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Simulating positive-operator-valued measures with projective measurements
Michał Oszmaniec, Leonardo Guerini, Peter Wittek, Antonio Acín
TL;DR
The paper asks which general quantum measurements can be simulated using projective measurements and classical processing. It proves a same-dimension ancilla construction and gives characterisations and bounds; its application improves the visibility bound for two-qubit Werner-state locality under all measurements.
Problem
It is unknown whether an arbitrary POVM offers an advantage over projective measurements or can be replaced by them, motivating a characterisation of projective-simulable measurements.
Source: p. 1Method
The paper combines a generalised Naimark construction with projective-simulability characterisations and semi-definite programming for qubits and qutrits.
Source: pp. 1, 3Results
The improved POVM-locality bound for two-qubit Werner states is p = t_Δ^2p* ≈ 0.4519, improving the previous best-known result.
Source: p. 4Takeaways & Limitations
The results provide partial conditions for projective simulation in arbitrary dimension and extend the visibility range for which qubit Werner states have a local hidden-variable model for all measurements.
Source: p. 1The scaling of the critical visibility t(d) for projective simulability remains an open question.
Source: p. 4
Abstract
from arXiv · showhide
Standard projective measurements represent a subset of all possible measurements in quantum physics, defined by positive-operator-valued measures. We study what quantum measurements are projective simulable, that is, can be simulated by using projective measurements and classical randomness. We first prove that every measurement on a given quantum system can be realised by classical processing of projective measurements on the system plus an ancilla of the same dimension. Then, given a general measurement in dimension two or three, we show that deciding whether it is projective-simulable can be solved by means of semi-definite programming. We also establish conditions for the simulation of measurements using projective ones valid for any dimension. As an application of our formalism, we improve the range of visibilities for which two-qubit Werner states do not violate any Bell inequality for all measurements. From an implementation point of view, our work provides bounds on the amount of noise a measurement tolerates before losing any advantage over projective ones.
Generalisation of the Naimark theorem for POVMs
The generalised Naimark theorem realises any POVM on a system as a projective-simulable POVM on that system plus an ancilla of the same dimension. The construction uses classical processing, and the ancilla dimension is minimal for this general guarantee.
Generalised Naimark dilation theorem
- Any n-outcome POVM on a d-dimensional system can be realised by a projective-simulable nd-outcome POVM on the system plus a d-dimensional ancilla.The simulation reproduces the original outcome probabilities for every input state.Source: Generalisation of the Naimark theorem… · p. 6
Simulation construction
- The construction first refines the POVM into a rank-one nd-outcome measurement, then decomposes it into extremal rank-one POVMs with at most d^2 nonzero effects.Each extremal component is implemented by a projective measurement on the system and d-dimensional ancilla; classical mixing and coarse-graining recover the original POVM.Source: Generalisation of the Naimark theorem… · pp. 6–7
Ancilla cost
- Dimension d is the minimum ancilla dimension that guarantees simulation of every POVM on the system, even with classical randomisation and post-processing.The minimality argument uses extremal rank-one POVMs with d^2 outcomes.Source: Generalisation of the Naimark theorem… · pp. 6–7
- For a POVM M with n ≤ d outputs, the proof’s construction implements it by a PM-simulable POVM on H ⊗ H′ using an ancilla H′ of dimension exactly n.Source: Generalisation of the Naimark theorem… · p. 7
Construction difference
- The construction avoids the standard requirement that the auxiliary basis vectors have a fixed product structure, which would force ancilla dimension to scale like d^2.The authors attribute the smaller ancilla cost to not imposing that structure.Source: Generalisation of the Naimark theorem… · p. 8
Proofs of technical results
This section collects technical results used in the main text or stated there without complete proofs.
Proofs of technical results
- The section supplies technical results used in the main text, including results whose proofs were omitted there.Source: Proofs of technical results · p. 8
Partial characterisations of PM-simulability
The paper gives general-dimensional sufficient conditions and visibility bounds for projective simulability, alongside explicit qubit examples. These include a depolarised tetrahedral POVM simulation and a lower bound on the critical visibility.
General-dimensional bound
- For every dimension d, the depolarised version Φ_1/d(M) of any POVM is projective-simulable, yielding t(d) ≥ 1/d.Source: Partial characterisations of… · p. 8
General-dimensional simulation
- For rank-one effects with trace 1/d, including tetrahedral and other symmetric informationally complete POVMs, uniform wrong-outcome relabelling simulates Φ_d/(d^2−1)(M).Source: Partial characterisations of… · p. 9
Tetrahedral POVM
- For the tetrahedral qubit POVM, six pairwise projective measurements chosen uniformly simulate Φ_√(2/3)(M_tetra), and the SDP establishes optimality at that visibility.Source: Optimal projective simulation of Mtetra( · p. 9
Characterisations of PM-simulable and m-chotomic-simulable POVMs
The paper characterises projective-simulable measurements through convex-hull descriptions, including a qutrit-specific result and an SDP-based test. It also gives an extremality criterion used in the supporting arguments.
Qutrit characterisation
- The extremal points among qutrit three-outcome measurements with unit-trace effects are precisely rank-one projective measurements.Source: Characterisations of PM-simulable and… · p. 10
Extremality proof
- For three-outcome qutrit POVMs, a measurement is non-extremal when a supported traceless Hermitian perturbation can be assigned to each effect so the perturbations sum to zero.Source: Characterisations of PM-simulable and… · p. 10
Qutrit characterisation
- For qutrits, projective-simulable measurements are the convex hull of measurements with at most two outcomes and three-outcome measurements whose effects have unit trace.Source: Characterisations of PM-simulable and… · p. 11
Efficient characterisation
- The qutrit characterisation is stated to have an efficient test, while a slight modification yields SDPs for computing t(M) when d = 2 or 3.Source: Characterisations of PM-simulable and… · pp. 11–12
Counterexamples for the generalisations of PM-simulation constructions for higher dimensions
Counterexamples show that dimension-matched POVMs need not be projective-simulable beyond qubits. The constructions use extremal, non-projective POVMs, including examples with unit-trace effects.
- For every d > 2, there exist d-outcome POVMs that are not PM-simulable.Source: Counterexamples for the… · p. 12
- In dimension three, the modified trine is extremal and therefore not PM-simulable; its extremality follows from that of the qubit trine.Source: Counterexamples for the… · p. 12
- For d > 3, there exist d-outcome non-PM-simulable POVMs whose effects all have unit trace.Source: Counterexamples for the… · p. 12
- In dimension four, two tetrahedral measurements on orthogonal two-dimensional subspaces yield an extremal non-projective POVM, and the construction extends to higher dimensions.The higher-dimensional extension supplements the example with a projective measurement on the orthogonal complement.Source: Counterexamples for the… · p. 12
SDP programs
The section gives semidefinite-programming formulations for testing simulability and computing critical visibility. It covers projective simulability for qubits and qutrits, as well as simulation using m-outcome measurements.
- For qubits and qutrits, the section supplies SDPs for deciding PM-simulability and computing critical visibility, and extends the approach to m-outcome simulation.Source: SDP programs · p. 13
- Membership in the convex hull of m-outcome measurements can be tested by an SDP, illustrated for four-outcome qubit POVMs.The formulation uses rescaled effects and probability variables.Source: SDP programs · p. 13
- Any PM-simulable four-outcome qubit POVM decomposes into six dichotomic measurements, each simulable by a projective measurement, so at most six projective measurements suffice.Source: SDP programs · p. 13
- For a qubit POVM, the modified SDP computes critical visibility, and PM-simulability is equivalent to critical visibility t(M) = 1.Source: SDP programs · pp. 13–14
- For qutrit POVMs, SDPs decide PM-simulability and compute critical visibility, with PM-simulability equivalent to t(M) = 1.Source: SDP programs · pp. 14–15
External polytopes approximating P(2, 4)
An outer polytope approximation reduces the qubit four-outcome robustness bound to a finite optimization over its vertices. Symmetries and geometric choices reduce the computation while retaining an approximation tailored to the tetrahedral measurement.
- The polytope relaxes positivity to finitely many pure-state inequalities while retaining normalization, so it contains the set of four-outcome qubit POVMs.Source: External polytopes approximating P(2, 4) · p. 15
- Unitary invariance, normalization, and rank-one extremal effects reduce the POVM representation from 16 parameters to 8.Source: External polytopes approximating P(2, 4) · p. 16
- The construction uses a polygon for the planar second-effect vector and adapts the polytope to be tangent near the tetrahedral measurement.Source: External polytopes approximating P(2, 4) · p. 16
- tΔ = 0.8152, from a polytope with approximately 850,000 extremal points, is a lower bound on t(2) and is close to t(Mtetra).Source: External polytopes approximating P(2, 4) · p. 16
Potytopes approximating covariant measurements for qutrits
For qutrit covariant POVMs, the construction approximates a class of measurements through a single seed effect and optimizes over a finite polytope. The resulting search finds covariant examples with substantial robustness, while general higher-dimensional PM-simulability remains uncharacterized.
- A positive trace-1/3 seed generates the nine effects of a qutrit covariant POVM by unitary conjugation; relaxing seed positivity produces covariant quasi-POVMs.Source: Potytopes approximating covariant… · p. 16
- A heuristic search found qutrit covariant POVMs with robustness t(M) ≈ 0.7934.Source: Potytopes approximating covariant… · p. 16
- Using 180 randomly chosen pure states, the construction produced a covariant quasi-POVM polytope with nearly 316,000 extremal points for optimization.Source: Potytopes approximating covariant… · p. 16
- For qutrit covariant POVMs, critical depolarization for PM-simulability matches that for 3-outcome simulability; the authors conjecture this relation extends to any covariant POVM.Source: Potytopes approximating covariant… · p. 17
Projective decomposition of trace-one, three-outcome qutrit POVMs
For trace-one, three-outcome qutrit POVMs, iterative rank reduction yields a convex decomposition into projective POVMs. The construction uses perturbations that preserve trace and normalization while reducing effect ranks.
- Perturbations X preserve trace-one normalization when tr(X_i)=0 and sum_i X_i=0; splitting M into M±X gives valid POVMs with lower effect ranks.Source: Projective decomposition of trace-one,… · pp. 17–18
- Starting from full-rank effects, an SDP-based decomposition produces a POVM whose effect ranks do not increase and at least one of which decreases.Source: Projective decomposition of trace-one,… · p. 17
- For rank patterns (3,3,2) or (3,2,2), a tailored decomposition extracts a projective POVM while lowering at least one effect rank; (2,1,3) cannot occur.Source: Projective decomposition of trace-one,… · p. 17
- In the (2,2,2) case, nine traceless operators supported on two-dimensional effect supports are linearly dependent because traceless Hermitian operators in dimension three form an eight-dimensional space.Source: Projective decomposition of trace-one,… · p. 17
- Iterating rank reduction yields a convex decomposition of the initial POVM into projective POVMs; the (1,2,2) rank case reduces to two outcomes on a subspace orthogonal to the first effect.Source: Projective decomposition of trace-one,… · p. 18