| Lüders rule | |
|---|---|
| Name | Lüders rule |
| Caption | Projection update in quantum measurement |
| Field | Quantum mechanics |
| Introduced | 1950 |
| Introduced by | Gerhart Lüders |
Lüders rule
Lüders rule is a prescription for updating the state of a quantum system after a projective measurement with degenerate eigenvalues. It refines the measurement update postulate of Quantum mechanics by specifying how a density operator transforms under ideal measurements that have degenerate spectral projectors. Lüders rule matters because it preserves coherence within degenerate subspaces and underpins rigorous treatments of quantum measurement in statistical ensembles, quantum information, and decoherence studies.
Lüders rule states that, given an observable represented by a self-adjoint operator A with spectral decomposition A = Σ_a a P_a where P_a are orthogonal projection operators (possibly of rank >1), a nonselective ideal measurement that yields outcome a updates an initial density operator ρ to
ρ → ρ_a = (P_a ρ P_a) / Tr(P_a ρ P_a).
For the nonselective (averaged) update over all outcomes the post-measurement state is ρ' = Σ_a P_a ρ P_a. This prescription contrasts with pure-state collapse schemes and is formulated naturally for mixed states and ensembles used in statistical descriptions by institutions such as the Statistical mechanics community and in textbooks by authors like J. von Neumann and P. Dirac.
The mathematical basis of Lüders rule relies on the spectral theorem for self-adjoint operators on a Hilbert space H and the theory of density operators (trace-class positive operators with unit trace). Starting from the assumption of repeatable, ideal projective measurements, one imposes that the update map be completely positive and trace-preserving on the subspace associated with the observed eigenvalue. Using the projection postulate leads to the Kraus operator form with Kraus operators equal to the projectors P_a; hence the nonselective map is a completely positive trace-preserving (CPTP) map ρ ↦ Σ_a P_a ρ P_a. This links Lüders rule to the theory of quantum channels and the Stinespring dilation theorem that characterize physically implementable transformations.
Key mathematical objects include the density matrix ρ, orthogonal projections P_a, and the trace functional Tr(·). Connections to operator algebras, particularly C*-algebra and von Neumann algebra techniques, clarify how degeneracy and commutation relations affect the update. The rule is consistent with Wigner's theorem on symmetry transformations where unitary or antiunitary maps preserve projector structure.
Lüders rule sits within the broader family of measurement postulates in foundations of quantum mechanics. It refines von Neumann's original collapse postulate by addressing degenerate measurements: whereas von Neumann emphasized collapse to an eigenvector, Lüders prescribes collapse to the corresponding eigenspace via projection. This preserves superpositions within the degenerate eigenspace, aligning with operational requirements in quantum information theory and experiment.
The rule is compatible with Born's rule for outcome probabilities, p(a) = Tr(P_a ρ), and with conditional state definitions used in sequential measurement scenarios and measurement-based quantum protocols such as quantum tomography and quantum error correction. Debates in the literature—invoking figures like John Bell and Eugene Wigner—highlight conceptual implications for realism and locality when combined with entanglement and nonlocal correlations exemplified by Bell's theorem and EPR paradox.
Lüders updates serve as idealized models for decoherence induced by projective interactions with a measurement apparatus or environment. In open quantum systems theory, the map ρ ↦ Σ_a P_a ρ P_a models complete dephasing in the eigenbasis of A and is a limiting case of dynamical semigroups described by the Lindblad equation. It is used to analyze loss of coherence in decoherence studies, pointer states in quantum measurement problem, and the emergence of classicality in models by researchers at institutions such as Los Alamos National Laboratory and Perimeter Institute.
Practical applications include modeling readout in quantum computing hardware (superconducting qubits at IBM and Google), state-preparation via measurement in quantum control protocols, and theoretical construction of measurement-based quantum gates. The rule also underlies statistical estimation procedures in quantum state estimation and ensemble descriptions in quantum statistical mechanics.
Compared with von Neumann's original projection, Lüders rule generalizes collapse to handle degeneracy without selecting an orthonormal basis within degenerate subspaces. In contrast to the much broader framework of positive operator-valued measures (POVMs), Lüders projections are a special case where measurement operators are orthogonal projectors and the instrument is repeatable. POVMs, described by nonorthogonal effects E_i, allow more general measurement outcomes and are represented via Kraus operators that need not be projectors; these are crucial in quantum optics (e.g., photodetection described by Glauber theory) and optimal discrimination tasks studied by groups at MIT and Caltech.
Operationally, Lüders maps are idempotent on the algebra generated by the P_a and commute with observables that share the same spectral projectors; POVM-based measurements need not share such commutation properties. The comparison elucidates when an ideal projective model suffices and when generalized measurement theory is required for realistic instruments.
Experimental implementations of Lüders-type projections occur in atomic, optical, and solid-state platforms where projective readout is approximated. Experiments with trapped ions at institutions like IonQ and NIST perform near-ideal projective measurements, demonstrating state update consistent with Lüders predictions. Superconducting qubit readout and single-photon polarization measurements in quantum optics laboratories verify selective and nonselective projection statistics and coherence retention within degenerate subspaces.
Tests often involve sequential measurement protocols that probe repeatability and disturbance, and tomography to reconstruct post-measurement states. Deviations from Lüders rule signal nonidealities such as measurement back-action beyond projection, detector inefficiency, or coupling to uncontrolled environments, motivating refined models using POVMs and quantum instrument frameworks developed by researchers at Oxford University and ETH Zurich.