Causal Order Cannot Be An Observable
Abstract
Recent developments in the formalisation of quantum causal structures have made it possible to test and compare hypotheses about causal structure empirically, rather than being a-priori assumptions. Such differences in causal structure may be leveraged to distinguish between the processes they belong to, akin to distinguishing between quantum states known to belong to different eigenspaces of an observable. So how far this analogy can be pushed? Can causal order be interpreted as a kind of observable? Can it be measured? To this end we construct a completely operational definition of observables in terms of `discrimination tasks'. This begins with some base set of classes (analogous to a quantum observable's eigenspaces) before we impose three conditions on how these classes relate under discrimination tasks. These conditions recover the properties of a standard observable, letting us apply this definition to non-state entities such as quantum processes, which naturally encode causal structure. These conditions can be described in plain language as (1) `members of any class are perfectly distinguishable from members of any other' (i.e. our classes are `sharp'), (2) `entities perfectly distinguishable pairwise are all perfectly distinguishable by a joint intervention', and (3) `if all members of a class are perfectly distinguishable from an entity, then the whole class is perfectly distinguishable from it jointly'. By analysing classes of quantum processes having strict causal orders, we find causal classes which are sharp but violate condition 2, and classes which violate condition 3. This implies that causal order cannot be an observable. We note cases in the literature which claim to prove the opposite, and discuss them. We find their analyses presume special circumstances equivalent to conditions (2) and (3), and so do not contradict our own findings.
Versions
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BibTeX
@misc{maguire2026causal,
title = {{Causal Order Cannot Be An Observable}},
author = {Declan Maguire and Fabio Costa},
howpublished = {arXiv},
year = {2026},
month = may,
doi = {10.48550/arxiv.2605.25302},
eprint = {2605.25302},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.25302},
}
RIS
TY - UNPB
TI - Causal Order Cannot Be An Observable
AU - Declan Maguire
AU - Fabio Costa
PY - 2026
DA - 2026/05/24
PB - arXiv
DO - 10.48550/arxiv.2605.25302
UR - https://arxiv.org/abs/2605.25302
AB - Recent developments in the formalisation of quantum causal structures have made it possible to test and compare hypotheses about causal structure empirically, rather than being a-priori assumptions. Such differences in causal structure may be leveraged to distinguish between the processes they belong to, akin to distinguishing between quantum states known to belong to different eigenspaces of an observable. So how far this analogy can be pushed? Can causal order be interpreted as a kind of observable? Can it be measured? To this end we construct a completely operational definition of observables in terms of `discrimination tasks'. This begins with some base set of classes (analogous to a quantum observable's eigenspaces) before we impose three conditions on how these classes relate under discrimination tasks. These conditions recover the properties of a standard observable, letting us apply this definition to non-state entities such as quantum processes, which naturally encode causal structure. These conditions can be described in plain language as (1) `members of any class are perfectly distinguishable from members of any other' (i.e. our classes are `sharp'), (2) `entities perfectly distinguishable pairwise are all perfectly distinguishable by a joint intervention', and (3) `if all members of a class are perfectly distinguishable from an entity, then the whole class is perfectly distinguishable from it jointly'. By analysing classes of quantum processes having strict causal orders, we find causal classes which are sharp but violate condition 2, and classes which violate condition 3. This implies that causal order cannot be an observable. We note cases in the literature which claim to prove the opposite, and discuss them. We find their analyses presume special circumstances equivalent to conditions (2) and (3), and so do not contradict our own findings.
ER -