Decomposition of any quantum measurement into extremals

Abstract

We design an efficient and constructive algorithm to decompose any generalized quantum measurement into a convex combination of extremal measurements. We show that if one allows for a classical post-processing step only extremal rank-1 POVMs are needed. For a measurement with $N$ elements on a $d$-dimensional space, our algorithm will decompose it into at most $(N-1)d+1$ extremals, whereas the best previously known upper bound scaled as $d^2$. Since the decomposition is not unique, we show how to tailor our algorithm to provide particular types of decompositions that exhibit some desired property.

10 pages

References in corpus (3)

• Quantum measurements on finite dimensional systems: relabeling and mixing
• Generalized Coherent States and Extremal Positive Operator Valued Measures
• A bound for the number of vertices of a polytope with applications

Cited by in corpus (12)

• Simulating positive-operator-valued measures with projective measurements
• Simulating all quantum measurements using only projective measurements and postselection
• Nonlocal Measurements via Quantum Erasure
• Measurement incompatibility and quantum advantage in communication
• Simulability of high-dimensional quantum measurements
• How much randomness can be generated from a quantum black-box device?
• Quantum measurement optimization by decomposition of measurements into extremals
• The complexity of compatible measurements
• Minimal scheme for certifying three-outcome qubit measurements in the prepare-and-measure scenario
• Entropic partial orderings of quantum measurements
• POVMs are equivalent to projections for perfect state exclusion of three pure states in three dimensions
• Extremal Steering Assemblages