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. \(d\)-undetermined [3] or \(d\)-maximally mixed [4]) 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 [5,6]. An AME state (a.k.a. maximally multi-partite entangled state [7,8]) 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 [10][9; 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 [11; Sec. 3.5].
Notes
Parent
- Planar-perfect-tensor code — Planar-perfect tensors are automatically perfect tensors.
Children
- \(((3,6,2))_{\mathbb{Z}_6}\) Euler code — The \(((3,6,2))_{\mathbb{Z}_6}\) Euler code is an example of a non-stabilizer perfect-tensor code [13].
- \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code — 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 [14; Thm. 13].
- Three-qutrit code — The three-qutrit code stems from the \([[4,0,3]]_3\) AME state [15–17].
Cousins
- Quantum maximum-distance-separable (MDS) code — AME states for even \(n\) are examples of quantum MDS codes with no logical qubits [1,18]. A family of conjectured perfect-tensor codes is quantum MDS [19].
- Maximum distance separable (MDS) code — MDS codes can be used to obtain perfect-tensor codes with minimal support [16,18,20].
- Combinatorial design — Combinatorial designs and \(d\)-uniform quantum states are related [16].
- Orthogonal array (OA) — Orthogonal arrays and \(d\)-uniform quantum states are related [2,21].
- Modular-qudit cluster-state code — 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 [22] (see also [23,24]), stabilizer AME states can be understood as modular-qudit cluster states [15].
- Galois-qudit GRS code — GRS codes can yield perfect tensors via a generalized Hermitian construction [25,26].
- Hermitian qubit code — The sole codeword of some \([[n,0,d]]\) Hermitian codes is an AME state [17].
- Pastawski-Yoshida-Harlow-Preskill (HaPPY) code — The encoding of a HaPPy code is a holographic tensor network consisting of pentagon and hexagon perfect tensors.
References
- [1]
- 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
- [2]
- D. Goyeneche and K. Życzkowski, “Genuinely multipartite entangled states and orthogonal arrays”, Physical Review A 90, (2014) arXiv:1404.3586 DOI
- [3]
- M. H. Hsieh, W. T. Yen, and L. Y. Hsu, “Undetermined states: how to find them and their applications”, The European Physical Journal D 61, 261 (2010) arXiv:0809.3081 DOI
- [4]
- L. Arnaud and N. J. Cerf, “Exploring pure quantum states with maximally mixed reductions”, Physical Review A 87, (2013) arXiv:1211.4118 DOI
- [5]
- G. Gour and N. R. Wallach, “Entanglement of subspaces and error-correcting codes”, Physical Review A 76, (2007) arXiv:0704.0251 DOI
- [6]
- F. Huber and M. Grassl, “Quantum Codes of Maximal Distance and Highly Entangled Subspaces”, Quantum 4, 284 (2020) arXiv:1907.07733 DOI
- [7]
- P. Facchi, G. Florio, G. Parisi, and S. Pascazio, “Maximally multipartite entangled states”, Physical Review A 77, (2008) arXiv:0710.2868 DOI
- [8]
- P. Facchi, G. Florio, U. Marzolino, G. Parisi, and S. Pascazio, “Classical statistical mechanics approach to multipartite entanglement”, Journal of Physics A: Mathematical and Theoretical 43, 225303 (2010) arXiv:1002.2592 DOI
- [9]
- A. Ketkar, A. Klappenecker, S. Kumar, and P. K. Sarvepalli, “Nonbinary stabilizer codes over finite fields”, (2005) arXiv:quant-ph/0508070
- [10]
- M. Grassl and M. Rotteler, “Quantum MDS codes over small fields”, 2015 IEEE International Symposium on Information Theory (ISIT) 1104 (2015) arXiv:1502.05267 DOI
- [11]
- D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
- [12]
- F. Huber, C. Eltschka, J. Siewert, and O. Gühne, “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
- [13]
- S. A. Rather, A. Burchardt, W. Bruzda, G. Rajchel-Mieldzioć, A. Lakshminarayan, and K. Życzkowski, “Thirty-six Entangled Officers of Euler: Quantum Solution to a Classically Impossible Problem”, Physical Review Letters 128, (2022) arXiv:2104.05122 DOI
- [14]
- E. M. Rains, “Nonbinary quantum codes”, (1997) arXiv:quant-ph/9703048
- [15]
- W. Helwig, “Absolutely Maximally Entangled Qudit Graph States”, (2013) arXiv:1306.2879
- [16]
- D. Goyeneche, D. Alsina, J. I. Latorre, A. Riera, and K. Życzkowski, “Absolutely maximally entangled states, combinatorial designs, and multiunitary matrices”, Physical Review A 92, (2015) arXiv:1506.08857 DOI
- [17]
- Z. Raissi, “Modifying Method of Constructing Quantum Codes From Highly Entangled States”, IEEE Access 8, 222439 (2020) arXiv:2005.01426 DOI
- [18]
- Z. Raissi, C. Gogolin, A. Riera, and A. Acín, “Optimal quantum error correcting codes from absolutely maximally entangled states”, Journal of Physics A: Mathematical and Theoretical 51, 075301 (2018) arXiv:1701.03359 DOI
- [19]
- M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004) arXiv:quant-ph/0312164 DOI
- [20]
- W. Helwig and W. Cui, “Absolutely Maximally Entangled States: Existence and Applications”, (2013) arXiv:1306.2536
- [21]
- D. Goyeneche, Z. Raissi, S. Di Martino, and K. Życzkowski, “Entanglement and quantum combinatorial designs”, Physical Review A 97, (2018) arXiv:1708.05946 DOI
- [22]
- D. Schlingemann, “Stabilizer codes can be realized as graph codes”, (2001) arXiv:quant-ph/0111080
- [23]
- 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
- [24]
- 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
- [25]
- M. Marcolli, “Holographic Codes on Bruhat--Tits buildings and Drinfeld Symmetric Spaces”, (2018) arXiv:1801.09623
- [26]
- M. Heydeman, M. Marcolli, S. Parikh, and I. Saberi, “Nonarchimedean Holographic Entropy from Networks of Perfect Tensors”, (2018) arXiv:1812.04057
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