Self-testing of quantum systems: a review

Ivan Šupić; Joseph Bowles

Source-linked AI summary

Self-testing of quantum systems: a review

Ivan Šupić, Joseph Bowles

arXiv:1904.10042v4Apr 22, 2019quant-ph

TL;DR

Self-testing infers the underlying physics of quantum experiments in a black-box scenario, providing a strong form of certification. This paper gives a thorough, self-contained review of self-testing, its techniques, extensions, applications, and experiments, while identifying open problems in higher-dimensional analytic methods.

Problem
Experimental noise and finite sample sizes make perfect self-testing impossible, motivating robust self-testing and statistical-inference tools.
Method
The paper provides a self-contained introduction, formal definitions, literature review of self-testing techniques, extensions, applications, experiments, and future directions.
Results
The review consolidates progress in self-testing and its applications to device-independent quantum information, quantum correlations, quantum gates, sequential protocols, and experiments.
Takeaways & Limitations
Self-testing serves as a strong certification framework for quantum systems and provides a reference for newcomers and researchers in the field.
Takeaways & Limitations
Measurement self-testing is limited by complex conjugation: the stated definition applies only to measurement sets invariant under conjugating all measurement operators.

Abstract

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Self-testing is a method to infer the underlying physics of a quantum experiment in a black box scenario. As such it represents the strongest form of certification for quantum systems. In recent years a considerable self-testing literature has been developed, leading to progress in related device-independent quantum information protocols and deepening our understanding of quantum correlations. In this work we give a thorough and self-contained introduction and review of self-testing and its application to other areas of quantum information.

1 Introduction

Self-testing emerged from Bell nonlocality as a device-independent way to identify quantum correlations with unique physical realisations. This review introduces the field, surveys its techniques and applications, and discusses experiments and open problems.

Bell nonlocality showed that entangled states and incompatible measurements can produce correlations stronger than classical theories allow.
Certain Bell correlations require particular entangled states and incompatible measurements, motivating self-testing as the study of uniquely realised quantum correlations.
Mayers and Yao’s 2004 work established the terminology and formalism of self-testing and helped introduce the device-independent paradigm.
Under an independent, identically distributed preparation assumption, self-testing provides black-box certification of quantum systems without specifying their physical implementation.
The review develops definitions, examples, literature surveys, applications across quantum information, experimental realisations, and future research directions.

2 Self-testing as a device-independent protocol

Device-independent self-testing treats laboratories and sources as black boxes, using observed correlations and Bell nonlocality to certify entanglement, states, and measurements. Practical certification requires robustness and statistical analysis because noise and finite data prevent ideal observations.

Device-independent analysis models each laboratory as a black box that maps settings to outcomes while making no assumptions about the underlying physical system.
Repeated experiments estimate p(a, b|x, y), allowing certification based on observable correlations despite ignorance of whether the systems are atoms, electrons, neutrinos, or other physical systems.
A Bell inequality bounds correlations from separable sources, so observing a violation certifies entanglement without knowing how the experiment was performed.
Maximal Bell-inequality violation can identify the source state up to local transformations and often determine the measurement operators producing the outcomes.
Because experimental noise and finite statistics prevent observing the maximum violation exactly, practical protocols combine robust self-testing with statistically valid probability estimates.
A single device cannot certify non-classical properties in a device-independent setting because a pre-programmed classical computer could simulate its observed statistics.
Multipartite Bell nonlocality is therefore crucial for ruling out local pre-programmed explanations, and self-testing supports many device-independent protocols.

3 Definitions

This section defines self-testing as inferring a quantum state and measurements from observed correlations alone under device-independent assumptions. Because correlations cannot determine an exact realization, self-testing identifies it up to local isometries and ancillary degrees of freedom, while practical noise and finite data prevent perfect certification.

3.2 The self-testing scenario: Self-testing infers the form of a state and measurements using only the correlations p(a, b|x, y) in a device-independent scenario.The scenario assumes a quantum description, separated laboratories without communication, freely and independently chosen settings, and i.i.d. experimental rounds.
3.2 The self-testing scenario: The physical and reference experiments are distinguished: the former contains the unknown state and measurements, while the latter specifies the target state and measurements.The reference state is pure, whereas the physical state may be mixed and can be represented using a purification.
3.2 The self-testing scenario: Exact inference from correlations is impossible because local unitary rotations and additional unused degrees of freedom can produce the same correlations.The equivalence relation therefore accounts for local unitary transformations and ancillary systems on which measurements act trivially.
3.2 The self-testing scenario: A local isometry is a tensor product of local isometries that embeds systems into larger spaces and applies local unitary transformations, typically using local ancillas.This construction is central to the definition of self-testing and is illustrated by adding |00⟩ ancillas before local unitaries act.
3.3 Self-testing of quantum states: The definition of pure-state self-testing requires every compatible physical realization and purification to map to the reference state under a suitable local isometry.The formal definition begins by quantifying over any state compatible with the observed correlations and any purification of that state.
3. Definitions: Perfect self-testing is unattainable in practice because experimental noise dampens correlations and finite samples make the probability distribution uncertain.Statistical inference addresses finite-sample uncertainty, while noise requires robust rather than perfect self-testing statements.
3.7.1 The issue of complex conjugation: For bipartite state self-testing, real reference states suffice, but the corresponding measurement limitation means Definition 2 cannot certify arbitrary measurement sets.The restriction arises because correlations are unchanged under complex conjugation, whereas measurements need not be invariant under it.

4 A first example

The review uses maximal CHSH violation to illustrate self-testing: it identifies anticommuting observables and constructs a local isometry extracting a maximally entangled state and measurements.

Setup: The example studies a two-qubit maximally entangled state with two binary inputs and outputs for each party.The observables are taken as Hermitian, unitary operators because the physical measurements are projective.
Anticommutativity: The geometric proof uses all correlations and Cauchy–Bunyakovsky–Schwarz saturation to show that the relevant vectors are parallel, yielding anticommutation.The argument first establishes unit norms and parallelism of the constructed vectors.
Anticommutativity: Maximal CHSH violation implies that Alice’s and Bob’s local observables anticommute on the support of the shared state.The review gives both geometric and algebraic arguments for this implication.
Anticommutativity: The algebraic proof starts from a sum-of-squares decomposition of the shifted CHSH Bell operator and derives the same anticommutation relation from maximal violation.SOS decompositions express the shifted Bell operator as sums of polynomial squares and can be found using the NPA hierarchy.
Measurement self-testing: The same construction self-tests the measurement observables, including an observable acting as σz on the support of the physical state.Combining state and measurement conclusions gives a full self-testing statement for the CHSH-maximizing realization.
State self-testing: A local isometry, typically implemented with ancillas and local unitaries, extracts the reference maximally entangled state from the physical state.For the partial-swap construction, the extracted state appears on ancilla registers while an unknown auxiliary state remains.

5 Self-testing of bipartite states

Self-testing of bipartite states develops from well-understood qubit protocols to higher-dimensional and repeated-state scenarios. Subspace methods complete the self-testing of bipartite pure states, while newer protocols address qudits, sequential robustness, parallel scaling, and approximate observables.

Scope and qubit foundations: All reviewed bipartite-state results concern pure states, with qubit protocols forming the best-understood starting point.Mixed states cannot be self-tested, according to the review’s stated scope.
Scope and qubit foundations: Maximal CHSH violation self-tests the maximally entangled pair of qubits, while Mayers–Yao provides an alternative protocol later made robust.Subsequent work also introduced self-testing involving σy observables and complex measurements.
Higher-dimensional states: Subspace methods self-test adjacent two-dimensional components of a Schmidt-decomposed state, then shifted components, completing self-testing for all bipartite pure states.The two steps use maximal violations of tilted CHSH inequalities.
Higher-dimensional states: Generalised CHSH inequalities analytically self-test maximally entangled qutrits and provide candidates for higher prime dimensions, although a complete self-testing statement remains lacking there.For d = 5 and d = 7, inequivalent realizations attain maximal violation, but all involve the corresponding maximally entangled state.
Higher-dimensional states: Weak projection pseudo-telepathy games can self-test maximally entangled finite-dimensional states, extending qudit self-testing through nonlocal-game constructions.Linear-constraint system games form another studied pseudo-telepathy class.
Multiple EPR pairs: Sequential and parallel protocols extend self-testing to multiple EPR pairs, with Pauli braiding and related tests maintaining robustness as the number of tested pairs increases.Parallel protocols use simultaneous input and output vectors, while overlapping-qubit analysis addresses approximate commutation between different observable pairs.

6 Self-testing of multipartite states

Multipartite self-testing is less complete than the bipartite pure-state case and uses several complementary strategies. The reviewed results include graph-state protocols, tailored Bell inequalities, reductions to bipartite tests, parallel repetition, and few-body-correlator approaches, with numerical scalability and experimental complexity remaining boundaries.

Overview: Multipartite states lack the simple Schmidt-decomposition characterization available for bipartite pure states, so only partial self-testing results are known.The review identifies five main approaches: graph-state stabilizers, tailored Bell inequalities, bipartite reductions, parallel self-testing, and marginal information.
Graph states: Every connected graph state admits a self-testing protocol using stabilizer-based reference measurements, with robustness to small imperfections.One party measures three observables while the others measure σx and σz; the isometry generalizes the Swap gate.
Tailored Bell inequalities: Bell inequalities tailored to stabilizer structure can self-test partially entangled GHZ states for any n ≥2, while other constructions target non-graph multipartite qubit states.The latter approach uses semidefinite programming to obtain robust fidelity bounds, but becomes infeasible as the number of parties increases.
Reductions to bipartite methods: Bipartite reductions self-test multipartite states by measurements that project remaining parties onto bipartite entangled states, as demonstrated for W states and broader families.For the three-qubit W state, computational-basis outcomes can leave the other two parties in a maximally entangled state testable by CHSH.
Parallel self-testing: Parallel repetition robustly self-tests n copies of the GHZ state, but it remains the only parallel self-test of a multipartite state reported in the review.The proof uses diagrammatic arguments based on categorical quantum mechanics.
Few-body correlators: Few-body correlators can reduce experimental demands: the tripartite W state is self-tested using two-body correlators, and the four-partite W state using three-body correlators.Most earlier multipartite protocols required full-body correlators, which are experimentally challenging.

7 Robust self-testing of states

Robust self-testing replaces ideal correlations with noise-dependent guarantees that approximately map an unknown physical state to a reference state. The review surveys five proof approaches, their applications, and limitations, including the distinction between robustness and noise tolerance for many entangled pairs.

7.1 Methods: The review identifies five main approaches: vector norm inequalities, Jordan’s lemma, operator inequalities, the numerical Swap method, and algebraic methods.These approaches are reviewed in Section 7.1.
7.1 Methods: Most protocols begin with observed probabilities or Bell-inequality violations, derive operator relations, and use an isometry—often a Swap gate—to recover the reference state.The operator relations may arise from geometric arguments, algebraic identities, or sum-of-squares decompositions.
7 Robust self-testing of states: Robust self-testing aims to infer approximate state fidelity from correlations that are close to ideal or from nearly maximal Bell-inequality violations.Approximate relations are propagated through an isometry to obtain noise-dependent fidelity bounds.
7.1.1 Norm inequalities method: Vector norm inequalities support robust proofs but typically yield weak bounds because of large constants.Their asymptotic robustness behavior is compared across protocols in Table 2.
7.1.4 Numerical Swap method: The numerical Swap method obtains lower bounds on minimum fidelity through semidefinite programs in the NPA hierarchy and supports self-tests of diverse states and measurements.Applications include singlets, partially entangled qubit pairs, qutrit states, graph-related states, GHZ and W states, and tensor products of singlets.
7.2 Robust certification of large entanglement: For n entangled pairs, robustness is distinct from noise tolerance because the fidelity of ρ^⊗n with the ideal tensor product can drop exponentially with n.The review also discusses protocols that certify large amounts of entanglement without explicitly stating a self-testing result.

8 Self-testing of measurements

Measurement self-testing extends state certification to local observables, generalized measurements, and entangling measurements. The review surveys qubit and qudit results, methods for handling complex conjugation, post-hoc certification, and analytic self-tests of Bell-state measurements.

8 Self-testing of measurements: Correlations that self-test a quantum state often also self-test the measurements applied to that state.The section reviews measurement results and the methods used to establish them.
8.1.1 Qubit measurements: Maximal CHSH violation can self-test qubit observables including σx, σz, and their rotated versions, while other inequalities certify broader measurement families.Chained, tilted, and weighted CHSH inequalities support additional qubit measurement self-tests.
8.1.2 Qudit measurements: Higher-dimensional measurement self-tests are less common, with results including mutually unbiased bases for d = 3 and tensor-product Pauli measurements in dimension 2^n.The Bell-state measurement has also been self-tested analytically.
8.1.3 Non-projective measurements: Non-projective measurement self-testing can be formulated as self-testing a Stinespring dilation, and the tetrahedral qubit POVM has analytic proofs and an experimental demonstration.Further work addresses measurements that are neither projective nor rank-one POVMs.
8.2.1 Complex measurements: Complex-valued measurements create a conjugation ambiguity: σz, σx, σy and σz, σx, −σy can produce identical observed correlations.The review describes controlled conjugation using ancillary registers to represent this ambiguity.
8.2.3 Post-hoc self-testing of measurements: Post-hoc self-testing uses already certified states and measurements to infer additional measurements, with informationally complete conditional states enabling measurement tomography.The method also supports higher-dimensional and non-projective measurement certification.
8.1.4 Entangling measurements: Analytic methods robustly self-test the Bell-state measurement in an entanglement-swapping scenario.Its measurement operators are non-separable across a suitable bipartition and have maximally entangled Bell states as eigenvectors.

9 Extensions of self-testing to other scenarios

Self-testing extends beyond standard state and measurement certification to quantum gates, semi-device-independent settings, and contextuality. These extensions introduce new certification targets or assumptions while enabling statements unavailable in fully device-independent scenarios.

9.1 Self-testing of quantum gates and circuits: Self-testing quantum gates certifies unitary transformations and can be extended from one-qubit gates to many-qubit gates and complete circuits.The circuit procedure self-tests input and output states before tomographic certification of each gate and is proven robust.
9.1 Self-testing of quantum gates and circuits: A channel self-test lower bounds fidelity to a reference channel by cos(arccos(Fi) + arccos(Fo)).The framework handles arbitrary quantum channels and provides an explicit solution for arbitrary two-qubit controlled gates.
9.2 Semi-device-independent approaches: Semi-device-independent self-testing adds assumptions to obtain more tractable or noise-tolerant certification and covers steering, prepare-and-measure, and contextuality scenarios.In one-sided device independence, recovering the shared state goes beyond merely witnessing entanglement.
9.2.2 Prepare-and-measure scenarios: Prepare-and-measure studies self-test non-projective measurements, including the extremal tetrahedral qubit POVM and arbitrary extremal qubit POVMs.These works also certify non-projective measurement structure, which is weaker than full self-testing.
9.2.3 Self-testing through noncontextuality inequalities: Contextuality-based self-testing uses compatibility assumptions because single-system statistics can always be classically simulated without them.The maximal KCBS violation robustly self-tests the corresponding strategy, with the proof linked to the Lovász theta number of an odd cycle.

10 Applications of self-testing

Self-testing supports theoretical results about quantum correlations and practical device-independent protocols. Its applications include randomness generation, cryptography, entanglement certification, and delegated quantum computing, while also informing separations between correlation sets.

10 Applications of self-testing: Self-testing provides theoretical characterisations of probability distributions that determine specific quantum states or measurements, including approximate statements.The review connects this theoretical work to progress in related areas.
10 Applications of self-testing: Self-testing contributes to device-independent entanglement certification and other applications reviewed across randomness, cryptography, delegated computing, and entanglement.The review presents these as practical applications of the technique.
10.1 Device-independent randomness generation: Self-testing certifies randomness because local measurements on a pure entangled state necessarily produce random outcomes.This relationship is used implicitly or explicitly in device-independent randomness protocols.
10.2 Device-independent quantum cryptography: Self-testing is intertwined with device-independent quantum key distribution, where certification of quantum resources appears implicitly in security proofs.The connection traces back to the Mayers–Yao work on self-checking and untrusted devices in cryptographic settings.
10.5 Applications to quantum correlations: Self-testing techniques have inspired strict inclusions between correlation sets and helped determine steered states in analyses of uncertainty relations.The review also records that whether Cqa ⊂ Cqc remains open.

11 Experiments

Experimental self-testing has progressed from mostly theoretical protocols toward demonstrations involving loophole-free Bell tests, ions, photons, qutrits, steering, and Bell-state measurements. Robustness and experimental coverage nevertheless remain limited.

11 Experiments: Most self-testing procedures remain theoretical because robust fidelity bounds often decrease rapidly with noise, although protocols tolerant to realistic noise have increased.Only a few experimental realisations are reported in the review.
11 Experiments: 398 meters and 55.54% fidelity marked the first implemented fully device-independent self-test free of detection and locality loopholes and the i.i.d. assumption.The experiment self-tested a Bell state through CHSH violation at 99% confidence.
11 Experiments: 0.958 fidelity was obtained for a maximally entangled pair of qubits encoded in two 9Be+ ions at 95% confidence.This experiment was based on a high CHSH violation but was not completely loophole-free.
11 Experiments: A fidelity ratio of approximately 0.998 between self-testing and tomography showed nearly matching conclusions for experimentally produced partially entangled qubit pairs.The comparison used photon-polarisation systems and standard tomographic methods.
11 Experiments: Numerical Swap-method bounds estimated fidelities of 0.799 for maximally entangled qutrits and 0.68 for a CGLMP-maximising state in an integrated optical chip.The states were encoded in photon mode degrees of freedom.
11 Experiments: Steering experiments estimated a GHZ-state fidelity lower bound of 0.7866, compared with tomographic fidelity 0.8725 ± 0.0034.The estimate was based on violation of Mermin’s steering inequality.

12 Concluding remarks and open questions

The review concludes that self-testing has expanded substantially but retains major unresolved problems. Open directions include higher-dimensional and multipartite systems, equivalence transformations, non-i.i.d. protocols, and improved robustness.

Analytic methods for dimension larger than 2: Many known protocols treat dimensions larger than two by adapting qubit methods, while genuinely higher-dimensional analytic self-testing remains largely unexplored.Analytic proofs are still lacking for examples including the CGLMP two-qutrit state and higher-dimensional SATWAP states.
Multipartite methods: General multipartite self-testing methods are needed because current techniques cover only restricted classes such as graph states.Hypergraph states are proposed as a richer target still admitting stabiliser-based descriptions.
Identifying the set of undetectable transformations: Higher-dimensional self-testing still lacks a complete characterisation of local transformations that leave correlations invariant.Complex conjugation already requires adapted definitions, while additional higher-dimensional invariances remain unknown.
Self-testing of a state or measurements only: It remains open to identify correlations that self-test a state but not its measurements, even after allowing complex-conjugation freedom.The review poses this as a distinct direction for self-testing states or measurements only.
Self-testing in non-i.i.d. scenarios: Removing the independent-and-identically-distributed assumption is an important practical direction for self-testing protocols.Sequential protocols and recently introduced non-i.i.d. techniques are suggested routes.
Improved robustness methods: General robustness bounds need improvement because poor noise tolerance currently hinders the applicability of most self-testing protocols.Known improvements for simple scenarios may not extend easily to more inputs and outputs.

A.1 Self-testing complex measurements

Self-testing of complex measurements requires a local isometry that maps every compatible physical realization to a reference state and measurements, up to auxiliary degrees of freedom and correlated complex conjugation.

A.1 Self-testing complex measurements: The definition requires that compatible states and measurements admit a local isometry mapping them to the reference state and measurements.The isometry has the form Φ = Φ_A ⊗ Φ_B and applies to purifications of potentially mixed physical states.
A.1 Self-testing complex measurements: The extracted realization may include an auxiliary state that accounts for degrees of freedom unrelated to the reference system.The auxiliary state appears alongside the reference state after applying the isometry.
A.1 Self-testing complex measurements: The measurement operators can implement effective controlled complex conjugations of the reference measurements.The conjugation is correlated between the parties, while its probability depends on the unknown auxiliary state.

A.2 Regularisation trick

The regularisation trick makes non-unitary operators suitable for Swap-isometry constructions by modifying their zero and nonzero eigenvalues while preserving their action on the physical state.

A.2 Regularisation trick: Regularisation addresses operators used in the Swap gate that are not unitary, including operators with zero eigenvalues.The procedure is motivated by operators such as Z_A = (A_0 + A_1)/2.
A.2 Regularisation trick: Zero eigenvalues are first replaced by 1, producing a modified operator Z*_A.This changes the operator on the kernel of Z_A.
A.2 Regularisation trick: The modified operator is then normalized so that the resulting operator is unitary by construction.The construction uses the absolute value of the modified operator for normalization.
A.2 Regularisation trick: The regularized operator must be shown to act on the physical state as the original operator did.The proof relies on the original operator differing from its zero-eigenvalue modification only on its kernel and on an operator inequality.
A.2 Regularisation trick: More generally, regularisation works when a unitary operator on one subsystem reproduces the action of the target operator on the state via the other subsystem.The required relation is A ⊗ 1 |ψ⟩ = 1 ⊗ U |ψ⟩.

A.3 Swap isometries

Swap-isometry constructions depend on the ancilla initialization and extend from qubits to bipartite qudits through Fourier-transform-based controlled operations satisfying generalized anticommutation conditions.

A.3 Swap isometries: When ancillas are not initialized in |0⟩, robust protocols use the full Swap gate rather than the partial Swap gate.For ancillas initialized in |0⟩, the full gate reduces to the partial Swap gate.
A.3 Swap isometries: The qudit construction uses a Fourier transform together with controlled operators C̄Z and C̄X.The Fourier transform is defined using the local dimension d and a d-th root of unity ω.
A.3 Swap isometries: For the qudit circuit to act as an effective Swap gate, the operators must satisfy conditions that mimic anticommutativity from the qubit case.The conditions involve the operators Z̄_A, Z̄_B, and powers or families of X̄ operators.
A.3 Swap isometries: The partial Swap gate is used in protocols for robust self-testing of qudit entangled states.Its qudit generalization targets states of the form |ψ⟩ = Σ_j λ_j |jj⟩ with positive real coefficients.
A.3 Swap isometries: The resulting qudit isometry outputs a state with the reference entangled state embedded alongside a normalized auxiliary state.The output has the form Σ_k λ_k |kk⟩_AB ⊗ |ξ⟩_AB.

A.4 Localising matrices in the Swap method

The Swap method must address non-unitary operators in robust self-testing, replacing them with nearby unitary operators and enforcing this proximity through localizing-matrix constraints.

A.4 Localising matrices in the Swap method: The numerical Swap method uses a Swap gate whose defining operators may fail to be unitary when built from sums or differences of observables.In the CHSH case, Z_A may be defined as (A_0 + A_1)/2.
A.4 Localising matrices in the Swap method: Ideal self-testing regularizes such operators, but robust protocols require a separate procedure for the Swap isometry.This robust procedure was introduced in prior work cited by the review.
A.4 Localising matrices in the Swap method: Two new unitary operators, A_2 and A_3, replace the non-unitary expressions while remaining close to (A_0 + A_1)/2 and its corresponding operator.The Swap isometry is then defined using Z_A = A_2 and X_A = A_3.
A.4 Localising matrices in the Swap method: Localizing moment matrices impose the required proximity constraints while preserving positive semidefiniteness.The condition using S = {1, A_0, A_1, A_2} is enforced by requiring the associated moment matrix to be positive semidefinite.
A.4 Localising matrices in the Swap method: The self-testing definitions allow mixed shared states by applying a purification and tracing out the purification space afterward.The same isometry maps the mixed state to the reference state because it acts trivially on the purification system.

B.2 Measurements

The section examines the projective-measurement assumption in self-testing and explains why it cannot be removed simply by applying a Naimark dilation. It outlines three possible ways forward, while noting that the most general route remains unresolved.

Projective-measurement assumption: Most self-testing proofs assume projective measurements; for example, projectivity is needed in one proof to establish observable anticommutation.Dropping this assumption leaves three possible strategies: prove compatible POVMs are projective, transfer self-testing from dilations, or treat projective measurements as fundamental.
Naimark dilation: The standard concatenation argument fails for POVMs because the proposed map to a Naimark dilation does not preserve inner products.For different outcomes, the corresponding vectors are orthogonal in the projective case but generally not for POVMs, so the map is not an isometry.
Naimark dilation: A self-test of dilated measurements does not automatically yield a self-test of the physical POVMs because the dilation space is explicitly used by the isometry.This differs from tracing out an ancillary purification space, which is not itself constructed from the measurements being self-tested.
Open problem: A general theorem transferring an isometry from any Naimark dilation to the original POVMs has not been proved and may not be possible.The proposed transfer would require an extension for which the residual ancilla contribution vanishes, but the existence of such an extension is unclear.
Interpretive stance: The projective-only stance can be defended by viewing nonprojective measurements as realizable through projective measurements on a dilated space, although this ontological argument is contested.Some researchers regard POVMs as equally fundamental rather than merely derived operationally from projective measurements and unitary evolution.