Bipartite Covariant Process

• Bipartite covariant processes are defined by their invariance under local symmetry transformations applied independently to each subsystem.
• They establish superselection rules by partitioning the process space into covariant and non-covariant sectors, ensuring that only physically realizable processes emerge.
• These processes enable causally separable dynamics and support scalable Bayesian hierarchical models in quantum information theory and probabilistic combinatorics.

A bipartite covariant process is a mathematical and operational framework for describing correlations, transformations, or evolutions involving two subsystems, often termed "parties" (e.g., Alice and Bob), under a covariance (symmetry) constraint. The notion appears in quantum information theory, quantum causality, mathematical physics, high-energy theory, and probabilistic combinatorics. Across these contexts, a unifying feature is the requirement that the process remains invariant under certain symmetry operations—typically local changes of reference frames or basis—on each party's systems or "wires." This symmetry principle leads to profound structural, operational, and computational consequences, ranging from superselection rules and causal separability in quantum process spaces to tractable hierarchical models for bipartite or multitask inference and ranking.

1. Symmetry Principles and Mathematical Invariance

The defining property of a bipartite covariant process is invariance under independent local transformations applied to each “wire” or subsystem. In the context of quantum process matrices (Ibnouhsein, 23 Aug 2025), consider a general bipartite process $W$ acting on Hilbert space

$$\mathcal{H} = \mathcal{H}_{A_I} \otimes \mathcal{H}_{A_O} \otimes \mathcal{H}_{B_I} \otimes \mathcal{H}_{B_O}$$

where $A$ and $B$ denote two local laboratories, with input and output spaces. The covariance requirement is stipulated as invariance under all local unitaries $V_1, V_2 \in U(d)$ acting independently on each wire:

$$T_\text{wire}(W) = \int_{U(d) \times U(d)} U_{V_1, V_2} W U_{V_1, V_2}^{\dagger} dV_1 dV_2 = W$$

with

$$U_{V_1, V_2} = V_1^{A_O} \otimes V_1^{* B_I} \otimes V_2^{B_O} \otimes V_2^{* A_I}$$

Such invariance projects $W$ onto an algebra generated by projectors onto local maximally entangled states (representing the "wires"), ensuring that any relative basis information is inaccessible—a principle called independent wire covariance.

In communication-theoretic contexts (Shadman et al., 2013), a bipartite covariant channel $\Lambda$ is likewise defined by its commutation with a complete orthogonal set of local unitaries $\{\tilde V_i\}$:

$$\Lambda(\tilde V_i \rho\, \tilde V_i^\dagger) = \tilde V_i\, \Lambda(\rho)\, \tilde V_i^\dagger$$

with each $\tilde V_i$ typically acting as $V_i^A \otimes I^B$ or similarly on the shared system.

2. Superselection Rules and Sector Decomposition

The invariance under independent frame changes induces a superselection rule: processes decomposed as $M = M_G + M_\perp$ with $T_\text{wire}(M) = M_G$ (the covariant component) and $T_\text{wire}(M_\perp) = 0$. Operations that respect the symmetry cannot convert between $M_G$ and $M_\perp$, partitioning the process space into distinct, operationally isolated sectors (Ibnouhsein, 23 Aug 2025). Thus, any physically realizable (by local agents lacking a shared reference) process must reside in the covariant sector.

This superselection structure has direct operational impact in quantum causality. For example, the maximally entangled (Choi) states representing circuit "wires" are invariant under all local basis choices; only processes compatible with this symmetry are physically embeddable in quantum circuits. Mathematical consequences include a basis for process matrices in this sector as linear combinations of tensor products of projectors $\{ \Pi, \Pi^\perp \}$ on each wire, forming a four-dimensional convex subset subject to positivity and normalization constraints.

3. Causal Separability and Operational Limitations

A key theorem (Ibnouhsein, 23 Aug 2025) is that every bipartite covariant process is causally separable: it can always be written as a convex mixture of fixed-order processes,

$$W_\text{sep} = p\, W_{A \prec B} + (1 - p)\, W_{B \prec A}, \quad p \in [0, 1]$$

where $W_{A \prec B}$ and $W_{B \prec A}$ correspond to definite causal orders from Alice to Bob and vice versa. Geometrically, the covariant process space’s allowed parameters form a polygon whose vertices are the fixed-order processes.

This constraint implies that all circuit-embeddable dynamics—including the quantum switch (when the control system is traced out)—cannot violate bipartite causal inequalities. Only processes outside the covariant sector (requiring correlated reference frames across wires) can exhibit "causal nonseparability" and thereby exceed bounds imposed by definite causal structure (as in the Oreshkov-Costa-Brukner process).

In communication theory, covariance often enables closed-form capacity results by exploiting symmetry—e.g., in super dense coding over covariant noisy channels, allowing entropic expressions that cleanly separate local and global noise effects (Shadman et al., 2013).

4. Variational and Bayesian Hierarchical Models in Bipartite Inference

The symmetry concepts underlying bipartite covariant processes also inform powerful statistical modelling strategies in machine learning. In multitask and bipartite ranking problems, hierarchical structures are captured through matrix-variate Gaussian processes (MV-GPs) with Kronecker covariance (product of task and example kernels) (Koyejo et al., 2013), reflecting assumed symmetries between rows and columns.

A bipartite covariant process in this context is understood as a hierarchical Bayesian model structured to respect correlations and invariances between two types of objects (e.g., genes and diseases). Imposing additional structure via trace-norm constraints or a spectral elastic net regularizer on the posterior mean promotes low rank, enhancing regularization, scalability, and interpretability. Such models guarantee joint convexity in the variational optimization of posterior mean, covariance, and ranking penalties, resulting in globally optimal estimates and robust parameter learning, even under extreme data sparsity.

5. Covariant Holographic Processes in Quantum Field Theory

In the AdS/CFT correspondence and quantum information theory, bipartite covariant processes arise in proposals and computations of entanglement measures for bipartite systems evolving in time-dependent or non-static backgrounds (Chaturvedi et al., 2016, Basak et al., 2021, Biswas et al., 2023). The central organizing feature is Lorentz (or diffeomorphism) covariance of the relevant bulk and boundary constructions.

For example, the covariant holographic negativity conjecture prescribes that the entanglement negativity (a quantifier of mixed-state entanglement) is computed as a covariant combination of extremal surface areas anchored on subsystems and their unions. This prescription

$\mathcal{E} = \frac{3}{2} E_W^\text{ext}(A:B)$

links the negativity to the minimal cross section of the entanglement wedge, with extremization enforcing covariance under Lorentz transformations. This approach naturally generalizes static "minimal surface" prescriptions to arbitrary time-dependent or rotating black hole spacetimes, and agrees (up to constant terms) with replica method results in conformal field theory.

Similar constructions appear for odd entanglement entropy (OEE), where extremal entanglement wedge cross sections yield covariant expressions that match CFT replica results in the large central charge limit (Biswas et al., 2023).

6. Emergence of Definite Causality via Measurement Restriction

Restricting allowable operations to fixed-basis measurements projects arbitrary bipartite process matrices into the covariant sector, forcibly eliminating all causal nonseparability (Baumann et al., 2016). Operationally, this is the manifestation of the von Neumann–Lüders update rule: a nonselective measurement in a fixed product basis "diagonalizes" the process matrix, rendering it causally separable:

$$W \mapsto W_\text{eff} = \sum_{n,m} (P_n \otimes P_m) W (P_n \otimes P_m)$$

In the bipartite case, the effective process matrix can always be written as a mixture of fixed-order processes, reinforcing the conclusion from symmetry-based arguments that bipartite covariant processes cannot exhibit indefinite causal order under experimentally feasible—“locally classical”—operations.

7. Relationships to Classical and Probabilistic Bipartite Processes

In combinatorial probabilistic settings, "bipartite covariant process" terminology can appear for stochastic constructions observing local constraints and symmetry between parts. For example, in the random greedy $K_{2,2}$-free process on $K_{n,n}$ (Bal et al., 2018), one iteratively introduces edges under a constraint that respects the bipartition symmetry, yielding graphs with properties optimized across both parts (in that case, improved lower bounds for bipartite Ramsey numbers). The underlying principles—symmetry, local invariance, and process-induced sectorization—echo the structural themes of bipartite covariant quantum processes, despite differing in formalism and application domain.

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In summary, bipartite covariant processes unify a variety of structural, operational, and statistical frameworks across mathematical physics, quantum information theory, and statistical modelling. Their defining feature—symmetry under independent local operations on subsystems or wires—drives sector partitioning (superselection), prohibits nonseparable causal structure, enables scalable and globally optimal variational inference, and grounds covariant holographic measures for quantum correlations. These constraints clarify which processes are physically realizable, dictate operational resource requirements, and facilitate tractable, symmetry-respecting model design in both quantum and classical bipartite systems.