Projection PostulateEdit
The projection postulate is a foundational rule in the standard formulation of quantum mechanics that describes how the state of a quantum system changes when a measurement is performed. It provides a bridge between the deterministic, unitary evolution that governs a system between measurements and the probabilistic outcomes that arise in measurements themselves. In its most common form, the postulate asserts that a measurement yields a definite eigenvalue of the measured observable and that the post-measurement state collapses to the corresponding eigenstate (or eigensubspace). The probabilities for the possible outcomes are given by the Born rule, and the state update follows a precise prescription.
In modern language, the projection postulate sits alongside the unitary evolution generated by the Schrödinger equation as part of the two-layer dynamics that characterize quantum processes. Between measurements, a closed system evolves deterministically under a Hamiltonian. At the moment of measurement, the state undergoes an abrupt update consistent with the observed value. This dichotomy is central to many discussions about the meaning of quantum theory and its interpretation, but the operational content of the postulate—how to predict outcomes and update the state—remains a robust part of the mathematical framework used in laboratories around the world quantum mechanics.
Formal statement
Let A be an observable with a spectral decomposition A = ∑_i a_i P_i, where the P_i are the projection operators onto the eigen-subspaces associated with the eigenvalues a_i. If the system is prepared in a pure state |ψ⟩, the probability of obtaining outcome a_i upon measuring A is
p_i = ⟨ψ| P_i |ψ⟩.
If the outcome a_i is observed, the post-measurement state is
|ψ'⟩ = P_i |ψ⟩ / sqrt(p_i),
which, in the case of a degenerate eigenvalue, places the state into the corresponding eigensubspace rather than a single eigenvector (this refinement is often attributed to the Lüders postulate). In the density-matrix formalism, the post-measurement state after obtaining outcome i is
ρ' = (P_i ρ P_i) / Tr(P_i ρ).
This formulation is the core of the projective measurement model, and it provides a precise, testable rule for how measurements update the state of a system.
The projection postulate is the simplest instantiation of a broader idea: measurements can be treated as physical processes that select a subset of the state space compatible with the observed value. In more general measurement schemes, described by generalized measurements, the update rule is captured by Kraus operators {M_k} with ∑_k M_k† M_k = I, leading to
ρ' = (M_k ρ M_k†) / Tr(M_k ρ M_k†)
for the observed outcome k. The projection postulate is the special case where each outcome corresponds to a single projector, i.e., M_k = P_k. This broader framework encompasses a wide range of measurement procedures, including those that are not perfectly projective in practice Kraus operators POVM Positive-operator-valued measure.
History and interpretations
The postulate originated in the mathematical treatment of quantum theory developed by von Neumann, who distinguished between the unitary, deterministic evolution of isolated systems and a nonunitary, discontinuous change associated with measurement. The resulting two-stage view—continuous evolution punctuated by abrupt state update—became a central feature of the traditional account of quantum phenomena. Over time, this led to a variety of interpretations about what the collapse actually represents and when it occurs.
Different interpretations offer distinct readings of the projection postulate. In the Copenhagen tradition, measurement plays a special role and the postulate is taken as a real physical update that reflects the acquisition of a definite outcome. Other viewpoints treat the collapse as an update of information or a Bayesian revision of an observer’s knowledge, rather than a physical process. The Many-Worlds Interpretation posits that all possible outcomes occur in branching universes and that no single collapse happens; what is observed is a single branch of a more complex, unitary evolution. Decoherence theory explains the appearance of collapse as a rapid loss of coherence due to interaction with an environment, without requiring a fundamental, nonunitary reduction at the level of the universal wavefunction. These interpretational debates are central to the philosophy of quantum mechanics and continue to influence how researchers frame experiments and analyze data Copenhagen interpretation Many-worlds interpretation decoherence measurement problem.
Generalizations and related rules
Lüders postulate: For degenerate eigenvalues, the state after a measurement with eigenvalue a_i is projected onto the full degenerate subspace, ensuring the post-measurement state remains pure if the pre-measurement state was pure.
Generalized measurements and quantum operations: The Kraus operator formalism extends the projection postulate to a broad class of measurement procedures. This framework is essential for describing imperfect measurements, indirect measurements, and measurements performed with ancillary systems Kraus operators POVM.
State-update rules for density operators: The density-matrix version of the postulate makes explicit how mixed states transform under measurement and how classical information about the outcome gets correlated with the quantum state.
Examples
Spin-1/2 along the z-axis: If the system is in |ψ⟩ = α|↑⟩ + β|↓⟩, measuring σ_z yields outcomes ↑ and ↓ with probabilities |α|^2 and |β|^2, respectively. If the outcome is ↑, the post-measurement state collapses to |↑⟩ (up to a phase), and similarly for ↓.
Energy measurement in a superposition: If the system is in a superposition of energy eigenstates |E_n⟩, a measurement of the Hamiltonian yields eigenvalues E_n with probabilities |⟨E_n|ψ⟩|^2, and the state collapses to the corresponding |E_n⟩.
Degenerate measurements: If an observable has a degenerate eigenvalue with projector P_i, the post-measurement state lies in the subspace P_i H, and the exact post-measurement state depends on the measurement details; Lüders rule specifies the standard update within that subspace.
Experimental status
The projection postulate provides predictions that have been repeatedly tested and are foundational to the practice of quantum mechanics. It underpins procedures in quantum state tomography, quantum computation, and quantum communication, where measurements are used to extract information and to condition subsequent operations. Modern experiments often employ generalized measurements and quantum channels that go beyond ideal projective measurements, yet the core update rule remains compatible with the broader Kraus-operator framework. The interplay between a system and its environment, captured by decoherence theory, helps explain why measurements yield well-defined outcomes without recourse to metaphysical claims, while still leaving room for interpretational questions about the nature of the wavefunction and collapse in different theoretical pictures quantum state tomography quantum computing decoherence.