Sequential Operationally No-Signalling Strategies

• Sequential operationally no-signalling strategies are frameworks that formalize the admissible compositions of quantum and generalized processes under strict causal and no-signalling conditions.
• They utilize higher-order quantum maps and the Choi–Jamiołkowski isomorphism to represent processes with indefinite causal order in quantum protocols and device-independent cryptographic settings.
• These strategies underpin secure sequential operations in quantum circuits and spacetime protocols by enforcing linear constraints that prevent causal loops and forbidden signalling.

Sequential operationally no-signalling strategies are a central concept at the intersection of higher-order quantum maps, quantum information theory, and the study of causality in quantum and operational probabilistic frameworks. They formalize the most general admissible patterns of composition for quantum and generalized processes—ranging from ordinary channels to supermaps and beyond—subject to the requirement that no forbidden causal or operational signalling occurs, even in scenarios with indefinite or dynamical causal structures.

1. Hierarchies of Higher-Order Maps and Types

The modern foundation for sequential operationally no-signalling strategies is the framework of higher-order quantum theory, formalized recursively through a hierarchy of types. Every finite-dimensional quantum system forms an elementary type (e.g., $A$, $B$, etc.), and composite types are built via type recursion: given types $x, y$, $(x \to y)$ is a higher-order type representing maps from $x$-type objects (which themselves may be maps) to $y$-type objects. This construction admits types of arbitrary order, for example $((A \to B) \to (C \to D)) \to E$ (Apadula et al., 2022).

Each type $x$ is associated with a Hilbert space $\mathcal H_x$, formed as the tensor product over all elementary systems in $x$. Higher-order maps of type $B$0 are linear maps sending deterministic maps of type $B$1 to those of type $B$2. The Choi–Jamiołkowski isomorphism is used extensively to translate maps into operators: for a map $B$3, the Choi operator $B$4 is $B$5 for a maximally entangled state $B$6 on $B$7.

Admissible maps are characterized as positive operators $B$8 bounded above by some deterministic $B$9 in the set $x, y$0 of normalized maps of type $x, y$1:

$x, y$2

Deterministic maps admit an explicit affine decomposition involving representation-specific subspaces $x, y$3 and normalization constants $x, y$4 recursively defined through the type structure.

2. Signalling, No-Signalling, and Causal Constraints

Causality is encoded via precise signalling constraints. For an ordinary quantum channel $x, y$5, $x, y$6 is said to be no-signalling from $x, y$7 to $x, y$8 if the marginal output $x, y$9 does not depend on the input $(x \to y)$0:

$(x \to y)$1

For higher-order maps of arbitrary type $(x \to y)$2, no-signalling from an elementary input $(x \to y)$3 to an output $(x \to y)$4 requires that, when regarded as a channel from all inputs to outputs,

$(x \to y)$5

for an appropriate choice of outputs to be traced.

Crucially, these constraints must be satisfied at the level of types (i.e., for all deterministic operators of a given type), leading to the notion $(x \to y)$6.

Full admissibility of a contraction (i.e., closing a wire between $(x \to y)$7 and $(x \to y)$8 in a process network) is possible if and only if no-signalling from $(x \to y)$9 to $x$0 is satisfied at the type level:

$x$1

This establishes the exact correspondence between allowed sequential operations and type-level no-signalling constraints (Apadula et al., 2022).

3. Admissible Composition and Sequential Strategies

The main compositional theorem for sequential strategies is as follows. Given two higher-order maps $x$2 and $x$3 of types $x$4 and $x$5 respectively, their composition (via the "link product" or contraction over shared systems $x$6) is admissible if and only if:

• All shared systems are contracted in directions consistent with input/output causality:

$x$7

• The resulting map after contraction remains a valid deterministic map of reduced type.

This ensures that every contraction or wire closure in a sequential strategy doesn't create a forbidden compositional causal loop (Apadula et al., 2022). For a multi-step sequential strategy, admissibility must be checked not merely pairwise but for the entire set of contractions—a stricter requirement than individual no-signalling per step.

A summary of key representational constraints is given in the table below.

Map Type / Operation	No-Signalling Condition	Admissibility Criterion
Ordinary channel $x$8	$x$9	Marginal on $y$0 not affected by $y$1
Higher-order, generic type $y$2	$y$3	No fixed input signals to forbidden output
Contraction $y$4	$y$5	Allowed iff no signalling from $y$6 to $y$7 at the type level

4. Operator and Process Tensor Formalisms

Sequential operationally no-signalling strategies can be equivalently characterized in the operator formalism via Choi operators and in the process tensor/quantum comb framework (Gutoski, 2010). An $y$8-round sequential strategy is encoded by a positive semidefinite Choi operator $y$9 on the product space of all $((A \to B) \to (C \to D)) \to E$0 rounds' inputs and outputs:

$((A \to B) \to (C \to D)) \to E$1

The recursive partial-trace/identity constraints encode the requirement of no backward (future-to-past) signalling:

$((A \to B) \to (C \to D)) \to E$2

recursively down to $((A \to B) \to (C \to D)) \to E$3. The subset of sequential no-signalling strategies is thus defined as the spectrahedral cone:

$((A \to B) \to (C \to D)) \to E$4

No-signalling constraints are imposed as additional linear equalities, e.g.,

$((A \to B) \to (C \to D)) \to E$5

for appropriate subsystems $((A \to B) \to (C \to D)) \to E$6. The intersection of comb and no-signalling constraints precisely captures the desired class.

5. Operational No-Signalling and Causal Dynamics

Recent work extends the notion of operational no-signalling to general spacetime-embedded protocols, including both spatial and temporal correlations (Eckstein et al., 29 Dec 2025, Eckstein et al., 2019). Here, each input/output is assigned to a spacetime event, and operational no-signalling is formulated as the condition that, for any operational separation of output and input events (i.e., when all outputs can be gathered at $((A \to B) \to (C \to D)) \to E$7 such that the relevant inputs are outside $((A \to B) \to (C \to D)) \to E$8's past), the output statistics are independent of those inputs.

Formally, for conditional probabilities $((A \to B) \to (C \to D)) \to E$9,

$x$0

for all $x$1 differing only on $x$2.

In the continuous-time dynamical case, the dynamical no-signalling condition restricts the effect of a local spacetime measurement: performing a measurement in region $x$3 at time $x$4 must not alter subsequent detection probabilities in any spacetime region $x$5 outside the causal future of $x$6 (Eckstein et al., 2019). Operational sequences (such as sequential measurements on a single system) are only admissible if the probability evolution respects the causal structure—i.e., satisfies the sequential operationally no-signalling criterion at each step.

6. Applications and Computational Consequences

Sequential operationally no-signalling strategies underlie the compositional calculus of quantum circuits with open slots, generalized quantum protocols with indefinite causal order, and time-ordered protocol security analyses. Notably:

• In device-independent randomness expansion/accumulation, the time-ordered no-signalling (TONS) constraints for multiple sequential rounds ensure that adversarial actions cannot exploit forbidden temporal signalling. They guarantee that conditional min-entropy grows at least linearly in the number of rounds, subject to underlying Bell-monogamy properties (Ramanathan et al., 20 Jun 2025).
• In quantum computation with indefinite causal structure, the precise characterization of which higher-order process compositions are admissible enables a rigorous analysis of physical implementability and rules out illegal operations that would permit "causal loops" or superluminal signalling (Apadula et al., 2022).
• In spacetime settings subject to general relativity or black-hole event horizons, operational no-signalling constraints clarify which nonlocal and temporal correlations are physically admissible and identify scenarios where nontrivial phenomena (e.g., "jamming") are possible without enabling observable signalling or paradox (Eckstein et al., 29 Dec 2025).

7. Conceptual and Methodological Insights

Sequential operationally no-signalling strategies reveal a deep equivalence between the compositional rules for physical processes and the underlying causal (no-signalling) structure. In the quantum hierarchy, the admissibility of sequential contraction and composition reduces entirely to the respect of no-signalling relations at the type and operator levels. This insight provides a unified algorithmic criterion—linear constraints on process operators or conditional distributions—for the construction, verification, and analysis of complex quantum, classical, and generalized operational protocols across arbitrary causal and spacetime backgrounds (Apadula et al., 2022, Eckstein et al., 29 Dec 2025, Gutoski, 2010).

The table below summarizes representative settings and the key sequential operationally no-signalling constraints:

Setting	Sequential No-Signalling Constraint	Consequence
Higher-order quantum maps	$x$7 for all contractions	Only causally admissible compositions allowed
Quantum combs/strategies	Recursive partial-trace identities plus operator NS	Operator set is a spectrahedron $x$8
Spacetime-embedded protocols	Output marginals independent of causally separated inputs	No operational superluminal signalling or causal loops
Device-independent cryptography	TONS constraints on multi-round behaviors	Linear min-entropy growth under no-signalling adversaries

The principle robustly generalizes classical, quantum, and post-quantum theories, always enforcing that every step in a sequential protocol—whether algebraic (type) or operational (spacetime statistics)—respects the mechanism of causality expressed through operational no-signalling conditions. Any violation leads directly to logical or physical pathologies, including superluminal communication or causal inconsistency.