Quantum Homeomorphisms in Quantum Hypergraphs

• Quantum homeomorphisms are structure-preserving maps between quantum hypergraphs that extend classical homomorphism concepts to the non-commutative setting.
• They utilize t-homomorphisms and perfect QNS correlation strategies to establish well-defined quantum channels under various operational resources.
• Their study reveals deep connections with operator space theory and TRO equivalence, offering a categorical framework that bridges quantum and classical structures.

A quantum homeomorphism refers to a family of structure-preserving maps—termed “t-homomorphisms” for various choices of operational context—between quantum hypergraphs. Quantum hypergraphs generalize classical hypergraphs to the non-commutative setting by encoding hyperedges as subspaces of linear operators between finite-dimensional Hilbert spaces. Quantum homeomorphisms are fundamentally characterized using the existence of perfect strategies for associated quantum non-local games; their structure and classification reveal parallels and distinctions with classical homomorphic concepts and operator space theory, particularly in the context of ternary rings of operators (TRO) equivalence (Hoefer et al., 2023).

1. Quantum Hypergraphs and Operator Spaces

A quantum hypergraph over finite sets $X,Y$ is defined as a linear subspace $U\subseteq L(\mathbb{C}^{X},\mathbb{C}^{Y})$, where $L(\mathbb{C}^{X},\mathbb{C}^{Y})$ denotes the space of linear operators from $\mathbb{C}^{X}$ to $\mathbb{C}^{Y}$. Elements $u\in U$ are regarded as quantum edges. The classical case arises as the special case where $U$ is invariant under left and right multiplication by diagonal algebras $D_Y$ and $D_X$ (the diagonal subalgebras of matrix algebras $M_X$ and $U\subseteq L(\mathbb{C}^{X},\mathbb{C}^{Y})$0). In this scenario, for a classical edge set $U\subseteq L(\mathbb{C}^{X},\mathbb{C}^{Y})$1, the corresponding quantum hypergraph is $U\subseteq L(\mathbb{C}^{X},\mathbb{C}^{Y})$2.

Operator-system-theoretic structures are instantiated by sets such as $U\subseteq L(\mathbb{C}^{X},\mathbb{C}^{Y})$3, which comprises matrices in $U\subseteq L(\mathbb{C}^{X},\mathbb{C}^{Y})$4 commuting with $U\subseteq L(\mathbb{C}^{X},\mathbb{C}^{Y})$5 and $U\subseteq L(\mathbb{C}^{X},\mathbb{C}^{Y})$6. These settings underpin the formalism of quantum no-signalling (QNS) correlations used in defining t-homomorphisms, with tensor products taken with respect to the minimal (i.e., “operator-system”) norm.

2. Canonical Non-local Games and Correlation Classes

Given two quantum hypergraphs $U\subseteq L(\mathbb{C}^{X},\mathbb{C}^{Y})$7 and $U\subseteq L(\mathbb{C}^{X},\mathbb{C}^{Y})$8, the associated non-local game $U\subseteq L(\mathbb{C}^{X},\mathbb{C}^{Y})$9 establishes a test for t-homomorphic relations. Alice and Bob receive indices from $L(\mathbb{C}^{X},\mathbb{C}^{Y})$0 and $L(\mathbb{C}^{X},\mathbb{C}^{Y})$1 respectively and must output elements in $L(\mathbb{C}^{X},\mathbb{C}^{Y})$2 and $L(\mathbb{C}^{X},\mathbb{C}^{Y})$3. A correlation $L(\mathbb{C}^{X},\mathbb{C}^{Y})$4 is said to provide a perfect type-$L(\mathbb{C}^{X},\mathbb{C}^{Y})$5 strategy if:

• $L(\mathbb{C}^{X},\mathbb{C}^{Y})$6 is contained in a prescribed class of QNS correlations $L(\mathbb{C}^{X},\mathbb{C}^{Y})$7 (for $L(\mathbb{C}^{X},\mathbb{C}^{Y})$8), corresponding to local, quantum (entanglement-assisted), approximate, commuting operator, or no-signalling resources, respectively;
• For all $L(\mathbb{C}^{X},\mathbb{C}^{Y})$9, $\mathbb{C}^{X}$0, meaning that the action of $\mathbb{C}^{X}$1 (expressed via the canonical flip isomorphism $\mathbb{C}^{X}$2) maps every $\mathbb{C}^{X}$3-edge into the span of $\mathbb{C}^{X}$4.

Equivalent formulations are given in terms of the Choi matrix $\mathbb{C}^{X}$5 of $\mathbb{C}^{X}$6 (with $\mathbb{C}^{X}$7, where $\mathbb{C}^{X}$8 is a subspace canonically constructed from $\mathbb{C}^{X}$9 and $\mathbb{C}^{Y}$0), or using Kraus (stochastic operator-matrix) representations, where each operator $\mathbb{C}^{Y}$1 for all $\mathbb{C}^{Y}$2.

3. Definition and Typology of Quantum Homeomorphisms

A t-homomorphism between quantum hypergraphs formalizes the notion of a structure-preserving quantum map tailored to operational constraints:

• $\mathbb{C}^{Y}$3 (t-quasi-homomorphic): Existence of a QNS correlation $\mathbb{C}^{Y}$4 satisfying $\mathbb{C}^{Y}$5;
• $\mathbb{C}^{Y}$6 (t-homomorphic): Existence of such a $\mathbb{C}^{Y}$7 which is a quantum channel (trace-preserving and completely positive).

The variants are summarized in the following table:

$\mathbb{C}^{Y}$8	Allowed Resources	Channel Structure
loc	Shared randomness (local)	Convex mixtures of product channels
q	Finite-dimensional entanglement	Local POVMs on entangled states
qa	Approximate quantum	Limits of $\mathbb{C}^{Y}$9-type
qc	Commuting measurements	Commuting operator algebras on possibly infinite $u\in U$0
ns	Full no-signalling	Arbitrary QNS correlations

Here, "channel structure" designates the affiliated operational implementation of the homomorphism.

4. Preorder Structure and Composition

Each relation $u\in U$1 is a preorder: it is reflexive (the identity channel constitutes a local perfect strategy) and transitive (composability follows from the channel-simulation and composition lemma). Specifically, if $u\in U$2 via $u\in U$3 and $u\in U$4 via $u\in U$5, then $u\in U$6 gives a $u\in U$7 t-homomorphism with type preserved [(Hoefer et al., 2023), Thm 3.3]. This establishes a categorical landscape for quantum hypergraph homomorphisms, with subclasses defined by the operational constraints on allowed strategies.

5. TRO Equivalence and Local Quantum Homeomorphisms

A critical structural insight links local quantum homeomorphisms to the theory of ternary rings of operators (TROs). For $u\in U$8, the dual space is $u\in U$9. TROs $U$0 are subspaces closed under $U$1 and are non-degenerate if their action densely spans $U$2 and $U$3.

Two operator spaces $U$4, $U$5 are TRO-equivalent, $U$6, if there exist non-degenerate TROs $U$7 and $U$8 such that $U$9 and $D_Y$0.

The main result (Theorem 5.3) asserts:

$D_Y$1

In addition, if the local channel can be chosen with invertible Choi matrix ("fully local" homomorphism), then the equivalence can be sharpened to a Morita-type equivalence with two-sided non-degenerate TROs. The construction of these TROs from the Kraus operators of the local channel, and vice versa, provides the precise mechanism for the correspondence.

6. Classical Case as a Specialization

Classical hypergraphs are naturally embedded in this quantum framework. For example, with $D_Y$2, $D_Y$3, $D_Y$4, and classical edge sets $D_Y$5, $D_Y$6, the corresponding operator subspaces are

• $D_Y$7,
• $D_Y$8.

Maps $D_Y$9 and $D_X$0 (with $D_X$1 and $D_X$2) generate a local channel $D_X$3 verifying $D_X$4. The corresponding TROs $D_X$5 and $D_X$6 are non-degenerate and realize the TRO equivalence of $D_X$7 and $D_X$8, illustrating the precise correspondence in Theorem 5.3.

7. Notational Summary and Operational Landscape

Key notational conventions include $D_X$9 for the existence of a QNS correlation, $M_X$0 for the existence of a quantum channel fitting the canonical game, $M_X$1 as the definition of a quantum hypergraph (with classical instance where $M_X$2 for an edge set $M_X$3), dual operator space $M_X$4, and TRO equivalence $M_X$5 as the local quantum homomorphism criterion. The operational typology (local, quantum, approximate, commuting, and no-signalling) systematically controls the permissible strategies and channel structures (Hoefer et al., 2023).