# Projective measurement (quantum mechanics)
A projective measurement is a [measurement in quantum mechanics](https://www.edgepedia.org/measurement-in-quantum-mechanics) whose possible outcomes are the eigenvalues of the measured observable and whose effect is to project the system's state onto the eigenspace matching the outcome, with probabilities given by the [Born rule](https://www.edgepedia.org/born-rule) (formula giving probabilities of quantum measurement outcomes). It is also called a von Neumann measurement, a name that reflects that the Born rule is written entirely in terms of projection operators.[1](http://www.qi.damtp.cam.ac.uk/files/QIC-3.pdf) After an outcome, immediate repetition of the measurement returns the same value for ideal projectors, a property called repeatability.[2](https://atom-tweezers-io.org/wp-content/uploads/2025/10/pqi2025-lecture5-measurement.pdf)
| Key fact | Detail |
|---|---|
| Outcomes | Eigenvalues of the measured observable, with probability p(aₙ) = ⟨ψ|Pₙ|ψ⟩[1](http://www.qi.damtp.cam.ac.uk/files/QIC-3.pdf) |
| State update | Lüders rule: ρₓ = EₓρEₓ / Tr[Eₓρ][3](https://ar5iv.labs.arxiv.org/html/2110.03219) |
| Repeatability | Guaranteed for ideal projectors; lost with realistically imperfect devices[2](https://atom-tweezers-io.org/wp-content/uploads/2025/10/pqi2025-lecture5-measurement.pdf) • [4](https://www.reed.edu/physics/faculty/wheeler/documents/Quantum%20Mechanics/Miscellaneous%20Essays/Generalized%20Quantum%20Measurement/Generalized%20Quantum%20Measurements.pdf) |
| Relation to POVMs | A projective measurement is the special POVM whose effects are orthogonal projectors, M̂_λ = M̂_λ†M̂_λ = Π̂_λ[5](https://arxiv.org/html/2608.09639v1) |
| Historical attribution | A 2024 historical analysis credits Dirac (1930) with the projection postulate, von Neumann (1932) with a different collapse theory, and Lüders (1951) with the degenerate-case rule[6](https://arxiv.org/html/2402.15280) |
| Practical stability | Projective measurements are unstable under detector imperfection and realistic only in special circumstances[7](https://arnold-neumaier.at/ms/BornM.pdf) |
| Superconducting readout | Mid-circuit measurement with 98.7% average QND fidelity and 200 ns feedback latency has been demonstrated[8](https://journals.aps.org/prl/abstract/10.1103/2c1p-8vx9) |
## How it works
A projective measurement is defined by a set of Hermitian projection operators Π = {Π₁, …, Πₘ}, each positive semidefinite and idempotent (Πᵢ² = Πᵢ), summing to the identity: Π₁ + ⋯ + Πₘ = I.[9](https://people.eecs.berkeley.edu/~jswright/quantumlearningtheory24/scribe%20notes/lecture02.pdf) Each projector corresponds to a distinct eigenvalue λᵢ of the observable, and the outcomes of the measurement are exactly those eigenvalues.[10](https://www.math.purdue.edu/~esampert/IQC/2024/3.1.pdf)
The Born rule supplies the statistics. For an observable A with eigenvalues aₙ and projectors Pₙ onto the corresponding eigenspaces, the probability of outcome aₙ in state |ψ⟩ is p(aₙ) = ⟨ψ|Pₙ|ψ⟩.[1](http://www.qi.damtp.cam.ac.uk/files/QIC-3.pdf) In the density-matrix form used for mixed states, Pr(λ) = Tr[Π̂_λ ρ̂].[5](https://arxiv.org/html/2608.09639v1) This textbook form of Born's rule applies to projective measurements; joint measurements of noncommuting observables require non-projective POVMs.[7](https://arnold-neumaier.at/ms/BornM.pdf)
**Collapse and repeatability.** After an outcome, the state is projected onto the corresponding eigenspace and renormalized, |ψ⟩ → P̂_j|ψ⟩/√(⟨ψ|P̂_j|ψ⟩).[2](https://atom-tweezers-io.org/wp-content/uploads/2025/10/pqi2025-lecture5-measurement.pdf) Because projectors are idempotent, applying the same measurement again returns the same eigenvalue. For a non-degenerate observable, von Neumann's repeatability hypothesis, which requires the state after measurement to be the eigenstate matching the outcome, uniquely fixes the state change.[3](https://ar5iv.labs.arxiv.org/html/2110.03219) For degenerate observables the eigenstate is not unique, and the update is given by the Lüders rule, ρₓ = EₓρEₓ / Tr[Eₓρ], where Eₓ is the eigenprojection for outcome x.[3](https://ar5iv.labs.arxiv.org/html/2110.03219) Born's rule together with Lüders' update implies repeatability when the measured projection commutes with the Hamiltonian, [Ĥ, Π̂_λ] = 0.[5](https://arxiv.org/html/2608.09639v1)
## How it is done
In an idealized implementation, the apparatus couples the system to a detector so strongly that the detector's final states for different outcomes are orthogonal; this strong-coupling limit is associated with quantum collapse.[11](https://iopscience.iop.org/article/10.1088/1367-2630/18/1/013016) A textbook example is a polarized beam splitter for photon polarization, which is nearly noise-free.[12](https://ar5iv.labs.arxiv.org/html/1609.06139) In superconducting circuits, a phase-sensitive homodyne scheme projectively measures a cavity field along a specific quadrature, which is how qubit observables such as \( \sigma_z \) are read out.[13](https://quantumsteampunk.umiacs.io/public/docs/Monroe_21_Weak.pdf) Continuous monitoring of a system can also implement a projective measurement in the long-time limit, but only if the monitoring is nondestructive, which is warranted when the jump operator L̂ is diagonalizable and commutes with the Hamiltonian.[5](https://arxiv.org/html/2608.09639v1)
Real devices rarely match the ideal. The original [Stern–Gerlach experiment](https://www.edgepedia.org/stern-gerlach-experiment), often presented as the canonical projective measurement of spin, in fact produced two overlapping rather than disjoint beams of silver, an outcome that cannot be described by a projective measurement and requires an unsharp POVM.[7](https://arnold-neumaier.at/ms/BornM.pdf) The correct POVM of a real device must be determined by quantum tomography.[7](https://arnold-neumaier.at/ms/BornM.pdf)
## Origin
The history of the projection postulate is often conflated. According to a 2024 historical analysis, what is now generally known as the projection postulate was framed as the measurement causing the minimum of disturbance to the system.[6](https://arxiv.org/html/2402.15280) A different theory, which the same analysis argues applies only in special and rather unusual cases, because it assumes each measurement is associated with a complete orthonormal basis of each eigenspace.[6](https://arxiv.org/html/2402.15280)
The rule, identical to Dirac's, applies irrespective of the method of measurement; this statement is now often called the Lüders rule.[6](https://arxiv.org/html/2402.15280) The combined update is often called the von Neumann–Lüders projection postulate.[3](https://ar5iv.labs.arxiv.org/html/2110.03219)
## Variants
**QND and first-kind measurements.** A measurement is of the first kind if a second measurement made immediately after gives the same value of the observable; it is quantum non-demolition (QND) if a later measurement at any time gives the same value. Any QND measurement is of the first kind, but the converse holds only when the observable is a constant of motion.[14](https://ar5iv.labs.arxiv.org/html/quant-ph/9804026) A QND measurement can be repeated many times, at arbitrary times, always returning the same result; standard measurements such as Stern–Gerlach, photodetection, and time-of-flight imaging are demolition measurements.[2](https://atom-tweezers-io.org/wp-content/uploads/2025/10/pqi2025-lecture5-measurement.pdf)
**Weak measurements.** In the weak-coupling, continuous-measurement limit of the von Neumann framework, the system is disturbed minimally and only partial information is obtained, recoverable by repeating the measurement on a large ensemble.[11](https://iopscience.iop.org/article/10.1088/1367-2630/18/1/013016) Weak values arise from weak measurement followed by strong postselected measurement, and their features are washed out in projective measurements.[11](https://iopscience.iop.org/article/10.1088/1367-2630/18/1/013016) • [15](https://mdpi-res.com/d_attachment/applsci/applsci-11-04260/article_deploy/applsci-11-04260.pdf?version=1620463137)
**POVMs.** A POVM on a [Hilbert space](https://www.edgepedia.org/hilbert-space) with n outcomes is a vector of non-negative operators M = (M₁, …, Mₙ) satisfying Σᵢ Mᵢ = I, with the Mᵢ called effects; outcome probabilities follow from the state and the effect.[12](https://ar5iv.labs.arxiv.org/html/1609.06139) Every POVM element admits a spectral development in terms of projections, so the projective case is the special case where the effects are the projectors themselves.[4](https://www.reed.edu/physics/faculty/wheeler/documents/Quantum%20Mechanics/Miscellaneous%20Essays/Generalized%20Quantum%20Measurement/Generalized%20Quantum%20Measurements.pdf) • [5](https://arxiv.org/html/2608.09639v1) Generalized measurements of this kind are known as POVMs.[7](https://arnold-neumaier.at/ms/BornM.pdf)
## Applications
Projective measurement underlies qubit readout, quantum error correction, and mid-circuit measurement with feedforward, where the true outcome s collapses the state to ρ_s ∝ Π_s ρ Π_s and enables ancillary [qubit reset](https://www.edgepedia.org/qubit-reset), circuit knitting, teleportation, and efficient state preparation.[16](https://arxiv.org/pdf/2406.07611) A superconducting quantum processor has demonstrated global mid-circuit measurement with an average QND fidelity of 98.7% and conditional feedback with a 200 ns real-time decision latency.[8](https://journals.aps.org/prl/abstract/10.1103/2c1p-8vx9)
Since 2023, measurement-induced entanglement and teleportation have been demonstrated on a noisy quantum processor using techniques to overcome the problem of post-selection.[17](https://www.nature.com/articles/s41586-023-06505-7) A 2025 experiment provided a postselection-free observation of the measurement-induced phase transition, avoiding the postselection overhead that becomes intractable when the number of mid-circuit measurement outcomes grows.[18](https://www.nature.com/articles/s42005-025-02443-0)
## Limitations and alternatives
Projective measurements are unstable under imperfection in the detector, so they are realistic only under special circumstances.[7](https://arnold-neumaier.at/ms/BornM.pdf) Prompt repeatability of outcomes is violated when measurements are performed with realistically imperfect devices.[4](https://www.reed.edu/physics/faculty/wheeler/documents/Quantum%20Mechanics/Miscellaneous%20Essays/Generalized%20Quantum%20Measurement/Generalized%20Quantum%20Measurements.pdf) Finite measurement efficiency and readout infidelity, for example from qubit energy leakage in superconducting circuits, distort the measured statistics.[13](https://quantumsteampunk.umiacs.io/public/docs/Monroe_21_Weak.pdf) The Wigner–Araki–Yanase theorem is cited as a general constraint on measurement, generally requiring a condition on the measuring apparatus.[19](https://arxiv.org/pdf/2404.05679) Unlike classical perturbations, a projective measurement dramatically affects the system state whatever the coupling strength to the measurement device.[2](https://atom-tweezers-io.org/wp-content/uploads/2025/10/pqi2025-lecture5-measurement.pdf) The nearest general alternative is the POVM: implementing one requires controlling and probing additional degrees of freedom beyond the system, so POVMs are typically noisier than nearly noise-free projective measurements such as a polarized beam splitter, although POVMs outperform projective measurements for some tasks, including quantum tomography, unambiguous state discrimination, state estimation, quantum cryptography, Bell inequalities, and device-independent protocols.[12](https://ar5iv.labs.arxiv.org/html/1609.06139)
## References
1. [Postulate (QM4): Quantum measurements (Cambridge QIC lecture notes)](http://www.qi.damtp.cam.ac.uk/files/QIC-3.pdf)
2. [Lecture 5: Quantum measurement (PQI 2025)](https://atom-tweezers-io.org/wp-content/uploads/2025/10/pqi2025-lecture5-measurement.pdf)
3. [Quantum Measurement Theory for Systems with Finite Dimensional State Spaces / On the von Neumann–Lüders projection postulate](https://ar5iv.labs.arxiv.org/html/2110.03219)
4. [Generalized Quantum Measurement (J. Wheeler, Reed College)](https://www.reed.edu/physics/faculty/wheeler/documents/Quantum%20Mechanics/Miscellaneous%20Essays/Generalized%20Quantum%20Measurement/Generalized%20Quantum%20Measurements.pdf)
5. [Conditions for implementing projective measurements through continuous monitoring](https://arxiv.org/html/2608.09639v1)
6. [Whose projection postulate?](https://arxiv.org/html/2402.15280)
7. [Born's rule and measurement (A. Neumaier)](https://arnold-neumaier.at/ms/BornM.pdf)
8. [Superconducting quantum processor with midcircuit measurement and feedback (PRL)](https://journals.aps.org/prl/abstract/10.1103/2c1p-8vx9)
9. [Berkeley scribe notes, lecture 2 (quantum learning theory, 2024)](https://people.eecs.berkeley.edu/~jswright/quantumlearningtheory24/scribe%20notes/lecture02.pdf)
10. [Projective measurements (Born rule), Purdue IQC notes](https://www.math.purdue.edu/~esampert/IQC/2024/3.1.pdf)
11. [Crossover between strong and weak measurement in interacting many-body systems (New J. Phys. 18, 013016, 2016)](https://iopscience.iop.org/article/10.1088/1367-2630/18/1/013016)
12. [Simulating positive-operator-valued measures with projective measurements](https://ar5iv.labs.arxiv.org/html/1609.06139)
13. [Weak Measurement of Superconducting Qubit Reconciles Incompatible Operators (Monroe et al., 2021)](https://quantumsteampunk.umiacs.io/public/docs/Monroe_21_Weak.pdf)
14. [Model of measurement (QND measurement review)](https://ar5iv.labs.arxiv.org/html/quant-ph/9804026)
15. [Protective Measurement, A New Quantum Measurement Paradigm: Detailed Description of the First Realization (Applied Sciences 11, 4260, 2021)](https://mdpi-res.com/d_attachment/applsci/applsci-11-04260/article_deploy/applsci-11-04260.pdf?version=1620463137)
16. [Probabilistic readout error mitigation (PROM) for mid-circuit measurements and feedforward](https://arxiv.org/pdf/2406.07611)
17. [Measurement-induced entanglement and teleportation on a noisy quantum processor (Nature, 2023)](https://www.nature.com/articles/s41586-023-06505-7)
18. [Postselection-free experimental observation of the measurement-induced phase transition in circuits with universal gates (Communications Physics, 2025)](https://www.nature.com/articles/s42005-025-02443-0)
19. [Measurement in quantum mechanics (review, 2024)](https://arxiv.org/pdf/2404.05679)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum phenomena and measurement › Measurement and decoherence*
*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026; Oct 25, 2026 · Last review: Sep 30, 2026*
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License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license