Factor Code Networks
Abstract
For a quantum code defined by an isometry T:Hcode → Hphys, an operator map is a *-homomorphism φ: B(Hcode) → B(Hphys) from the space of operators on the logical Hilbert space to the space of operators on the physical Hilbert space satisfying φ(cal O) T = T cal O for every operator cal O on Hcode. This can additionally be taken to be unital φ(Icode) = Iphys if and only if dim Hcode divides dim Hphys. In this case, φ(B(Hcode) is a von Neumann factor subalgebra of B(Hphys) and we call the pair (T,φ) a {\it factor code}. A code network is a family of Hilbert spaces {cal Hi} indexed by a partially ordered set with isometries Tji:cal Hi → cal Hj for i < j satisfying Tkj Tji = Tki for i < j < k. In this paper, we ask when this can be promoted to a {\it factor code network} by defining unital operator maps φji so that (Tji, φji) is a factor code and φkj φji = φki. We show that the divisibility conditions on Hilbert space dimensions are sufficient for any code network with a tree structure but not in general. For a diamond-shaped network, we show that promotion is possible if and only if the principle angles between T42 H2 and T43 H3 all have multiplicity divisible by H1. For incomparable i1,i2 < j, we characterize when φj i1 (B(Hi1)) and φj i2 (B(Hi2)) commute, when they generate all of B(Hj), and when they intersect only on multiples of the identity. For i < j1,j2 < k, we characterize when φk j1 (B(Hj1)) ∩ φk j2 (B(Hj2)) = φk i (B(Hi)) and when these algebras together with B(Hk) form a nondegenerate commuting square.
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BibTeX
@misc{sahu2026factor,
title = {{Factor Code Networks}},
author = {Abhisek Sahu and Jeremy van der Heijden and Mark Van Raamsdonk and Rana Zibakhsh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00469},
eprint = {2610.00469},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00469},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Factor Code Networks
AU - Abhisek Sahu
AU - Jeremy van der Heijden
AU - Mark Van Raamsdonk
AU - Rana Zibakhsh
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00469
UR - https://arxiv.org/abs/2610.00469
AB - For a quantum code defined by an isometry T:H_code → H_phys, an operator map is a *-homomorphism φ: B(H_code) → B(H_phys) from the space of operators on the logical Hilbert space to the space of operators on the physical Hilbert space satisfying φ(cal O) T = T cal O for every operator cal O on H_code. This can additionally be taken to be unital φ(I_code) = I_phys if and only if dim H_code divides dim H_phys. In this case, φ(B(H_code) is a von Neumann factor subalgebra of B(H_phys) and we call the pair (T,φ) a {\it factor code}. A code network is a family of Hilbert spaces {cal H_i} indexed by a partially ordered set with isometries T_ji:cal H_i → cal H_j for i < j satisfying T_kj T_ji = T_ki for i < j < k. In this paper, we ask when this can be promoted to a {\it factor code network} by defining unital operator maps φ_ji so that (T_ji, φ_ji) is a factor code and φ_kj φ_ji = φ_ki. We show that the divisibility conditions on Hilbert space dimensions are sufficient for any code network with a tree structure but not in general. For a diamond-shaped network, we show that promotion is possible if and only if the principle angles between T_42 H_2 and T_43 H_3 all have multiplicity divisible by H_1. For incomparable i_1,i_2 < j, we characterize when φ_j i_1 (B(H_i_1)) and φ_j i_2 (B(H_i_2)) commute, when they generate all of B(H_j), and when they intersect only on multiples of the identity. For i < j_1,j_2 < k, we characterize when φ_k j_1 (B(H_j_1)) ∩ φ_k j_2 (B(H_j_2)) = φ_k i (B(H_i)) and when these algebras together with B(H_k) form a nondegenerate commuting square.
ER -