The polyperspective formalism of Polo-Gómez, Perche, and Martín-Martínez (2025) resolves the tension between covariant, causal state updates and the preservation of multipartite correlations after selective measurements in relativistic spacetime by replacing the single quantum state with a multiplet of density operators (the polystate), generated from an initial state by a family of completely positive spacetime-local maps ΨS\Psi_S. However, these maps have been written down explicitly only for ideal projective measurements on finite-dimensional qubit systems. The open problem is to derive and explicitly construct the family ΨS\Psi_S from first principles for a physically realistic model — for example, a quantum field probed by localized detectors in flat or curved spacetime, or an interacting quantum field theory — including, where the model involves them, its dependence on detector coupling, switching, renormalization, and field backreaction, and to verify that the resulting polystate retains the formalism's defining properties: covariance, no-signalling in individual sectors, preservation of multipartite correlations in joint sectors, and conservation of global charges across arbitrary spacetime foliations.

Background

In non-relativistic quantum mechanics, a selective measurement (one whose outcomes are post-selected) updates the state of all entangled subsystems at once through the Lüders projection rule, the standard prescription in which the pre-measurement density operator is projected onto the eigenspace corresponding to the observed outcome. In a relativistic spacetime this prescription becomes ill-defined: for spacelike-separated parties there is no fact about which measurement happened first, and no preferred spacelike hypersurface on which to apply the update. The problem has a long history. Hellwig and Kraus (1970) and Aharonov and Albert (1980) formulated measurement updates for local observables on hypersurfaces in quantum field theory (QFT); Aharonov and Albert (1980) in particular showed that collapse-like updates lead to inconsistencies for non-local observables such as the total charge integrated over a hypersurface. Sorkin (1993), in his “impossible measurements” argument, showed that a naively localized projection postulate in QFT would enable faster-than-light signalling; Dowker (2011) pressed the measurement problem on the relativistic quantum information (RQI) community. Jubb (2022) and Albertini and Jubb (2023) showed that even ideal measurements of local smeared field operators violate causality. Several frameworks now address local measurement in QFT — the algebraic measurement schemes of Fewster and Verch (2020), detector-based measurement theory (Polo-Gómez, Garay, Martín-Martínez 2022; Pranzini and Keski-Vakkuri 2025), and Oeckl's positive formalism (a reformulation of quantum theory in which probabilities are assigned to measurements in finite spacetime regions rather than on global time slices) — each with different update rules.

Against this backdrop, Polo-Gómez, Perche, and Martín-Martínez (2025) argued that every single-state prescription is unsatisfactory: updating along the future lightcone of the measurement respects the causal propagation of information but destroys correlations between parties spacelike separated from the measurement, while updating along the past lightcone amounts to a retroactive update. Both fail to conserve global charges on some hypersurfaces. Their proposal, the polyperspective formalism, replaces the state of a multipartite system with a polystate

ρ~(τA,τB)=ρ^A(τA)⊕ρ^B(τB)⊕ρ^AB(τA,τB),\tilde{\rho}(\tau_A,\tau_B)=\hat{\rho}_A(\tau_A)\oplus\hat{\rho}_B(\tau_B)\oplus\hat{\rho}_{AB}(\tau_A,\tau_B),

a direct sum of individual and joint density operators, each sector carrying the information available to the corresponding party or coalition. The sectors are generated from an initial joint state ρ^AB\hat{\rho}_{AB} by a family of completely positive maps ΨS\Psi_S (completely positive means the map remains positive when acting on part of an entangled larger system, the standard consistency condition for quantum operations) labelled by causally convex spacetime regions SS (a region is causally convex if every causal curve with endpoints in it stays inside it); ΨS\Psi_S encodes all evolution and measurements occurring in SS. For instance ρ^A(τA)∝TrB ΨJ−(xA(τA))(ρ^AB)\hat{\rho}_A(\tau_A)\propto\mathrm{Tr}_B\,\Psi_{\mathcal{J}^{-}(\mathsf{x}_A(\tau_A))}(\hat{\rho}_{AB}), where J−(x)\mathcal{J}^{-}(\mathsf{x}) is the causal past of the spacetime point x\mathsf{x} and τA\tau_A the proper time along party A's worldline. Joint observables — those requiring data from several parties — are evaluated in a processing region, a spacetime region in the common causal future of the local operations where outcomes can be gathered, using the joint sector. A foliation is a decomposition of spacetime into spacelike Cauchy hypersurfaces Σt\Sigma_t (hypersurfaces on which initial data determine the state everywhere in spacetime); the foliation-dependent state ρ^Σ(t)∝ΨJ−(Σt)(ρ^0)\hat{\rho}_{\Sigma}(t)\propto\Psi_{\mathcal{J}^{-}(\Sigma_t)}(\hat{\rho}_0) generally differs between foliations, while the polystate itself is covariant. The formalism reproduces no-signalling statistics for individual parties, preserves correlations in joint sectors, and conserves global charges across every foliation.

What the paper works out explicitly, however, covers only finite-dimensional systems: the maps ΨS\Psi_S are constructed only for ideal projective measurements on qubits. The authors state that the observer-dependent generalization is “particularly convenient for spatially extended non-relativistic systems and quantum field theories”, deferring a QFT implementation to separate work that has not appeared, and Appendix B notes that applying the recipe to each concrete scenario “amounts to identifying the appropriate family of maps ΨS\Psi_S for that scenario”. How to identify, derive, and characterize those maps for realistic systems — extended detectors, quantum fields, curved backgrounds, with coupling, renormalization, and backreaction — is the unfinished part of the program.

Problem Statement

For the polyperspective formalism of Polo-Gómez, Perche, and Martín-Martínez (arXiv:2506.18906), construct explicitly — deriving it from first principles of the underlying physical model rather than postulating it — the family of completely positive spacetime-local maps ΨS\Psi_S (a completely positive map is one that remains positive when applied to part of an entangled larger system; a spacetime-local map acts only on data from a causally convex spacetime region, i.e. one that contains every causal curve joining any two of its points), together with the sector rule of Eq. (10)–(12) of that paper, generalized as in its Eq. (22), for at least one physically realistic model class beyond finite-dimensional qubits, for example: (a) a quantum field in flat or curved spacetime probed by one or more localized non-relativistic detectors (such as an Unruh–DeWitt two-level detector linearly coupled to a scalar field), where the maps encode the physical ingredients of the model — detector coupling, switching (the time profile of the coupling), and, to the extent the model includes them, the renormalization of the field observables being measured and the backreaction of the measurement on the field; or (b) an interacting quantum field theory in a curved spacetime. The construction must be shown to retain the formalism's defining properties: (i) covariance, i.e. ΨS\Psi_S and the induced polystate (the multiplet of individual and joint density-operator sectors) depend only on the causal structure of the spacetime region, not on a chosen foliation (a decomposition of spacetime into spacelike Cauchy hypersurfaces, i.e. hypersurfaces on which initial data determine the state everywhere); (ii) no-signalling — individual sectors reproduce the statistics available to each party alone, with no faster-than-light signalling; (iii) preservation of multipartite correlations — joint sectors reproduce the correct joint statistics for measurements aggregated in any admissible processing region (a region in the common causal future of the local operations); and (iv) conservation of global charges (e.g. total charge integrated over arbitrary Cauchy hypersurfaces) across all foliations. Secondary sub-questions, useful but not required for completion, are the formulation of operational rules telling experimenters which sector to use and how to select processing regions in multi-step distributed protocols, and the analysis of whether foliation-dependent states are empirically distinguishable.

References

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