POVM construction: a simple recipe with applications to symmetric states

TL;DR

This paper introduces a straightforward method for constructing POVMs using orthonormal matrix bases, with applications to symmetric states in quantum systems, and demonstrates physical realization and entanglement generation.

Contribution

It presents a novel, simple recipe for POVM construction based on orthonormal matrix sets, with practical implementation for symmetric quantum states.

Findings

Constructed POVMs using irreducible spherical tensors.

Realized POVMs physically via Clebsch--Gordan decomposition.

Generated entangled states from separable states.

Abstract

We propose a simple method for constructing POVMs using any set of matrices which form an orthonormal basis for the space of complex matrices. Considering the orthonormal set of irreducible spherical tensors, we examine the properties of the construction on the $N+1$-dimensional subspace of the $2^N$-dimensional Hilbert space of $N$ qubits comprising the permutationally symmetric states. Similar in spirit to Neumark's result on realization of a POVM as a projective measurement, we present a method to physically realize the constructed POVMs for symmetric states using the Clebsch--Gordan decomposition of the tensor product of irreducible representations of the rotation group. We illustrate the proposed construction on a spin-1 system, and show that it is possible to generate entangled states from separable ones.