A well-recognized open conceptual problem in relativistic quantum field theory concerns the relation between measurement and causality. Naive generalizations of quantum measurement rules can allow for superluminal signaling (``impossible measurements''). This raises the problem of delineating physically allowed quantum measurements and operations. We analyze this issue in a recently proposed framework in which local measurements (in possibly curved spacetime) are described physically by coupling the system to a probe. We show that the state-update rule in this setting is consistent with causality provided that the coupling between the system and probe is local. Thus, by establishing a well-defined framework for successive measurements, we also provide a class of physically allowed operations. Conversely, impossible measurements can only be performed using impossible (nonlocal) apparatus.
A well-recognized open conceptual problem in relativistic quantum field theory concerns the relation between measurement and causality. Naive generalizations of quantum measurement rules can allow for superluminal signaling (``impossible measurements''). This raises the problem of delineating physically allowed quantum measurements and operations. We analyze this issue in a recently proposed framework in which local measurements (in possibly curved spacetime) are described physically by coupling the system to a probe. We show that the state-update rule in this setting is consistent with causality provided that the coupling between the system and probe is local. Thus, by establishing a well-defined framework for successive measurements, we also provide a class of physically allowed operations. Conversely, impossible measurements can only be performed using impossible (nonlocal) apparatus.
The topic of measurement in relativistic quantum field theory is addressed in this article. Some of the long standing problems of this subject are highlighted, including the incompatibility of an instantaneous ``collapse of the wavefunction'' with relativity of simultaneity, and the difficulty of maintaining causality in the rules for measurement highlighted by ``impossible measurement'' scenarios. Thereafter, the issue is considered from the perspective of mathematical physics. To this end, quantum field theory is described in a model-independent, operator algebraic setting, on generic Lorentzian spacetime manifolds. The process of measurement is modelled by a localized dynamical coupling between a quantum field called the ``system'', and another quantum field, called the ``probe''. The result of the dynamical coupling is a scattering map, whereby measurements carried out on the probe can be interpreted as measurements of induced observables on the system. The localization of the dynamical coupling allows it to derive causal relations for the induced observables. It will be discussed how this approach leads to the concept of selective or non-selective system state updates conditioned on the result of probe measurements, which in turn allows it to obtain conditional probabilities for consecutive probe measurements consistent with relativistic causality and general covariance, without the need for a physical collapse of the wavefunction. In particular, the problem of impossible measurements is resolved. Finally, there is a brief discussion of accelerated detectors and other related work.
Naive attempts to put together relativity and quantum measurements lead to signaling between space-like separated regions. In QFT, these are known as impossible measurements . We show that the same problem arises in non-relativistic quantum physics, where joint nonlocal measurements (i.e., between systems kept spatially separated) in general lead to signaling, while one would expect no-signaling (based for instance on the principle of no-nonphysical communication ). This raises the question: Which nonlocal quantum measurements are physically possible? We review and develop further a non-relativistic quantum information approach developed independently of the impossible measurements in QFT, and show that these two have been addressing virtually the same problem. The non-relativistic solution shows that all nonlocal measurements are localizable (i.e., they can be carried out at a distance without violating no-signaling) but they (i) may require arbitrarily large entangled resources and (ii) cannot in general be ideal , i.e., are not immediately reproducible. These considerations could help guide the development of a complete theory of measurement in QFT.
Quantum operations that are perfectly admissible in non-relativistic quantum theory can enable signalling between spacelike separated regions when naively imported into quantum field theory (QFT). Prominent examples of such "impossible measurements" include certain unitary kicks and projective measurements. It is generally accepted that only those quantum operations whose physical implementation arises from a fully relativistically covariant interaction, between the quantum field and a suitable probe, should be regarded as admissible. While this idea has been realised at the level of abstract algebraic QFT, or via particular measurement models, there is still no general set of operational criteria characterising which measurements are physically implementable. In this work we adopt the local S-matrix formalism to describe local operations, and make use of a hierarchy of factorisation conditions that exclude both superluminal signalling and retrocausality, thereby providing such a criterion. Realising the local S-matrices through explicit interactions between smeared field operators and a pointer degree of freedom, we further derive local causality conditions for the induced Kraus operators, which guarantee the absence of signalling in "impossible measurement" scenarios. Finally, we show that the accuracy with which local field observables can be measured is fundamentally limited by the retarded propagator of the field, which plays an essential role in the factorisation identity we prove for the field Kraus operators.
It is by now well recognized that the naive application of the projection postulate on composite quantum systems can induce signalling between their constituent components, indicative of a breakdown of causality in a relativistic spacetime context. Here we introduce a necessary and sufficient condition for an ideal measurement of an observable to be nonsignalling. As well as being particularly simple, it generalizes previous no-signalling conditions in that it allows for degeneracies and can be applied to all bounded self-adjoint operators. The condition is used to establish that arbitrary sums of local observables will not signal, in accordance with our expectations from relativistic quantum field theory. On the other hand, it is shown that the measurement of the tensor product of commuting local observables, for example bipartite operators of the form $A\ensuremath{\bigotimes}B$, can in fact signal, contrary to the widely held belief that such measurements are always locally realizable. The implications for the notion of measurement in relativistic quantum field theory are addressed; it appears that the most straightforward application of the standard quantum formalism generically leads to violations of causality. We conclude that either the class of observables that can be measured should be restricted and/or that the naive translation of the measurement framework of quantum theory, in particular the projection postulate, to quantum field theory should be reevaluated.
In this essay, we explore the limits imposed by the impossibility of superluminal signalling on the class of physically realisable quantum operations, focusing on the difference in approaches one can take towards this problem in Hilbert space quantum theory and algebraic quantum field theory (AQFT). We review a recent proposal by Fewster and Verch for a generally covariant measurement scheme in AQFT, which we call the FV scheme; along the way, we argue that the framework as set up in its original proposal can be greatly simplified without losing generality or physical motivation. We then discuss the problem, introduced by Sorkin, that naive generalisations of the notion of operation to the relativistic context can lead to signalling between spacelike observers by introducing a third observer. We consider approaches to tackle this problem in Hilbert space quantum theory and argue that these are sometimes ambiguous, since the physical (im)possibility of operations depends in part on the specific spacetime embodiment of the quantum system under consideration, which lies outside the Hilbert space formalism. In the AQFT context, we review and generalise a recent result showing that operations induced by the FV measurement scheme do not enable superluminal signalling. We connect this result to Hilbert space by introducing a concrete model of AQFT, which we call the `hybrid model' as it provides a way to explicitly embed Hilbert space quantum theory into spacetime using the formalism of AQFT. Finally, we show that in this model, under suitable necessary and sufficient conditions, the converse result also holds: every operation that does not enable superluminal signalling can be implemented in the FV framework.
We provide a framework and explicit construction for the regularized measurement of a large class of spacetime-localized observables in bosonic quantum field theory. The measurements fully satisfy relativistic causality and causal transparency, i.e., avoid unphysical superluminal signaling. We show explicitly how the measurement of time-extended observables back-reacts on itself and induces correlations between other measurements in its causal future. Our framework is fully compositional in spacetime and extends previous results on the measurement of instantaneous observables.