Quantum Correlations with No Causal Order

• Quantum correlations with no causal order are defined by the absence of a global causal structure, allowing quantum operations to occur without a fixed time sequence.
• The process-matrix formalism provides a framework where such correlations violate causal inequalities, as demonstrated by experiments like the quantum switch.
• These phenomena act as an operational resource in quantum information processing, challenging conventional causality and offering insights into quantum gravity.

Quantum correlations with no causal order involve the realization of correlations among quantum systems that cannot be embedded in any definite global temporal sequence of local operations. Unlike conventional quantum theory, where spacetime defines a fixed causal structure governing all operational correlations, this phenomenon operationalizes scenarios where quantum mechanics governs each local laboratory but makes no global assumption about causal relations among events. Such correlations fundamentally challenge the classical and relativistic paradigm of spacetime and have profound implications for quantum information theory, foundational physics, and the interface with quantum gravity.

1. Formalism and Characterization of Indefinite Causal Order

The key mathematical framework for studying quantum correlations without a fixed causal structure is the process-matrix formalism. Consider $N$ parties, each localized in a shielded laboratory, denoted $S_i$, with incoming and outgoing Hilbert spaces $\mathcal H^{I_i}$ and $\mathcal H^{O_i}$. Each party applies a completely positive (CP), trace-non-increasing map (labelled instrument) to their incoming system, described via Choi–Jamiołkowski isomorphism by positive semidefinite operators $M^{I_i O_i}_{a_i|x_i}$.

A global process matrix $W$ is a positive semidefinite operator on $$\bigotimes_i (\mathcal H^{I_i} \otimes \mathcal H^{O_i})$$, encoding all correlations among the parties. The joint conditional probability for outcomes is given by the generalized Born rule: $$P(a_1,\dots,a_N|x_1,\dots,x_N) = \mathrm{Tr}\left[ W \cdot \bigotimes_{i=1}^N M_{a_i|x_i}^{I_i O_i} \right]\,,$$ where $W$ must satisfy positivity and normalization constraints ensuring that $P$ is a valid conditional probability for any choice of local CP maps (Oreshkov et al., 2011, Ibnouhsein, 2015, Kunjwal et al., 2023). Crucially, no global background time or causal structure is assumed in this framework.

A process is causally separable if $S_i$0 can be decomposed as a convex combination of process matrices implementing definite global causal orders (e.g., $S_i$1 and $S_i$2 for two parties). If such a decomposition is impossible, $S_i$3 is causally nonseparable, and may exhibit indefinite causal order.

2. Causal Inequalities and Their Violation

When operations are embedded in an a priori definite causal structure—or any probabilistic mixture thereof—the allowed correlations satisfy causal inequalities. These are linear constraints on observed probabilities, analogous to Bell inequalities but targeting causal rather than local realism (Oreshkov et al., 2011, Brukner, 2014, Liu et al., 2024). The paradigmatic example is the Oreshkov–Costa–Brukner (OCB) game, with average success probability

$S_i$4

where $S_i$5 are guesses by Alice and Bob and $S_i$6 selects the task. This bound is tight for all definite-order (including probabilistic) strategies.

Quantum process matrices with indefinite causal order allow violations of causal inequalities. The maximal quantum violation, analogous to the Tsirelson bound for Bell inequalities, is (Brukner, 2014, Liu et al., 2024): $S_i$7 which is strictly less than the algebraic maximum $S_i$8.

The violation of a causal inequality operationally certifies that no global causal order can explain the observed correlations, in direct analogy to the device-independent witness of Bell nonlocality (Oreshkov et al., 2011, Ibnouhsein, 2015, Ibnouhsein et al., 2014).

3. Physical Realizations and Experimental Platforms

Correlations with no definite causal order can be physically realized, for instance, via the quantum switch—a higher-order operation in which two quantum channels are applied in a coherent superposition of alternative orders (Goswami et al., 2018, Chiribella et al., 2018, Zhao et al., 9 Oct 2025). In a typical implementation, a control qubit determines whether channel $S_i$9 precedes $\mathcal H^{I_i}$0 or vice versa: $\mathcal H^{I_i}$1 State-of-the-art photonic experiments demonstrate indefinite causal order by measuring a causal witness—a Hermitian operator whose expectation value is non-negative for all causally separable processes, but negative for causally nonseparable ones. Measured values exceeding the definite-order bound by many standard deviations unambiguously demonstrate indefinite causal order in a loophole-free manner (Goswami et al., 2018).

Another realization employs Gaussian-localized field-theoretic models within Minkowski spacetime. Here, “closed laboratories” are idealized as mode-selective mirrors, with parties acting on temporally and spatially delocalized wave packets. Despite all dynamics unfolding within ordinary spacetime, such protocols can violate the causal inequality, with the magnitude optimized by tuning the wavepacket spread $\mathcal H^{I_i}$2 and temporal delays between parties (Ho et al., 2018). This demonstrates that causal inequality violation need not require exotic spacetime structure: quantum field theory with temporally extended operations suffices.

4. Quantum Advantage, Limitations, and Resource Theory

Quantum correlations with no causal order represent a new operational resource distinct from entanglement and standard nonlocality. Indefinite causal order can provide concrete advantages in information processing, notably:

• One-shot classical/quantum communication: Certain pairs of noisy zero-capacity channels, when placed in indefinite causal order (e.g., via the quantum switch), achieve perfect channel fidelity—a phenomenon impossible even for superpositions of spatial paths (Chiribella et al., 2018, Zhao et al., 9 Oct 2025).
• Computation and channel discrimination: Indefinite-order processes enable discrimination between commutative and noncommutative operations with a single query and can reduce communication complexity in distributed tasks (Ibnouhsein, 2015).

However, these advantages are not generic. In the asymptotic (many-use) regime, the classical and quantum channel capacities of processes with maximal causal uncertainty (50:50 mixtures) reduce to those of the equivalent erasure channel (classical capacity $\mathcal H^{I_i}$3, quantum capacity $\mathcal H^{I_i}$4, with $\mathcal H^{I_i}$5 the bias towards one order) (Jia et al., 2019, Zhao et al., 9 Oct 2025). Entanglement assistance offers strictly more power than indefinite order in channel simulation tasks.

In resource-theoretic language, causal order emerges as a resource. Causally separable processes are “free states”; monotones such as the robustness of causal nonseparability quantify the degree of indefinite order. The quantum switch achieves indefinite order but always generates causal correlations in device-independent scenarios—its causal nonseparability can only be certified in semi-device-independent or trusted-instrument settings (Bavaresco et al., 2019, Steffinlongo et al., 21 Feb 2025). Device-independent causal-inequality violations require processes with more “genuine noncausality,” which currently lack a physical realization in quantum mechanics.

5. Multipartite and Classical Indefinite Causal Structures

Indefinite causal order admits rich generalizations beyond bipartite quantum scenarios:

• Multipartite noncausality: In the multipartite process-matrix setting, one can define “genuinely $\mathcal H^{I_i}$6-partite noncausal” correlations—those not reproducible by any partitioning of $\mathcal H^{I_i}$7 parties into causally ordered groups. The convex set of $\mathcal H^{I_i}$8-causal correlations is a polytope, and linear “$\mathcal H^{I_i}$9-causal” inequalities tightly characterize boundaries; violations by quantum processes establish genuine multipartite noncausality (Abbott et al., 2017).
• Classical indefinite-order processes: While the classical limit of the bipartite process-matrix framework always reproduces a definite causal order, in the multipartite (e.g., $\mathcal H^{O_i}$0) setting there exist classical, locally consistent protocols violating global causal order, strongly violating multipartite causal inequalities. Thus, indefinite causal order is not uniquely quantum but can also emerge in suitably structured classical communication scenarios (Baumeler et al., 2014, Kunjwal et al., 2023).

6. Hierarchies, Physical Principles, and Open Problems

The space of correlations realizable in “no global causal order” scenarios admits a strict hierarchy:

• Deterministically consistent (nomic) $\mathcal H^{O_i}$1 probabilistically consistent (classical) $\mathcal H^{O_i}$2 quantum-process correlations $\mathcal H^{O_i}$3 quasi-consistent (Kunjwal et al., 2023).
• “Antinomicity,” defined as the device-independent nonclassicality not explainable by any classical deterministic-local process function, refines causal-inequality violation. All antinomic correlations violate some causal inequality, but not all causal-inequality violations are antinomic (Kunjwal et al., 2023).

The quantum set is sharply bounded: Tsirelson-type bounds on causal-inequality violation restrict quantum correlations with indefinite order below the algebraic maxima, determined by semidefinite programming (SDP) methods. For all paradigmatic scenarios, these bounds have been established and are not saturated by known physically allowed procedures (Liu et al., 2024, Brukner, 2014).

Fundamental open problems include:

• Identification of physical or information-theoretic principles (analogous to information causality or macroscopic locality) that single out the quantum boundary for indefinite causality.
• Classification of causally nonseparable processes with and without device-independent causal-inequality violation, and the identification of physically realizable processes in higher-partite or more general settings (Feix et al., 2016, Steffinlongo et al., 21 Feb 2025).
• Connection of indefinite causal order to quantum gravity, emergent spacetime, and the operational foundations of quantum field theory (Ho et al., 2018, Ibnouhsein, 2015).

7. Broader Implications

Quantum correlations with no causal order fundamentally alter the conceptual structure of quantum theory. They force a reconsideration of causality, spacetime, and the operational capabilities of quantum information processing devices:

• Causal order, traditionally a background structure, is rendered an emergent or resource-like property of the operational layer, much like entanglement or coherence (Jia et al., 2019).
• Physical realizations necessarily involve temporally extended modes or coherent control of communication order, as in quantum switches or temporally delocalized field-theoretic operations (Goswami et al., 2018, Ho et al., 2018).
• Experimental demonstrations of causally nonseparable processes and the tight separation between semi-device-independent and device-independent certifications highlight both the physical achievability and absolute operational boundaries of indefinite causal order in quantum theory (Goswami et al., 2018, Lugt et al., 2023, Bavaresco et al., 2019).

Indefinite causal order thus stands as a fundamentally quantum—and potentially broader—phenomenon, with operational, foundational, and resource-theoretic consequences for quantum science and the quest for a deeper understanding of spacetime.