Why causal order may not behave like a quantum observable

Quantum observables are often pictured as labels that a measurement can reveal: spin is up or down, for example. In quantum theories where the order of events is not fixed in advance, it is tempting to imagine that causal order could work similarly. An experiment might determine whether one operation came before another, or which ordering applies among several possibilities.

Declan Maguire and Fabio Costa ask whether this analogy holds for quantum processes, mathematical objects that describe how several interventions fit together and what probabilities they produce. Their answer is that causal order, as they formalise it, cannot be a sharp observable in the same way as spin or energy.

What would it mean to measure an order?

The authors define an observable through a single-shot discrimination task. An experimenter receives one unknown object, knows it belongs to one of several possible classes, and must identify its class from a measurement. For an ordinary sharp observable, one measurement can correctly identify the value for any state in the relevant class.

The authors translate this idea into three conditions for classes of processes. First, any two members from different classes must be perfectly distinguishable: the test never confuses them. But that pair-by-pair ability is not enough. A single intervention must also be able to distinguish a collection of objects drawn from different classes all at once. And if one object can be distinguished from every member of a class individually, a single intervention must be able to distinguish it from that class as a whole. Together, the conditions capture the authors’ operational meaning of an observable.

This framework matters because processes are not quantum states. For states, perfect pairwise distinguishability corresponds to orthogonality, which lets pairwise distinctions be combined into a measurement of the observable. That implication does not automatically hold for processes or channels.

Constructing the counterexamples

The paper builds examples using processes with a definite sequence of sites—locations where operations are applied—and channels that carry information between them. The examples are restricted to Markovian processes, where influence from one site to the next is mediated by the connecting channel. The authors also require those channels to transmit at least one bit perfectly, so the examples have a clear, detectable causal order rather than an order obscured by complete noise.

In one construction, the researchers take processes with the same local system dimensions and rearrange the order of the sites. They show that any two processes with different orders can be distinguished perfectly. But for sufficiently many sites, a dimension argument shows that no single intervention can map all the possible orders into mutually orthogonal measurement outcomes. The set of pairwise distinctions therefore does not add up to one measurement that identifies every order.

A second construction concerns just two sites, but allows different channels between them. Here, processes with opposite orders can be perfectly distinguished member by member. Yet mixing over the possible connecting channels can make the two opposite-order classes produce the same effective process. If the classes can overlap in this way, no single intervention can distinguish the classes in every case. The authors also construct examples that violate both of these measurement requirements while retaining pairwise distinguishability.

The result is not that causal order can never be identified in an experiment. If the connecting operations are known, the paper says, it can be possible to distinguish two processes with opposite orders. The problem is the broader demand that one measurement work across all members of each causal class—or across many possible orders. The order cannot, on this account, be detached from the specific transformations linking the events.

What the conclusion does—and does not—say

The authors argue that their examples are enough to rule out causal order as an observable under their definition. They also contend that moving the processes into another representation cannot repair the failure: a linear mapping of the processes has a corresponding action on the possible interventions, so the missing discrimination ability is not simply a problem with their chosen representation.

This is a mathematical argument about a defined measurement task, not an experimental demonstration that causal order is unknowable. The paper explicitly allows for methods that witness causal properties and for indirect reconstruction from correlations. It also does not disprove indefinite causal order or establish that every notion of causal structure in every theory must fail as an observable.

The scope deserves care. The counterexamples use specially constructed, causally strict processes, and the paper’s broader conclusion depends on the authors’ operational definition and on their arguments about how failures carry through mappings and class restrictions. The supplied source is a preprint, and its claims should be read as research results awaiting the scrutiny associated with publication and further work. The useful takeaway is narrower than the title may suggest: being able to tell two causal orders apart in selected cases does not guarantee that causal order behaves like a standard quantum observable across a whole family of processes.