Multiqubit orthogonal product bases

Abstract: We study complete orthogonal product bases (OPBs) of an $n$-qubit system via the formal-matrix formalism for multiqubit OPBs. To each OPB formal matrix we associate an edge-colored complete multigraph, and we prove that two formal matrices are equivalent if and only if their multigraphs are isomorphic, thereby reducing the classification of OPBs to graph isomorphism. Via subcube partitions and a lemma of Tarsi, we upper bound the number of variables by $2^n - 1$. We also study lower and upper bounds on the number $a_n$ of equivalence classes of $n$-qubit OPBs, by showing that $$\binom{a_{n-1}+1}{2} \le a_n \le B_{2<sup>{n-1}}<sup>n$$, where $B_m$ denotes the number of partitions of an $m$-element set. These bounds yield the asymptotic behavior $a_n = 2<sup>{2<sup>{n+o(n)}}$. Finally, we obtain a two-phase algorithm that decides whether two OPB formal matrices are equivalent, together with its correctness proof and complexity analysis. The complexity is exponential in the worst case.

• The paper reduces equivalence testing for complete multiqubit orthogonal product bases to isomorphism of edge-colored complete multigraphs, recovering variable pairs and direction flips from graph components.
• It proves structural and enumerative results, including the variable bound v ≤ 2^n − 1, the exact value a₄ = 27,385, and doubly exponential growth aₙ = 2^{2^{n+o(n)}} in the number of equivalence classes.
• It introduces a two-phase exact algorithm that combines fast invariant checks with structure-aware backtracking, avoiding enumeration of all 2^n! row permutations while retaining worst-case exponential complexity.

Overview

This paper develops a combinatorial classification theory for complete orthogonal product bases (OPBs) of an $n$-qubit system, building on the formal-matrix framework of Chen and Đoković. The central contribution is a reduction of OPB equivalence to graph isomorphism: each OPB formal matrix is encoded as an edge-colored complete multigraph, and two formal matrices are equivalent exactly when their multigraphs are isomorphic. On top of this encoding, the paper derives structural constraints (balancedness, a bound on the number of variables), quantitative bounds on the number of equivalence classes with doubly exponential asymptotics, and an exact two-phase algorithm for testing equivalence.

The motivation comes from quantum information: although OPBs contain no entanglement, they can exhibit "nonlocality without entanglement," failing to be perfectly distinguishable by LOCC, and they are closely tied to unextendible product bases (UPBs), bound entanglement, quantum information hiding, and secret sharing (2608.18421). A systematic enumeration of essentially different OPBs is therefore a foundational prerequisite for further study of these phenomena.

Formal matrices and the edge-colored multigraph encoding

An $n$-qubit OPB formal matrix is an $N \times n$ array over an alphabet of variable pairs $\{x_\alpha, x_\alpha^\perp\}$, where $N = 2^n$, such that each row contains $n$ distinct letters, any two rows are orthogonal in some column via complementary letters, and the rows are realizable as an orthonormal product basis of $(\mathbb{C}^2)^{\otimes n}$. Equivalence permits row permutations, column permutations, variable renamings, and direction flips—precisely the transformations that leave the physical content of the basis unchanged.

The graph construction assigns to each column $j$ a color layer $\Gamma_j(M)$, the disjoint union of balanced complete bipartite graphs $K(P_\alpha, N_\alpha)$, one per variable pair occurring in that column; merging the $n$0 layers yields an edge-colored complete multigraph whose edges cover exactly those of $n$1. Crucially, variable renamings and direction flips induce no graph-level change at all: a flip merely swaps the two parts of a connected bipartite component, whose bipartition is unique up to swapping. This observation underlies the equivalence theorem:

$n$2

with equivalence of graphs defined via vertex and color permutations. The forward direction is immediate from the correspondence between operations; the converse recovers variable pairs as connected components of each layer and direction choices from the uniqueness of bipartitions. This theorem reduces the classification of multiqubit OPBs to isomorphism testing within this structured graph family.

Balancedness and the variable bound

Two structural results constrain all OPB formal matrices. First, the balancedness lemma shows that whenever a variable pair occurs in a column, its two symbols appear equally often: $n$3. The proof is a dimension count: truncating rows containing $n$4 or $n$5 yields orthogonal subspaces $n$6 and $n$7 of dimensions $n$8 and $n$9, and every remaining row must have its truncated factor in $N \times n$0, forcing $N \times n$1 with $N \times n$2; combining with $N \times n$3 gives equality throughout, hence $N \times n$4.

Second, the number $N \times n$5 of free variables satisfies $N \times n$6. Orienting each variable pair maps row $N \times n$7 to a partial assignment fixing exactly $N \times n$8 coordinates, hence to a codimension-$N \times n$9 subcube of $\{x_\alpha, x_\alpha^\perp\}$0. Because the color layers cover $\{x_\alpha, x_\alpha^\perp\}$1, these subcubes are pairwise disjoint and partition $\{x_\alpha, x_\alpha^\perp\}$2; since every variable occurs, the partition is tight and minimal, so Tarsi's lemma ($\{x_\alpha, x_\alpha^\perp\}$3 subcubes in a tight minimal cover of $\{x_\alpha, x_\alpha^\perp\}$4) yields $\{x_\alpha, x_\alpha^\perp\}$5. The paper notes that an equivalent bound was previously known via color forests of admissible hypercube colorings; the contribution here is a proof internal to the multigraph language.

Counting layers and bounds on the number of classes

For a single color layer with spectrum $\{x_\alpha, x_\alpha^\perp\}$6, the paper obtains the exact closed form

$\{x_\alpha, x_\alpha^\perp\}$7

together with a convolution recurrence for the total number $\{x_\alpha, x_\alpha^\perp\}$8 of labeled layers, whose first values are $\{x_\alpha, x_\alpha^\perp\}$9, $N = 2^n$0, $N = 2^n$1, $N = 2^n$2, $N = 2^n$3.

Letting $N = 2^n$4 denote the number of equivalence classes of $N = 2^n$5-qubit OPBs, the paper establishes:

$N = 2^n$6

where $N = 2^n$7 is the $N = 2^n$8-th Bell number. The lower bound comes from stacking two $N = 2^n$9-qubit OPBs under a new variable pair; inequivalence follows because the first layer is the unique one containing a $n$0 component, so any isomorphism must preserve it and recover the unordered pair $n$1. Iterating from the exact value $n$2 gives $n$3.

The upper bound is the more delicate argument. Coarsening each layer into a single balanced cut $n$4, the side markings $n$5 turn out to be injective—and hence bijective—on $n$6. Consequently each coarsened layer forces a perfect matching $n$7 of $n$8 edges, and every forced matching edge lies in the original layer. The variable branches then partition $n$9, and conversely any set partition of $(\mathbb{C}^2)^{\otimes n}$0 determines a valid layer structure, giving at most $(\mathbb{C}^2)^{\otimes n}$1 combinations. A three-qubit example illustrates the mechanism concretely: with $(\mathbb{C}^2)^{\otimes n}$2, each layer admits at most $(\mathbb{C}^2)^{\otimes n}$3 branch structures.

Combining both bounds with de Bruijn's asymptotics $(\mathbb{C}^2)^{\otimes n}$4 yields

$(\mathbb{C}^2)^{\otimes n}$5

i.e., the number of essentially different multiqubit OPBs grows doubly exponentially. Numerically, for $(\mathbb{C}^2)^{\otimes n}$6 the lower bound is already about $(\mathbb{C}^2)^{\otimes n}$7 while the upper bound reaches $(\mathbb{C}^2)^{\otimes n}$8; exact values are known only through $(\mathbb{C}^2)^{\otimes n}$9. The gap between the exponents' second-order terms ($j$0 versus $j$1) remains substantial.

Equivalence-testing algorithm

The algorithm exploits the layered structure directly rather than delegating to generic graph-isomorphism solvers, which do not exploit the fact that each layer is a disjoint union of balanced complete bipartite blocks—a relevant concern since the vertex count $j$2 grows quickly.

Phase 1 compares necessary invariants: size, number of variables, global spectrum, multiset of column spectra, multiset of pairwise edge multiplicities, and multiset of vertex degrees. Failure of any invariant certifies inequivalence; agreement does not suffice.

Phase 2 performs a backtracking search over layer mappings, within-layer component mappings, and flip indicators—the only three branching ingredients, per the equivalence theorem. Components are matched only when sizes and profiles relative to current row classes agree (up to flipping), and after each mapped layer the row classes are refined; inconsistent refinements prune the branch. Column selection uses the minimum-remaining-values heuristic. Once all layers are matched, pairing source rows with target rows of equal class produces the row permutation implicitly, avoiding any enumeration of the $j$3 row permutations.

Correctness is proved in both directions: soundness because a completed search assembles exactly the operations of Definition of equivalence; completeness because the true equivalence always constitutes a feasible branch that no refinement check rejects. Phase 1 runs in time polynomial in the input size, dominated by $j$4 for edge multiplicities; Phase 2's cost is proportional to visited branches times near-polynomial per-branch work. The worst case remains exponential—for highly symmetric instances with many identical column spectra and weakly distinguishing profiles—but the layered pruning is substantially stronger than brute force. Space complexity is $j$5.

Limitations and open questions

The paper is explicit about several constraints on its results. The upper bound $j$6 discards illegal partition combinations and quotienting by isomorphisms only crudely, so it is far from tight; correspondingly, the asymptotic statement fixes only the leading behavior $j$7, leaving the second-order term undetermined. Exact values of $j$8 are known only for $j$9, so the recursive lower bound rests on the external computation $\Gamma_j(M)$0. The algorithm's worst-case exponential behavior on highly symmetric instances is conceded rather than resolved, and no comparison against general-purpose isomorphism tools such as nauty or Traces is carried out. Two problems are stated as open: tightening the bounds on $\Gamma_j(M)$1, and constructing an effective algorithm that enumerates all equivalence classes for a given $\Gamma_j(M)$2, which would extend exact classifications beyond four qubits.

Conclusion

The paper converts the classification of multiqubit OPBs into a well-defined graph-isomorphism problem via edge-colored complete multigraphs, proves the resulting equivalence theorem, rederives the variable bound $\Gamma_j(M)$3 through subcube partitions and Tarsi's lemma, and establishes doubly exponential growth $\Gamma_j(M)$4 for the number of equivalence classes. The accompanying two-phase algorithm decides equivalence exactly while avoiding row-permutation enumeration, at the price of worst-case exponential search. The principal unresolved issues are the gap between the established lower and upper bounds on $\Gamma_j(M)$5 and the absence of a practical enumeration procedure for complete classifications at larger qubit numbers.