Let A be a finite abelian group, B=A×A, and W=B⋊⟨τ⟩=A wr S₂, where τ swaps the two factors. For an arbitrary subgroup S≤B, consider the induced permutation module C[W/S] with its natural sheet/base tensor factorization C²⊗C[B/S]. We derive a closed formula for the maximum operator-Schmidt rank attained by the image of the complex group algebra C[W]. The formula depends only on three orbit statistics (d,p,u) describing how the annihilator Q=S⊥ intersects the fixed points and two-cycles of the swap involution. The non-swap-invariant case introduces half-orbits, encoded by u. We also prove that the same maximum is attained by unitary elements of the image algebra. This yields, in Nielsen's exact one-use coherent communication model, a symmetry-determined Hartley lower bound on the communication cost of unitaries in the class. The magnitude is necessarily at most two qubits for the present C² sheet factor; the point is the exact symmetry-forced value, not a large asymptotic bound. Supplementary files: bilingual (English/French) README and two Python verification scripts covering 131 exhaustive subgroup cases for the theorem and 390 for the unitary corollary.
Let A be a finite abelian group, B=A×A, and W=B⋊⟨τ⟩=A wr S₂, where τ swaps the two factors. For an arbitrary subgroup S≤B, consider the induced permutation module C[W/S] with its natural sheet/base tensor factorization C²⊗C[B/S]. We derive a closed formula for the maximum operator-Schmidt rank attained by the image of the complex group algebra C[W]. The formula depends only on three orbit statistics (d,p,u) describing how the annihilator Q=S⊥ intersects the fixed points and two-cycles of the swap involution. The non-swap-invariant case introduces half-orbits, encoded by u. We also prove that the same maximum is attained by unitary elements of the image algebra. This yields, in Nielsen's exact one-use coherent communication model, a symmetry-determined Hartley lower bound on the communication cost of unitaries in the class. The magnitude is necessarily at most two qubits for the present C² sheet factor; the point is the exact symmetry-forced value, not a large asymptotic bound. Supplementary files: bilingual (English/French) README and two Python verification scripts covering 131 exhaustive subgroup cases for the theorem and 390 for the unitary corollary.
The operator-Schmidt decomposition is useful in quantum information theory for quantifying the nonlocality of bipartite unitary operations. We construct a family of unitary operators on C^n tensor C^n whose operator-Schmidt decompositions are computed using the discrete Fourier transform. As a corollary, we produce unitaries on C^3 tensor C^3 with operator-Schmidt number S for every S in {1,...,9}. This corollary was unexpected, since it contradicted reasonable conjectures of Nielsen et al [Phys. Rev. A 67 (2003) 052301] based on intuition from a striking result in the two-qubit case. By the results of Dur, Vidal, and Cirac [Phys. Rev. Lett. 89 (2002) 057901 quant-ph/0112124], who also considered the two-qubit case, our result implies that there are nine equivalence classes of unitaries on C^3 tensor C^3 which are probabilistically interconvertible by (stochastic) local operations and classical communication. As another corollary, a prescription is produced for constructing maximally-entangled operators from biunimodular functions. Reversing tact, we state a generalized operator-Schmidt decomposition of the quantum Fourier transform considered as an operator C^M_1 tensor C^M_2 --> C^N_1 tensor C^N_2, with M_1 x M_2 = N_1 x N_2. This decomposition shows (by Nielsen's bound) that the communication cost of the QFT remains maximal when a net transfer of qudits is permitted. In an appendix, a canonical procedure is given for removing basis-dependence for results and proofs depending on the "magic basis" introduced in [S. Hill and W. Wootters, "Entanglement of a pair of quantum bits," Phys Rev. Lett 78 (1997) 5022-5025, quant-ph/9703041 (and quant-ph/9709029)].
The operator-Schmidt decomposition is useful in quantum information theory for quantifying the nonlocality of bipartite unitary operations. We construct a family of unitary operators on n ⊗ n whose operator-Schmidt decompositions are computed using the discrete Fourier transform. As a corollary, we produce unitaries on 3 ⊗ 3 with operator-Schmidt number S for every S ∊ {1, ..., 9}. This corollary was unexpected, since it contradicted reasonable conjectures of Nielsen et al (2003 Phys. Rev. A 67 052301) based on intuition from a striking result in the two-qubit case. By the results of Dür et al (2002 Phys. Rev. Lett. 89 057901), who also considered the two-qubit case, our result implies that there are nine equivalence classes of unitaries on 3 ⊗ 3 which are probabilistically interconvertible by (stochastic) local operations and classical communication. As another corollary, a prescription is produced for constructing maximally-entangled unitaries from biunimodular functions. Reversing tact, we state a generalized operator-Schmidt decomposition of the quantum Fourier transform considered as an operator M 1 ⊗ M 2 → N 1 ⊗ N 2 , with M 1 M 2 = N 1 N 2 . This decomposition shows (by Nielsen's bound) that the communication cost of the QFT remains maximal when a net transfer of qudits is permitted. In an appendix, a canonical procedure is given for removing basis-dependence for results and proofs depending on the 'magic basis' introduced in S Hill and W Wootters (1997 Entanglement of a pair of quantum bits Phys Rev. Lett. 78 5022–5).
Code surgery measures a logical operator of a quantum LDPC code with O(d) rounds of syndrome extraction. Lifted surgery amortises this cost on Abelian group-algebra codes by measuring an orbit of logical operators under a translation symmetry in one merged code. We extend it to two-block group-algebra codes over non-Abelian groups and ask whether non-commutativity lets one merged code measure more operators than any Abelian group of its symmetries. We classify the block-affine automorphisms of these codes and find that, measured against this full group, most apparent non-Abelian gains disappear. We prove that the gain is at most the index of a largest Abelian subgroup. We find codes with exact distance up to 13 where the gain is two, including codes in which one merged code reads out every logical qubit, and codes over products of A4, S4 and SL(2,3) with gain three for a best logical, up to [[336,26,12]]. A merged-distance lemma gives a simple condition for the gadget to preserve the code distance. In circuit-level simulations with Relay-BP decoding, at the error rates we can resolve, the non-Abelian gadget is as reliable as, or more reliable than, the Abelian gadgets it replaces within statistical error, while using two to three times fewer rounds.
Bargmann invariants constructed from a bipartite density operator and its two lifted marginals are polynomial invariants of local-unitary conjugation. We determine the precise information encoded in these invariants. Whenever one subsystem is a qubit, the ordinary marginal-word family determines the full spectrum of the partial transpose and hence decides whether the state has the positive-partial-transpose property. In 2⊗2 and 2⊗3 systems, this yields complete separability criteria. For the two-qubit system, a finite subfamily additionally separates local-unitary orbits, and a finite extension generates the polynomial invariant ring. These three tasks already diverge for qubit-qutrit states: we exhibit full-rank, locally maximally mixed states that agree on all ordinary marginal-word invariants yet have different operator-Schmidt ranks, together with a quartic correlation invariant that separates them. When both local dimensions are at least three, the analogous collapse on the locally maximally mixed sector produces isospectral pairs consisting of one separable state and one entangled state with negative partial transpose. The ordinary Bargmann algebra therefore coincides with the full local unitary invariant ring if and only if both subsystems are qubits. The missing data are geometric: they encode the placement of global eigenspaces relative to the tensor-product decomposition
We give an exact method for enumerating every codeword of weight at most W in the kernel of a block matrix over S[x]/(xM+1), where S is a commutative GF(2)-algebra with a fixed basis and M = m0·2L. The class contains quasi-cyclic codes of any index, block-circulant matrices with several block rows, generalized and bivariate bicycle codes, and two-block group-algebra codes over abelian groups with a cyclic factor of order m0·2L. The method rests on x2m+1 = (xm+1)2 in characteristic two: the kernel folds onto half the length without increasing weight, and every low-weight word is recovered exactly from its fold. The paper proves the recursion in this generality, reduces it to one representative per symmetry orbit, specifies the lift step in full, proves every implementation step and the whole algorithm correct, gives a dimension formula (k = 2 deg gcd(ru, rv, xm+1) for two-block CSS codes) and a certified-distance procedure for CSS codes, degenerate ones included, with certificates checkable by a proof assistant, and a random-code heuristic that predicts where the cost concentrates. A worked [[24,4,4]] example and a case study at length 832 are included: the method certifies d = 15 for a [[832,10]] code in about one second per sector, three separately written implementations agree, and the dimension formula explains the whole tower and proves k = 10. Code. The reference implementation, the tests and the example scripts are in the companion record doi:10.5281/zenodo.23466024. Related records. Case study paper v2.1: doi:10.5281/zenodo.23466181 (version 1.0: doi:10.5281/zenodo.23327681). Verification bundle 1.2.2: doi:10.5281/zenodo.23463892. License. CC BY 4.0. Copyright (c) 2026 Guillaume Lessard.
This work studies bipartite Schmidt rank in continuously weighted diagonal phase networks, with particular emphasis on the distinction between interaction support, parameter freedom, and higher-order algebraic dependencies. For a fixed bipartition and a full-support product input, the Schmidt rank is reduced exactly to the complex rank of a cross-cut phase kernel. In the independently weighted pairwise qubit sector, the main result gives an exact generic matching law: rgen=2ν(G×), where ν(G×) is the maximum matching number of the crossing bipartite interaction graph. The symbolic rank, maximum attainable rank, and almost-everywhere physical rank coincide. The paper then shows that the interaction support alone does not determine the generic Schmidt rank once the parameter family is constrained. For a uniformly weighted complete bipartite network Kp,q, the exact rank is rankKθ=min{p+1,q+1,ord(eiθ)}, with infinite multiplicative order interpreted in the natural way. The K2,2 example therefore exhibits a rank stratification (4→3→2) when passing from independently weighted phases to a uniform one-parameter family and then to the Clifford point. For genuine higher-order phase interactions, the cross-cut kernel is factorized through a union-coefficient matrix. This reveals a different obstruction: even with independent hyperedge weights, repeated algebraic use of the same phase parameters can force the symbolic rank below the structural matching rank. The paper proves that no such obstruction occurs with at most two crossing hyperedges and gives an explicit three-hyperedge example with rstr=4,rsym=3. Exact finite enumeration further certifies, within the stated simple crossing-hypergraph model, that five total cut vertices are the smallest size at which this structural-versus-symbolic rank mismatch occurs. The results distinguish three mechanisms controlling cross-cut Schmidt-rank capacity: pairwise interaction combinatorics, external parameter tying, and intrinsic higher-order union-algebraic dependencies. The manuscript is accompanied by exact verification code and reproducibility checks for the finite classification statements. Version: v0.2r3