The process-matrix framework describes quantum correlations without presupposing a global causal order, yet its standard formulation implicitly relies on background structure through a fixed Choi-Jamiolkowski identification of local input-output spaces. We analyze how such background assumptions can be treated operationally relative to a fixed device-independent interface defined by a causal game. We impose local-frame covariance, requiring invariance under independent actions of a physical symmetry group $G$ on each laboratory, thereby excluding symmetry-breaking background resources. Covariance induces a representation-theoretic decomposition into symmetry sectors and symmetry-invariant multiplicity subsystems, introducing physical degrees of freedom that lie outside the declared device-independent interface. We then analyze causal-inequality signatures at the level of interface-observable statistics and identify when symmetry-induced, interface-inaccessible degrees of freedom undermine device-independent certification. A certification is called background-free if it arises from a locally covariant implementation and does not rely on hidden control mediated by interface-excluded degrees of freedom. We prove that background-free certifications cannot yield device-independent violations of bipartite causal inequalities in the multiplicity-free regime or when all multiplicity subsystems are classical-classical (CC). Such violations necessarily require non-CC multiplicity, with a concrete sufficient route provided by input-output embeddability of an effective process-matrix structure into non-CC blocks. These results delineate which device-independent causal signatures remain certifiable once both symmetry-breaking background structure and interface-level hidden control are excluded.
The process-matrix framework describes quantum correlations without presupposing a global causal order, yet its standard formulation implicitly relies on background structure through a fixed Choi-Jamiolkowski identification of local input-output spaces. We analyze how such background assumptions can be treated operationally relative to a fixed device-independent interface defined by a causal game. We impose local-frame covariance, requiring invariance under independent actions of a physical symmetry group G on each laboratory, thereby excluding symmetry-breaking background resources. Covariance induces a representation-theoretic decomposition into symmetry sectors and symmetry-invariant multiplicity subsystems, introducing physical degrees of freedom that lie outside the declared device-independent interface. We then analyze causal-inequality signatures at the level of interface-observable statistics and identify when symmetry-induced, interface-inaccessible degrees of freedom undermine device-independent certification. A certification is called background-free if it arises from a locally covariant implementation and does not rely on hidden control mediated by interface-excluded degrees of freedom. We prove that background-free certifications cannot yield device-independent violations of bipartite causal inequalities in the multiplicity-free regime or when all multiplicity subsystems are classical-classical (CC). Such violations necessarily require non-CC multiplicity, with a concrete sufficient route provided by input-output embeddability of an effective process-matrix structure into non-CC blocks. These results delineate which device-independent causal signatures remain certifiable once both symmetry-breaking background structure and interface-level hidden control are excluded.
While the standard formulation of quantum theory assumes a fixed background causal structure, one can relax this assumption within the so-called process matrix framework. Remarkably, some processes, termed causally nonseparable, are incompatible with a definite causal order. We explore a form of certification of causal nonseparability in a semi-device-independent scenario where the involved parties receive trusted quantum inputs, but whose operations are otherwise uncharacterized. Defining the notion of causally nonseparable distributed measurements, we show that certain causally nonseparable processes that cannot violate any causal inequality, including the canonical example of the quantum switch, can generate noncausal correlations in such a scenario. Moreover, by imposing some further natural structure to the untrusted operations, we show that all bipartite causally nonseparable process matrices can be certified with trusted quantum inputs.
One of the key ways in which quantum mechanics differs from relativity is that it requires a fixed background reference frame for spacetime. In fact, this appears to be one of the main conceptual obstacles to uniting the two theories. Additionally, a combination of the two theories is expected to yield non-classical, or "indefinite", causal structures. In this paper, we present a background-independent formulation of the process matrix formalism – a form of quantum mechanics that allows for indefinite causal structure – while retaining operationally well-defined measurement statistics. We do this by postulating an arbitrary probability distribution of measurement outcomes across discrete "chunks" of spacetime, which we think of as physical laboratories, and then requiring that this distribution be invariant under any permutation of laboratories. We find (a) that one still obtains nontrivial, indefinite causal structures with background independence, (b) that we lose the idea of local operations in distinct laboratories, but can recover it by encoding a reference frame into the physical states of our system, and (c) that permutation invariance imposes surprising symmetry constraints that, although formally similar to a superselection rule, cannot be interpreted as such.
Causal nonseparability is the property underlying quantum processes incompatible with a definite causal order. So far it has remained a central open question as to whether any process with a clear physical realisation can violate a causal inequality, so that its causal nonseparability can be certified in a device-independent way, as originally conceived. Here we present a method solely based on the observed correlations, which certifies the causal nonseparability of all the processes that can induce a causally nonseparable distributed measurement in a scenario with trusted quantum input states, as defined in [Dourdent et al., Phys. Rev. Lett. 129, 090402 (2022)]. This notably includes the celebrated quantum switch. This device-independent certification is achieved by introducing a network of untrusted operations, allowing one to self-test the quantum inputs on which the effective distributed measurement induced by the process is performed.
Quantum processes can exhibit indefinite causal order, allowing coherent superpositions of distinct orderings of events. Whether such processes generate genuinely noncausal statistics, however, depends not only on their global causal structure but also on the local operations available to the parties. Here, we restrict communication to the classical outcomes of local operations, yielding a causal polytope with new causal inequalities that can be violated using measure-and-reprepare, entanglement-breaking instruments requiring no coherent transmission through the laboratories. Using a semidefinite-programming framework, we bound the achievable violations and reveal noncausal correlations that satisfy all conventional input-signalling inequalities, demonstrating that restricting the information available for communication can uncover forms of indefinite causal order inaccessible under less restrictive operational assumptions.
Causal nonseparability is the property underlying quantum processes incompatible with a definite causal order. So far it has remained a central open question as to whether any process with a clear physical realisation can violate a causal inequality, so that its causal nonseparability can be certified in a device-independent way, as originally conceived. Here we present a method solely based on the observed correlations, which certifies the causal nonseparability of all the processes that can induce a causally nonseparable distributed measurement in a scenario with trusted quantum input states, as defined in [Dourdent et al., Phys. Rev. Lett. 129, 090402 (2022)]. This notably includes the celebrated quantum switch. This device-independent certification is achieved by introducing a network of untrusted operations, allowing one to self-test the quantum inputs on which the effective distributed measurement induced by the process is performed.
Bell non-local correlations cannot be naturally explained in a fixed causal structure. This serves as a motivation for considering models where no global assumption is made beyond logical consistency. The assumption of a fixed causal order between a set of parties, together with free randomness, implies device-independent inequalities—just as the assumption of locality does. It is known that local validity of quantum theory is consistent with violating such inequalities. Moreover, for three parties or more, even the (stronger) assumption of local classical probability theory plus logical consistency allows for violating causal inequalities. Here, we show that a classical environment (with which the parties interact), possibly containing loops, is logically consistent if and only if whatever the involved parties do, there is exactly one fixed-point, the latter being representable as a mixture of deterministic fixed-points. We further show that the non-causal view allows for a model of computation strictly more powerful than computation in a world of fixed causal orders.