{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "This notebook shows how we can check how much depolarizing noise a qubit positive operator-valued measure (POVM) can take before it becomes simulable by projective measurements. The general method is described in [arXiv:1609.06139](https://arxiv.org/abs/1609.06139). The question of simulability by projective measurements boils down to an SDP problem. Eq. (8) from the paper defines the noisy POVM that we obtain subjecting a POVM $\\mathbf{M}$ to a depolarizing channel $\\Phi_t$:\n", "\n", "$\\left[\\Phi_t\\left(\\mathbf{M}\\right)\\right]_i := t M_i + (1-t)\\frac{\\mathrm{tr}(M_i)}{d} \\mathbb{1}$.\n", "\n", "If this visibility $t\\in[0,1]$ is one, the POVM $\\mathbf{M}$ is simulable.\n", "\n", "We will use Convex.jl to solve the SDP problem." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "ExecuteTime": { "end_time": "2017-01-06T22:57:42.633901", "start_time": "2017-01-06T21:57:39.471Z" }, "collapsed": false }, "outputs": [], "source": [ "using Convex" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For the qubit case, a four outcome qubit POVM $\\mathbf{M} \\in\\mathcal{P}(2,4)$ is simulable if and only if \n", "\n", "$M_{1}=N_{12}^{+}+N_{13}^{+}+N_{14}^{+},$\n", "\n", "$M_{2}=N_{12}^{-}+N_{23}^{+}+N_{24}^{+},$\n", "\n", "$M_{3}=N_{13}^{-}+N_{23}^{-}+N_{34}^{+},$\n", "\n", "$M_{4}=N_{14}^{-}+N_{24}^{-}+N_{34}^{-},$\n", "\n", "where Hermitian operators $N_{ij}^{\\pm}$ satisfy $N_{ij}^{\\pm}\\geq0$ and $N_{ij}^{+}+N_{ij}^{-}=p_{ij}\\mathbb{1}$, where $i