Perfect-tensor code 

Also known as AME code.

Description

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].

Notes

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

Parent

Children

Cousins

References

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A. J. Scott, “Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions”, Physical Review A 69, (2004) arXiv:quant-ph/0310137 DOI
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[3]
L. Arnaud and N. J. Cerf, “Exploring pure quantum states with maximally mixed reductions”, Physical Review A 87, (2013) arXiv:1211.4118 DOI
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P. Facchi et al., “Classical statistical mechanics approach to multipartite entanglement”, Journal of Physics A: Mathematical and Theoretical 43, 225303 (2010) arXiv:1002.2592 DOI
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A. Ketkar et al., “Nonbinary stabilizer codes over finite fields”, (2005) arXiv:quant-ph/0508070
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M. Grassl and M. Rotteler, “Quantum MDS codes over small fields”, 2015 IEEE International Symposium on Information Theory (ISIT) (2015) arXiv:1502.05267 DOI
[10]
D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
[11]
F. Huber et al., “Bounds on absolutely maximally entangled states from shadow inequalities, and the quantum MacWilliams identity”, Journal of Physics A: Mathematical and Theoretical 51, 175301 (2018) arXiv:1708.06298 DOI
[12]
S. A. Rather et al., “Thirty-six Entangled Officers of Euler: Quantum Solution to a Classically Impossible Problem”, Physical Review Letters 128, (2022) arXiv:2104.05122 DOI
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W. Helwig, “Absolutely Maximally Entangled Qudit Graph States”, (2013) arXiv:1306.2879
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[16]
Z. Raissi, “Modifying Method of Constructing Quantum Codes From Highly Entangled States”, IEEE Access 8, 222439 (2020) arXiv:2005.01426 DOI
[17]
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M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004) arXiv:quant-ph/0312164 DOI
[19]
W. Helwig and W. Cui, “Absolutely Maximally Entangled States: Existence and Applications”, (2013) arXiv:1306.2536
[20]
D. Goyeneche et al., “Entanglement and quantum combinatorial designs”, Physical Review A 97, (2018) arXiv:1708.05946 DOI
[21]
D. Schlingemann, “Stabilizer codes can be realized as graph codes”, (2001) arXiv:quant-ph/0111080
[22]
M. Van den Nest, J. Dehaene, and B. De Moor, “Graphical description of the action of local Clifford transformations on graph states”, Physical Review A 69, (2004) arXiv:quant-ph/0308151 DOI
[23]
M. Grassl, A. Klappenecker, and M. Rotteler, “Graphs, quadratic forms, and quantum codes”, Proceedings IEEE International Symposium on Information Theory, arXiv:quant-ph/0703112 DOI
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Zoo Code ID: ame

Cite as:
“Perfect-tensor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/ame
BibTeX:
@incollection{eczoo_ame, title={Perfect-tensor code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/ame} }
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“Perfect-tensor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/ame

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/block/tensor_network/single_tensor/ame.yml.