The Lüders (projection) postulate assigns the post-measurement state PxρPx/Tr(Pxρ)P_x\rho P_x/\mathrm{Tr}(P_x\rho) after a projective measurement with outcome PxP_x. Fiorentino and Weigert (Phys. Rev. A 113, 012204 (2026)) introduced a framework of generalised state-update rules: hypothetical theories that keep quantum states, projective measurements, and the Born rule, but replace the Lüders assignment by any update rule satisfying six operational requirements A1–A6 (completeness, context-independence, local covariance, self-consistency, no-signalling, local commutativity) together with Born-rule consistency and 1-homogeneity. Within this framework, coherence uniquely yields the Lüders rule for single systems, and coherence plus composition compatibility for composite systems; non-Lüders alternatives such as correlation-free, passive, depolarising, and probability-amplifying update rules are admissible. The open problem posed by that work is to settle the operational status of the projection postulate: whether the linearity of state updates — an assumption built into essentially every reconstruction of quantum theory — can itself be derived from operational principles rather than assumed; whether other natural principles, such as constraints on the degree of nonlocality, single out the Lüders rule; and whether admissible non-Lüders rules can reproduce or even exceed quantum correlations.

Background

The state-update postulate. In quantum mechanics, a projective measurement of a discrete observable is described by a projection-valued measure (PVM) — a family of orthogonal projectors {Px}\{P_x\} with ∑xPx=I\sum_x P_x=\mathbb{I}. Given the pre-measurement density operator ρ\rho (a positive operator of unit trace encoding preparation statistics), the Born rule assigns outcome xx probability Tr(Pxρ)\mathrm{Tr}(P_x\rho), and the state-update postulate fixes the post-measurement state. The Lüders rule assigns the normalised post-measurement state PxρPx/Tr(Pxρ)P_x\rho P_x/\mathrm{Tr}(P_x\rho); von Neumann's original projection postulate additionally chooses a basis inside each degenerate eigenspace. In the quantum-instrument formalism, each outcome acts on states through a quantum operation — a linear, completely positive, trace-non-increasing map, where complete positivity means the map remains positive when tensored with the identity on an arbitrary ancillary system, ensuring consistent behaviour on subsystems.

Foil theories. Whether a feature of quantum theory is forced or merely postulated is studied through foil theories: internally consistent alternatives differing from quantum theory in specific postulates. Generalised probabilistic theories (GPTs) modify the state space, effects, and composition rules; some GPTs exhibit super-quantum correlations, i.e. Bell-inequality violations beyond the Tsirelson bound of quantum theory (e.g. Popescu–Rohrlich boxes). Fiorentino and Weigert instead keep the Hilbert-space kinematics and the Born rule and vary only the update rule. A generalised state-update rule is a family of maps ωAB:P(HA)×Sˉ(HA⊗HB)→Sˉ(HA⊗HB)\omega_{AB}:\mathcal{P}(\mathcal{H}_A)\times\bar{\mathcal{S}}(\mathcal{H}_A\otimes\mathcal{H}_B)\to\bar{\mathcal{S}}(\mathcal{H}_A\otimes\mathcal{H}_B) (outcomes are projectors, states may be sub-normalised, i.e. of trace at most one) satisfying: (i) consistency with the Born rule, Tr[ωAB(PxA,ρAB)]=Tr(PxA⊗IB ρAB)\mathrm{Tr}[\omega_{AB}(P_x^A,\rho_{AB})]=\mathrm{Tr}(P_x^A\otimes\mathbb{I}_B\,\rho_{AB}); (ii) 1-homogeneity, ωAB(PxA,λρAB)=λ ωAB(PxA,ρAB)\omega_{AB}(P_x^A,\lambda\rho_{AB})=\lambda\,\omega_{AB}(P_x^A,\rho_{AB}) for λ∈[0,1]\lambda\in[0,1]; and six operational assumptions: A1 completeness (every outcome specifies a unique post-measurement state); A2 context-independence (the outcome and pre-measurement state alone determine the post-measurement state); A3 local covariance (local changes of reference frame commute with the update); A4 self-consistency (the update does not depend on whether the system is regarded as part of a larger composite); A5 no-signalling (an unconditioned local measurement leaves the reduced state of unmeasured subsystems unchanged); A6 local commutativity (local updates on spacelike-separated subsystems commute, so their order is irrelevant). A theory obtained by replacing the projection postulate with such an ω\omega is a generalised state-update theory (GUT). Unlike quantum instruments, these updates need be neither linear nor completely positive; consequently density operators may cease to describe systems completely, because a non-linear update can distinguish a proper mixture (classical ignorance about preparations) from an improper one (the reduced state of an entangled system).

Known results. Four admissible non-Lüders examples exist: correlation-free quantum theory (Lüders on single systems, but updates that map entangled joint states to product states, destroying all correlations); passive quantum theory (measurements leave the state unchanged; the rule ωP(P,ρ)=Tr(Pρ)ρ\omega^{\mathsf{P}}(P,\rho)=\mathrm{Tr}(P\rho)\rho is non-linear, and the theory permits tomographic reconstruction and cloning of individual systems and more efficient computation, but no Bell-inequality violations); depolarising measurements (any outcome sends the state to the maximally mixed state); and probability-amplifying projective updates. Coherence — the requirement that the outcome statistics of any observable be unchanged by a prior measurement of a coarse-graining of that observable — uniquely singles out the Lüders rule for non-composite systems (with historical antecedents surveyed in the paper's Appendix A), and coherence plus composition compatibility (the composite update is the standard tensor extension of the single-system rule) singles it out in general. However, essentially all reconstructions of quantum theory assume, explicitly or implicitly, that the state update is (convex-)linear in the pre-measurement state. Masanes–Galley–Müller claimed to deduce the measurement postulates from the remaining postulates plus an operational assumption; Fiorentino–Weigert, following Flatt–Barnett–Croke, observe that this assumption is mathematically equivalent to convex-linearity, and Kent has exhibited non-quantum measurement and state-update rules satisfying all of Masanes–Galley–Müller's assumptions, so the dispute remains unresolved. Related constraints are known to be inconclusive: complete positivity has been described as 'physically unfitting' as a constraint on nonlinear maps (Czachor–Kuna); non-linear single-system dynamics need not enable signalling (Kent 2005); and signalling arguments of the kind Gisin used for non-linear time evolution do not settle the update-rule case.

Problem Statement

Within the framework of generalised state-update theories — hypothetical theories sharing quantum theory's Hilbert-space states, projective (PVM) measurements, and Born rule, with post-measurement states assigned by an update rule ω\omega satisfying Born-rule consistency, 1-homogeneity, and assumptions A1–A6 (completeness, context-independence, local covariance, self-consistency, no-signalling, local commutativity; Eqs. (6)–(13) of Fiorentino–Weigert, Phys. Rev. A 113, 012204 (2026)) — the following interlinked questions are open:

(1) Linearity. Can the (convex-)linearity of the state-update rule in the pre-measurement state be derived from operational principles rather than assumed? The principles must be stated, like A1–A6, as constraints on observable statistics or on the behaviour of ω\omega, and must not covertly encode the conclusion; A1–A6 cannot suffice, since non-linear rules such as passive quantum theory already satisfy them.

(2) Exclusion principles. Coherence plus composition compatibility singles out the Lüders rule. Are there other physically natural operational principles — in particular constraints on the degree of nonlocality or on achievable correlations — that also single out the Lüders rule among all admissible generalised state-update theories, and, if so, what is the smallest set of principles that does so without composition compatibility — in particular, can a single principle beyond A1–A6 and coherence replace it?

(3) Correlations. Does there exist an admissible update rule, differing from the Lüders rule on composite-system post-measurement states, that nevertheless reproduces the correlations of quantum theory (the same Bell-inequality violations, e.g. the Tsirelson bound for the CHSH (Clauser-Horne-Shimony-Holt) expression)? Can any admissible rule produce super-quantum correlations, exceeding the quantum bound while respecting no-signalling? Which constraints, such as no-signalling combined with specified degrees of nonlocality, rule such alternatives in or out?

The three questions are independent sub-questions of a common objective and are separately resolvable: answering any one of them decisively, with the principles or rules it invokes stated explicitly, constitutes a complete solution, and they need not all be answered. Two of them bundle several distinct asks — (2) two (an exclusion principle, and the smallest set achieving it) and (3) three (reproducing quantum correlations, producing super-quantum correlations, and the constraints ruling alternatives in or out) — and settling one ask resolves that question only in part; a solution should state which question, and, within (2) and (3), which ask, it settles. The common objective is to settle the operational status of the projection postulate: either the Lüders rule and its linearity prerequisite are forced by operational principles (question (1)), or the Lüders rule is singled out by some further natural operational principle (question (2)), or the consistent alternatives are mapped out together with the correlation signatures that would distinguish them from quantum theory (question (3)).

References

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