Non-Signalling Tensor Product

• Non-Signalling Tensor Product is a mathematical framework enforcing that subsystem marginals remain invariant under distant measurement choices, distinguishing between classical, quantum, and generalized correlations.
• It is formulated using operator systems, quantum logics, ordered vector spaces, and Banach-space tensor norms to reveal distinct operational and compositional implications.
• The framework underpins studies of non-local phenomena and resource theories by classifying correlations and ensuring strict adherence to the no-signalling principle.

A non-signalling tensor product provides a rigorous mathematical framework for composing physical, probabilistic, or operator-theoretic systems in a manner that enforces the no-signalling principle: the marginal statistics of any subsystems cannot be affected by the measurement choices on other subsystems. This notion emerges in several mathematical languages—operator systems, quantum logics, ordered vector spaces, and Banach-space tensor norms—each highlighting key features of non-signalling compositionality, its distinction from classical and quantum tensor products, and its foundational role in general probabilistic theories and the study of non-local correlations.

1. Operator-System Framework for Non-Signalling Tensor Products

For finite sets $X,Y$ of questions and $A,B$ of answers (or inputs/outputs), define the universal operator system $T_{X,A}$ as the span in a C*-algebra $C_{X,A}$ of elements $e_{x,x';a,a'}=v_{a,x}v_{a',x'}^*$, subject to the relations

$$e_{x,x';a,a'}^* = e_{x',x;a',a},\quad \sum_{a\in A} e_{x,x;a,a}=1, \quad e_{x,x';a,a'}\ge 0.$$

Given $T_{Y,B}$ similarly, there exist three canonical tensor products on operator systems:

• The minimal (spatial) tensor product $T_{X,A}\otimes_{\min}T_{Y,B}$,
• The commuting tensor product $T_{X,A}\otimes_c T_{Y,B}$,
• The maximal tensor product $T_{X,A}\otimes_{\max}T_{Y,B}$.

These tensor products strictly correspond to natural physical and resource-theoretic classes of correlations:

• Maximal: all quantum no-signalling (QNS) correlations are in bijection with states on $T_{X,A}\otimes_{\max}T_{Y,B}$;
• Commuting: quantum commuting QNS with states on $T_{X,A}\otimes_c T_{Y,B}$;
• Minimal: quantum (finite-dimensional) QNS with states on $T_{X,A}\otimes_{\min}T_{Y,B}$;
• Local: local QNS with states on $$\mathrm{OMIN}(T_{X,A})\otimes_{\min}\mathrm{OMIN}(T_{Y,B})$$ (Todorov et al., 2020, Lupini et al., 2018).

The assignment $s\mapsto T_s$ produces affine homeomorphisms between these state spaces and the corresponding QNS classes. The operator-system tensor product enforces non-signalling by encoding marginal constraints at the level of operator system structure: for a transformation $T$, the marginalization (e.g., partial trace) on one subsystem yields zero whenever traced over traceless elements of the other, as a consequence of the functoriality and order-injectivity of the tensor product.

2. Tensor Products in Quantum Logic and Generalized Probability

In the quantum-logic and generalized probabilistic theories (GPT) context, the non-signalling tensor product is realized as follows.

Quantum-logic picture: The propositional system for a single box is given by the Boolean algebra generated by the elementary events $[a=\alpha]$, where $a$ is an input and $\alpha$ an output. The composite logic for two boxes is constructed as the sublogic generated by atoms $[a=\alpha,b=\beta]$. The key result is that for non-signalling boxes (e.g., PR boxes), the composite logic admits a strong tensor product: any extremal (two-valued) state is a product state,

$$\chi([a=\alpha, b=\beta]) = \mu([a=\alpha]) \nu([b=\beta]),$$

where $\mu$, $\nu$ are states on the single-box logics. This strong tensor product property holds only for non-signalling boxes and not for quantum projection logics, for which only the weak tensor product (the projection lattice on $\mathcal H_1\otimes\mathcal H_2$) is possible (Tylec et al., 2016).

GPT/Ordered-vector-space picture: Given systems $A$ and $B$ with effect vector spaces $(V_A, V_B)$ and cones $V_A^+$, $V_B^+$, the non-signalling tensor product is the minimal (projective) tensor product cone,

$$V^+_{\min} = \operatorname{cone}\{f\otimes g:\, f\in V_A^+,\, g\in V_B^+\}.$$

States on the composite are the linear functionals on $V_A\otimes V_B$ that are positive on $V^+_{\min}$ and normalized. This construction admits every convex combination of product states and enforces that all effects and states are locally accessible and non-signalling (Tylec et al., 2015).

3. Banach-Space and Tensor Norm Approaches

Non-signalling distributions can also be characterized in Banach-space language. For sets $X$ and outputs $A$, one sets $V = \ell_\infty^N(\ell_1^K)$ where $N = |X|, K = |A|$, and similarly for $B$ and $Y$. Three physically significant tensor norms arise:

• Injective ($\varepsilon$) norm corresponds to classical/local correlations,
• Operator-space minimal corresponds to quantum correlations,
• Non-signalling tensor norm $||\cdot||_{DNS}$ defines the set of all non-signalling behaviors (Amr et al., 2019).

Explicitly, for a bipartite table $P(a,b|x,y)$, the non-signalling polytope is the unit ball for the dual norm

$$\|P\|_{NS} = \max\{\max_x \sum_{a,b,y}|P(a,b|x,y)|,\, \max_y \sum_{a,b,x}|P(a,b|x,y)|\}.$$

The non-signalling tensor product is then the unique abstract completion that contains precisely all non-signalling tables as its normalized elements. This is not merely a convex closure: it imposes a tensor-norm geometry aligned with operational non-signalling constraints. In this language, maximal Bell inequality violations by non-signalling theories over classical ones are given by the ratio of the corresponding tensor norms (Amr et al., 2019).

4. Generalized Tensors, Consistency, and Non-Signalling

The generalized tensor product construction, applicable in settings such as dynamic or logical system partitioning, defines a bilinear "tensor" $\tensorchi$ parameterized by a restriction $\chi$ which determines the system split. The key point is the introduction of a consistency condition: only those pairs of pure states $(|\psi\rangle, |\phi\rangle)$ for which the induced glueing $\tensorchi$ does not produce forbidden configurations are permitted. Consistency preservation is equivalent to the absence of signalling; an operation is non-signalling if it never renders a $\chi$-consistent pair inconsistent under action.

Tracing out a subsystem and defining local operations are compatible with this construction. The partial trace $\rho_{|\chi}$ is a completely positive map, and trace-preserving under natural conditions. Causal unitaries are precisely those that, when acting on the composite, do not enable signalling between $\chi$ and its complement. This generalized approach recovers the standard quantum tensor product when the partition is fixed to the designated tensor factor (Arrighi et al., 2022).

5. Classical-to-Quantum and Synchronous Non-Signalling Tensor Products

The operator-system framework extends directly to mixtures of classical and quantum systems. For classical-to-quantum scenarios, the universal system $R_{X,A}$ (a free product of matrix algebras or their operator-system coproduct) replaces $T_{X,A}$, and non-signalling classes again correspond to states on $R_{X,A}\otimes_\tau R_{Y,B}$ for $\tau\in\{\max,c,\min,\mathrm{OMIN}-\min\}$. Restricting further to "diagonal" operator systems such as $S_{X,A}$ identifies purely classical non-signalling tensor products and the standard local, quantum, quantum commuting, and NS classes (Todorov et al., 2020, Lupini et al., 2018).

This framework enables a hierarchy:

Non-signalling class	Operator-system tensor product	Correlation type
Maximal	$T_{X,A}\otimes_{\max} T_{Y,B}$	Quantum no-signalling
Commuting	$T_{X,A}\otimes_c T_{Y,B}$	Quantum commuting
Minimal (spatial)	$T_{X,A}\otimes_{\min} T_{Y,B}$	Finite-dimensional quantum
Local (OMIN-minimal)	OMIN$(T_{X,A})\otimes_{\min}$OMIN$(T_{Y,B})$	Local/classical

The correspondence is an affine homeomorphism, and in the classical limit, the extreme points are deterministic (product) distributions.

6. Structural and Operational Consequences

A central structural property of the non-signalling tensor product is that, in box-world models, every extremal state is a product state, a feature formalized as strong tensor product in the sense of quantum logic (Tylec et al., 2016). This property does not extend to quantum systems, which admit entangled pure states that are not product states, necessitating the use of weak tensor products (projection lattices on $\mathcal H_1\otimes\mathcal H_2$).

Operationally, non-signalling tensor products are the unique composite structures enabling all local measurements, strict enforcement of no-signalling, and full convex closure. They fail to capture quantum entanglement structure in the logical sense and admit extremal PR-box-type correlations not possible in quantum theory but constrained in classical models. The set of non-signalling boxes forms a convex polytope strictly larger than both the quantum and classical sets, as realized at the level of tensor product geometry in Banach or operator systems (Tylec et al., 2015, Amr et al., 2019).

Non-signalling tensor products provide the foundational structure for the study of non-local games, perfect strategies, and the algebraic and convexity-geometric separation of locality, quantum, and general non-signalling resources. They are indispensable in distinguishing classical, quantum, and supra-quantum behaviors in generalized probabilistic frameworks and categorical quantum mechanics.