Genuinely Non-Projective Quantum Measurements

• Genuinely non-projective measurements are defined as quantum measurements that cannot be simulated by mixtures of projective measurements combined with classical randomness and post-processing.
• Certification methods, including device-independent tests, steering, and dimension-bounded prepare-and-measure schemes, empirically distinguish these measurements from projective alternatives.
• Operational advantages of non-projective measurements manifest in enhanced randomness generation, improved state discrimination, and robust self-testing protocols in finite-dimensional quantum systems.

Genuinely non-projective measurements are quantum measurements that cannot be simulated by projective measurements together with classical randomness and post-processing, and therefore occupy a distinct operational class within the POVM formalism. In the resource-theoretic viewpoint, the central question is not merely whether a measurement is written as a POVM, but whether it lies outside the convex set of projectively simulable measurements. This distinction has become consequential in self-testing, device-independent and semi-device-independent certification, randomness generation, state discrimination, measurement discrimination, and the geometry of quantum measurements in finite dimension (Cobucci et al., 5 Aug 2025, Brinster et al., 6 Nov 2025, Tavakoli et al., 2018).

1. Formal notion and quantitative characterization

A projective measurement is specified by orthogonal projectors summing to the identity. A POVM $\mathbf{E}=\{E_a\}_{a=1}^n$ is projectively simulable if it can be realized by mixing projective measurements and applying classical post-processing. One explicit formulation is

$$E_a=\sum_{\lambda} q_\lambda \sum_k p(a|k,\lambda) P_{k|\lambda},$$

and, without loss of generality, this can be reduced to the simpler convex-mixture form

$E_a=\sum_\lambda q_\lambda P_{a|\lambda}.$

A measurement is genuinely non-projective precisely when no such decomposition exists (Cobucci et al., 5 Aug 2025). An equivalent notation used elsewhere is

$M_i=\sum_{k=1}^N p_k P_i^k,$

with projectively simulable POVMs forming a convex subset $\mathcal{P}(d,n)$ of all POVMs on a $d$-dimensional Hilbert space (Brinster et al., 6 Nov 2025).

A standard quantifier is the critical visibility, obtained by applying depolarizing noise. With

$$\Phi_v(X)=vX+\frac{1-v}{d}\operatorname{tr}(X)\mathbb{I},$$

the threshold

$$v^*(\mathbf{E})=\max\{v\mid \{\Phi_v(E_a)\}_a\in\mathcal{P}\}$$

measures how much noise a POVM tolerates before becoming projectively simulable; lower $v^*$ means a measurement is more non-projective (Cobucci et al., 5 Aug 2025). A closely related notation is

$$t(\mathcal{M}) := \max \left\{ t \;\middle|\; \Phi_t(\mathcal{M}) \in \mathcal{P}(d, n),\; t \leq 1 \right\},$$

where smaller $$E_a=\sum_{\lambda} q_\lambda \sum_k p(a|k,\lambda) P_{k|\lambda},$$0 again indicates greater robustness against simulation by projective schemes (Brinster et al., 6 Nov 2025).

For qubits, the distinction is especially sharp. A projective qubit measurement has at most two outcomes, whereas rank-one extremal POVMs with three or four outcomes are genuinely non-projective. In particular, extremal three-outcome qubit POVMs, such as trine-type constructions with effects of the form

$$E_a=\sum_{\lambda} q_\lambda \sum_k p(a|k,\lambda) P_{k|\lambda},$$1

cannot be written as orthogonal projectors and cannot be reduced to mixtures of binary projective measurements in the same Hilbert space (Sarkar, 2022, Gómez et al., 2017).

The dimension assumption is essential. In prepare-and-measure certification, the fixed-dimension hypothesis is natural because any measurement can be made projective by artificially increasing the Hilbert space dimension; accordingly, non-projectivity is meaningful only relative to the physical system dimension being certified (Tavakoli et al., 2018). This same issue motivates ancilla-assisted extensions of simulability analyses, where projective measurements act on system plus ancilla and only the system outcome is retained (Brinster et al., 6 Nov 2025).

2. Certification paradigms

Several certification regimes have been developed, differing primarily in which devices are trusted and whether the Hilbert space dimension is assumed.

Framework	Characteristic criterion	Source
Device-independent Bell test	Violation of a Bell-like inequality valid for all binary measurements	(Gómez et al., 2016)
One-sided device-independent steering	Algebraic maximum of a steering functional achievable only with non-projective POVMs	(Sarkar, 2022)
Prepare-and-measure with dimension bound	Witness-based robust self-testing of extremal qubit POVMs	(Tavakoli et al., 2018)
Randomness-free dimension-bounded task	Output correlations exceeding the projective-simulable bound	(Rout et al., 2024)

In a fully device-independent photonic experiment on entangled qubits, the key Bell-like quantity was

$$E_a=\sum_{\lambda} q_\lambda \sum_k p(a|k,\lambda) P_{k|\lambda},$$2

with an all-binary-quantum bound $$E_a=\sum_{\lambda} q_\lambda \sum_k p(a|k,\lambda) P_{k|\lambda},$$3 and quantum maximum $$E_a=\sum_{\lambda} q_\lambda \sum_k p(a|k,\lambda) P_{k|\lambda},$$4. The reported experimental value,

$$E_a=\sum_{\lambda} q_\lambda \sum_k p(a|k,\lambda) P_{k|\lambda},$$5

exceeded the binary bound by more than $$E_a=\sum_{\lambda} q_\lambda \sum_k p(a|k,\lambda) P_{k|\lambda},$$6 standard deviations, and, combined with a device-independent qubit characterization via CHSH, yielded the first device-independent certification of a nonprojective qubit measurement (Gómez et al., 2016).

In the one-sided device-independent setting, steering is particularly important because the maximal quantum value in a Bell test is always achievable with projective measurements, whereas steering can enforce genuinely non-projective structure. A steering functional tailored to three three-outcome extremal POVMs attains $$E_a=\sum_{\lambda} q_\lambda \sum_k p(a|k,\lambda) P_{k|\lambda},$$7, while the local-hidden-state bound satisfies $$E_a=\sum_{\lambda} q_\lambda \sum_k p(a|k,\lambda) P_{k|\lambda},$$8, and the maximal value is achievable only with non-projective measurements (Sarkar, 2022). This establishes a certification mode that is unavailable in the same form in standard Bell scenarios.

In prepare-and-measure experiments with bounded dimension, witness constructions provide robust self-testing of target POVMs. For qubits, this has been carried out for extremal three-outcome and four-outcome POVMs, including certification that an uncharacterized measurement is non-projective and, in the four-outcome case, that it is a genuine four-outcome qubit POVM (Tavakoli et al., 2018).

A further development removes measurement-choice randomness altogether. In the task $$E_a=\sum_{\lambda} q_\lambda \sum_k p(a|k,\lambda) P_{k|\lambda},$$9, spatially separated parties use fixed $E_a=\sum_\lambda q_\lambda P_{a|\lambda}.$0-outcome devices on a shared state of known local dimension. The operational criterion is whether the observed correlations can be reproduced by projective-simulable measurements. In the qubit case, the projective-simulable and classical bounds coincide, so any violation certifies genuine non-projectivity without input randomness (Rout et al., 2024).

3. Self-testing and canonical qubit constructions

The most explicit qubit self-tests use extremal POVMs with three outcomes. In the steering construction for the two-qubit maximally entangled state, Alice and Bob employ three settings $E_a=\sum_\lambda q_\lambda P_{a|\lambda}.$1, each with effects

$E_a=\sum_\lambda q_\lambda P_{a|\lambda}.$2

where the vectors $E_a=\sum_\lambda q_\lambda P_{a|\lambda}.$3 are chosen so that the POVMs are extremal and non-projective. Bob’s optimal measurements are

$E_a=\sum_\lambda q_\lambda P_{a|\lambda}.$4

The steering functional

$E_a=\sum_\lambda q_\lambda P_{a|\lambda}.$5

has algebraic maximum $E_a=\sum_\lambda q_\lambda P_{a|\lambda}.$6. If that value is observed, then, up to a local isometry on Bob’s side, the shared state is the maximally entangled state

$E_a=\sum_\lambda q_\lambda P_{a|\lambda}.$7

and Bob’s measurements are exactly the corresponding three-outcome extremal non-projective POVMs (Sarkar, 2022).

This qubit self-test is also robust. Under white noise in the measurements,

$E_a=\sum_\lambda q_\lambda P_{a|\lambda}.$8

or white noise in the state,

$E_a=\sum_\lambda q_\lambda P_{a|\lambda}.$9

the functional obeys

$M_i=\sum_{k=1}^N p_k P_i^k,$0

showing linear dependence on noise (Sarkar, 2022).

A distinct route uses a parity-oblivious communication game. For three dichotomic observables per party, the Bell expression

$M_i=\sum_{k=1}^N p_k P_i^k,$1

has local bound $M_i=\sum_{k=1}^N p_k P_i^k,$2, preparation-non-contextual bound $M_i=\sum_{k=1}^N p_k P_i^k,$3, and optimal quantum value $M_i=\sum_{k=1}^N p_k P_i^k,$4. The optimal strategy self-tests the maximally entangled state together with trine observables. When Alice is given an additional input corresponding to a three-outcome POVM, the optimal statistics self-test

$M_i=\sum_{k=1}^N p_k P_i^k,$5

and the resulting local randomness is

$M_i=\sum_{k=1}^N p_k P_i^k,$6

bits (Pan, 2021).

Prepare-and-measure self-testing supplies a complementary viewpoint. Extremal qubit POVMs can be represented as

$M_i=\sum_{k=1}^N p_k P_i^k,$7

with $M_i=\sum_{k=1}^N p_k P_i^k,$8 and $M_i=\sum_{k=1}^N p_k P_i^k,$9. In that setting, trine and SIC-POVMs appear as canonical three-outcome and four-outcome targets, and witness values can be converted into lower bounds on the worst-case fidelity between the implemented measurement and the target POVM (Tavakoli et al., 2018).

4. Operational roles and demonstrated advantages

One of the clearest experimental advantages of non-projective measurements is certified randomness generation. In a photonic Bell-test setup using a maximally entangled two-qubit state and a three-outcome non-projective measurement, the certified device-independent randomness was

$\mathcal{P}(d,n)$0

compared with

$\mathcal{P}(d,n)$1

for projective measurements, a gain of about $\mathcal{P}(d,n)$2. In the one-sided device-independent steering scenario, the certified randomness reached

$\mathcal{P}(d,n)$3

about a $\mathcal{P}(d,n)$4 gain over projective measurements (Gómez et al., 2017).

State discrimination provides another operational benchmark. For two non-orthogonal pure states, a three-outcome task with outcomes “correct guess”, “incorrect guess”, and “non-guess” leads to the cost function

$\mathcal{P}(d,n)$5

Optimizing over projective measurements defines a modified Helstrom bound, while a nonprojective POVM can achieve a strictly smaller minimum cost for suitable $\mathcal{P}(d,n)$6. The Ivanovic-Dieks-Peres unambiguous state discrimination protocol is recovered in the limit $\mathcal{P}(d,n)$7, and the paper further shows that intermediate choices of $\mathcal{P}(d,n)$8 yield robust, experimentally appreciable advantages even when the ideal IDP advantage is washed out by any amount of noise (Dressel et al., 2014).

When post-measurement states are included, the relevant object is a quantum instrument rather than only a POVM. Projective simulation of instruments takes the form

$\mathcal{P}(d,n)$9

and qubit instruments admit a complete characterization via Choi-based criteria. This framework reveals non-projective advantages in the information-disturbance trade-off: projective instruments satisfy

$d$0

whereas quantum instruments can reach

$d$1

thereby outperforming all projective implementations. The same framework also yields improved sequential Bell inequality violations under projective measurements and shows scalable noise-advantages for high-dimensional Lüders instruments under dephasing noise (Khandelwal et al., 2 Mar 2025).

Joint entangled measurements also admit genuinely non-projective forms. A non-projective Bell state measurement on $d$2 is defined as a POVM

$d$3

whose outcomes form an equiangular tight frame of maximally entangled states. For two qubits, there exists an explicit five-outcome Bell state measurement, while no six-outcome Bell state measurement exists. In entanglement-assisted communication, the geometry of the five-outcome construction yields a success probability

$d$4

for a randomized matching task (Wei et al., 2024).

5. Higher-dimensional structure, SDP methods, and experiments

The geometry of maximally non-projective measurements becomes subtler beyond qubits. For qubits, the unique SIC-POVM is maximally non-projective, with global minimum critical visibility

$d$5

For qutrits, the Hesse SIC-POVM has

$d$6

For ququarts, the Weyl-Heisenberg SIC has

$d$7

but a flagged qutrit Hesse SIC embedded in $d$8 reaches

$d$9

This shows that, beyond qubits, the SIC property is not in general associated with the most non-projective measurement (Cobucci et al., 5 Aug 2025).

To analyze such questions systematically, semidefinite programming has become central. A necessary condition for projective simulability of a POVM can be written using positive semidefinite operators $$\Phi_v(X)=vX+\frac{1-v}{d}\operatorname{tr}(X)\mathbb{I},$$0 satisfying

$$\Phi_v(X)=vX+\frac{1-v}{d}\operatorname{tr}(X)\mathbb{I},$$1

For qutrits this criterion is also sufficient, while for ququarts and above it is generally necessary but not always sufficient (Cobucci et al., 5 Aug 2025). A later development introduced a hierarchy of SDPs, $$\Phi_v(X)=vX+\frac{1-v}{d}\operatorname{tr}(X)\mathbb{I},$$2, giving efficiently computable upper bounds on critical visibility and frequently becoming tight at level $$\Phi_v(X)=vX+\frac{1-v}{d}\operatorname{tr}(X)\mathbb{I},$$3 (Brinster et al., 6 Nov 2025).

The duals of these SDPs yield non-simulability witnesses. In one formulation, Hermitian operators $$\Phi_v(X)=vX+\frac{1-v}{d}\operatorname{tr}(X)\mathbb{I},$$4 certify non-projectivity whenever

$$\Phi_v(X)=vX+\frac{1-v}{d}\operatorname{tr}(X)\mathbb{I},$$5

This was implemented experimentally on a trapped-ion qudit processor for a qubit SIC-POVM and a qutrit $$\Phi_v(X)=vX+\frac{1-v}{d}\operatorname{tr}(X)\mathbb{I},$$6-effect informationally complete real POVM, with robust certification against state-preparation errors and statistical uncertainty at confidence exceeding $$\Phi_v(X)=vX+\frac{1-v}{d}\operatorname{tr}(X)\mathbb{I},$$7 (Brinster et al., 6 Nov 2025).

High-dimensional photonic certification has also been achieved with multiport beamsplitters. In a four-dimensional system, a seven-outcome POVM implemented using a $$\Phi_v(X)=vX+\frac{1-v}{d}\operatorname{tr}(X)\mathbb{I},$$8 multiport beamsplitter and a seven-core multicore optical fiber reached fidelity above $$\Phi_v(X)=vX+\frac{1-v}{d}\operatorname{tr}(X)\mathbb{I},$$9. In a communication task with score

$$v^*(\mathbf{E})=\max\{v\mid \{\Phi_v(E_a)\}_a\in\mathcal{P}\}$$0

the projective limit was

$$v^*(\mathbf{E})=\max\{v\mid \{\Phi_v(E_a)\}_a\in\mathcal{P}\}$$1

the optimal nonprojective protocol achieved

$$v^*(\mathbf{E})=\max\{v\mid \{\Phi_v(E_a)\}_a\in\mathcal{P}\}$$2

and the experiment reported

$$v^*(\mathbf{E})=\max\{v\mid \{\Phi_v(E_a)\}_a\in\mathcal{P}\}$$3

more than $$v^*(\mathbf{E})=\max\{v\mid \{\Phi_v(E_a)\}_a\in\mathcal{P}\}$$4 standard deviations above the projective limit (Martínez et al., 2022).

6. Limitations, extensions, and broader contexts

Genuinely non-projective measurements do not confer an unrestricted advantage in every setting. In Bell-state discrimination under linear evolution and local measurement, general LELM measurements still cannot reliably distinguish all four qubit Bell states. For qutrit Bell states of bosons, projective LELM measurements distinguish at most three of the nine Bell states, while generalized LELM measurements can distinguish at most five; the non-projective extension strengthens rather than overturns the underlying no-go structure (Leslie et al., 2019).

A common misconception is that “non-projective” simply means “having more outcomes than the dimension.” In fixed qubit dimension this is often a useful sufficient indicator, but the more general notion is non-simulability by projective measurements. Another misconception is that all highly symmetric POVMs are maximally non-projective; qubit SIC-POVMs do have that status, but higher-dimensional results show that flagged SIC constructions can be more non-projective than standard SICs (Cobucci et al., 5 Aug 2025). A further subtlety is that, although Bell-based device-independent tests can witness non-binarity, the fully device-independent maximal quantum value of a Bell inequality remains achievable with projective measurements, which is why steering-based certification occupies a special role (Sarkar, 2022).

The concept has also spread beyond standard static measurement scenarios. In temporal quantum correlations, sequential projective qubit measurements lead to correlations that factorize into pairwise terms, precluding genuine multi-point temporal correlations. Generalized measurements with update rule

$$v^*(\mathbf{E})=\max\{v\mid \{\Phi_v(E_a)\}_a\in\mathcal{P}\}$$5

remove that restriction, enabling temporal analogues of GHZ and cluster-state correlations and supporting universal one-way quantum computing with repeated application of a $$v^*(\mathbf{E})=\max\{v\mid \{\Phi_v(E_a)\}_a\in\mathcal{P}\}$$6d-cluster-gate supplemented by general measurements (Markiewicz et al., 2013).

In quantum thermodynamics, unsharp non-projective energy measurements modify the Jarzynski-type equality. A general measurement operator

$$v^*(\mathbf{E})=\max\{v\mid \{\Phi_v(E_a)\}_a\in\mathcal{P}\}$$7

leads to

$$v^*(\mathbf{E})=\max\{v\mid \{\Phi_v(E_a)\}_a\in\mathcal{P}\}$$8

where $$v^*(\mathbf{E})=\max\{v\mid \{\Phi_v(E_a)\}_a\in\mathcal{P}\}$$9 depends on the measurement resolution, induced noise, and coherence developed during the process. In the high-resolution limit the projective case is recovered, whereas overlapping outcome distributions generate additional corrections (Alonso et al., 2022).

At the most general operational level, randomness-free detection tasks show that output correlations alone can certify non-projective measurements under local dimension assumptions, and analogous sharp-simulability notions in general probabilistic theories have been used to argue that square-bit or box-world models are unphysical in this setting (Rout et al., 2024). Continuous and weak measurements have also been incorporated into nonlocality analysis, with Bell-CHSH estimators derived directly from continuous records and governed by the two resources

$v^*$0

thereby extending the discussion of genuine non-projectivity beyond sharp, discrete-output measurement models (Singh et al., 23 Dec 2025).

Genuinely non-projective measurements are therefore best understood not as a marginal refinement of the POVM formalism, but as a structurally distinct resource whose detectability, robustness, and utility depend on the operational scenario, the available dimensional assumptions, and the chosen notion of simulation. Across qubits, qudits, and instruments, the field has moved from existence proofs to quantitative certification, experimentally robust witnesses, and dimension-sensitive structure theorems, while leaving open a broader classification problem for higher-dimensional and ancilla-assisted measurement architectures (Brinster et al., 6 Nov 2025, Khandelwal et al., 2 Mar 2025).