ec669 Mon Sep 17 00:00:00 2001 From: "Victor V. Albert" Date: Sun, 12 Jan 2025 09:41:42 -0500 Subject: [PATCH] ~ --- .../block/tensor_network/single_tensor/ame.yml | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/codes/quantum/properties/block/tensor_network/single_tensor/ame.yml b/codes/quantum/properties/block/tensor_network/single_tensor/ame.yml index cc93c3bbd..afd14194a 100644 --- a/codes/quantum/properties/block/tensor_network/single_tensor/ame.yml +++ b/codes/quantum/properties/block/tensor_network/single_tensor/ame.yml @@ -9,7 +9,7 @@ code_id: ame name: 'Perfect-tensor code' alternative_names: - - 'AME code' + - '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. @@ -17,7 +17,7 @@ description: | \begin{defterm}{Absolutely maximally entangled (AME) state} \label{topic:ame} - A state on \(n\) subsystems is \(d\)\textit{-uniform} \cite{arXiv:quant-ph/0310137,arxiv:1404.3586} (a.k.a. \(d\)-undetermined \cite{arxiv:0809.3081} or \(d\)-maximally mixed \cite{arxiv:1211.4118}) if all reduced density matrices on up to \(d\) subsystems are maximally mixed. + A state on \(n\) subsystems is \(d\)\textit{-uniform} \cite{arxiv:quant-ph/0005031,arXiv:quant-ph/0310137,arxiv:1404.3586} (a.k.a. \(d\)-undetermined \cite{arxiv:0809.3081} or \(d\)-maximally mixed \cite{arxiv:1211.4118}) 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 \hyperref[topic:quantum-weight-enumerator]{pure} \(((n,K,d))_q\) code \cite{arxiv:0704.0251,arxiv:1907.07733}. An AME state (a.k.a. maximally multi-partite entangled state \cite{arxiv:0710.2868,arxiv:1002.2592}) is a \(\lfloor n/2 \rfloor\)-uniform state, corresponding to a \hyperref[topic:quantum-weight-enumerator]{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 \textit{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 \(|A|\leq|A^{\perp}|\). @@ -41,18 +41,18 @@ relations: - code_id: quantum_mds detail: '\hyperref[topic:ame]{AME states} for even \(n\) are examples of quantum MDS codes with no logical qubits \cite{arXiv:quant-ph/0310137,arxiv:1701.03359,arxiv:1907.11253}. A family of conjectured perfect-tensor codes is quantum MDS \cite{arxiv:quant-ph/0312164}.' - - code_id: mds - detail: 'MDS codes can be used to obtain perfect-tensor codes with minimal support \cite{arxiv:1306.2536,arxiv:1506.08857,arxiv:1701.03359,arxiv:1706.08318}.' - code_id: combinatorial_design detail: 'Combinatorial designs and \(d\)-uniform quantum states are related \cite{arXiv:1506.08857}.' - code_id: orthogonal_array detail: 'Orthogonal arrays and \(d\)-uniform quantum states are related \cite{arXiv:1404.3586,arxiv:1708.05946}.' + - code_id: mds + detail: 'MDS codes can be used to obtain cluster states that are AME with minimal support \cite{manual:{A. V. Thapliyal, Multipartite maximally entangled states, minimal entanglement generating states and entropic inequalities unpublished presentation (2003).},arxiv:1306.2536,arxiv:1306.2879,arxiv:1506.08857,arxiv:1701.03359,arxiv:1706.08318}.' - code_id: qudit_cluster_state - detail: 'Since any modular-qubit stabilizer code is equivalent to a graph quantum code for \(G=\mathbb{Z}_q\) via a single-modular-qudit Clifford circuit \cite{arxiv:quant-ph/0111080} (see also \cite{arxiv:quant-ph/0308151,arxiv:quant-ph/0703112}), \hyperref[topic:ame]{stabilizer AME states} can be understood as modular-qudit cluster states \cite{arxiv:1306.2879}.' - - code_id: galois_grs - detail: 'GRS codes can yield perfect tensors via a generalized Hermitian construction \cite{arxiv:1801.09623,arxiv:1812.04057}.' + detail: 'MDS codes can be used to obtain cluster states that are AME with minimal support \cite{manual:{A. V. Thapliyal, Multipartite maximally entangled states, minimal entanglement generating states and entropic inequalities unpublished presentation (2003).},arxiv:1306.2536,arxiv:1306.2879,arxiv:1506.08857,arxiv:1701.03359,arxiv:1706.08318}.' + - code_id: galois_polynomial + detail: '\hyperref[topic:ame]{AME states} for even \(n\) are examples of quantum MDS codes with no logical qubits \cite{arXiv:quant-ph/0310137,arxiv:1701.03359,arxiv:1907.11253}. MDS RS codes can yield perfect tensors via the CSS and Hermitian constructions \cite{arxiv:quant-ph/0312164} (see also Refs. \cite{arxiv:1801.09623,arxiv:1812.04057}).' - code_id: quantum_secret_sharing - detail: 'Perfect tensors are useful for quantum secret sharing and state teleportation \cite{arxiv:1204.2289}.' + detail: 'Perfect tensors are useful for quantum secret sharing and state teleportation \cite{arxiv:1204.2289,arxiv:1306.2536}.' - code_id: qubit_stabilizer detail: 'The codespace of a qubit stabilizer code with \hyperref[topic:quantum-weight-enumerator]{pure distance} \(d_{\textnormal{pure}}\) is a \((d_{\textnormal{pure}}-1)\)-uniform space.'