{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Quantum State Discrimination\n", "\n", "\n", " Copyright (c) 2021 Institute for Quantum Computing, Baidu Inc. All Rights Reserved. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Overview\n", "\n", "Quantum state discrimination (QSD) [1-2] is a fundamental question in quantum communication, quantum computation, and quantum cryptography. In this tutorial, we will explain how to discriminate two orthogonal bipartite pure states $\\lvert\\psi\\rangle$ and $\\lvert\\phi\\rangle$, which satisfies $\\langle\\psi\\lvert\\phi\\rangle=0$, under the constraint of Local Operations and Classical Communication (LOCC). We refer all the theoretical details to the original paper [3]." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## QSD Protocol\n", "\n", "Firstly, we want to make the problem definition clear. Consider two spatially separated parties $A$ (Alice) and $B$ (Bob) share a given two-qubit system, the system state is $\\lvert\\varphi\\rangle$ previously distributed by another party $C$ (Charlie). Alice and Bob were only notified that $\\lvert\\varphi\\rangle$ is either $\\lvert\\psi\\rangle$ or $\\lvert\\phi\\rangle$ (both are pure states), satisfying $\\langle\\psi\\lvert\\phi\\rangle=0$. Then, Charlie provides many copies of $\\lvert\\psi\\rangle$ and $\\lvert\\phi\\rangle$ to them, and he asks Alice and Bob to cooperate with each other to figure out which state they are actually sharing.\n", "\n", "\n", "Solving this problem under our LOCCNet framework is trivial. As always, let's start with the simplest one-round LOCC protocol with a QNN architecture shown in Figure 1. Then, the difficult lies in the design of an appropriate loss function $L$. Since we choose to let both parties to measure their subsystem, there will be four possible measurement results $m_Am_B\\in\\{00, 01, 10, 11\\}$. To distinguish $\\lvert\\psi\\rangle$ and $\\lvert\\phi\\rangle$, we will label the former state with measurement results $m_Am_B\\in\\{00, 10\\}$ and the latter with $m_Am_B\\in\\{01, 11\\}$. This step can be understood as adding labels to the data in supervised learning. With these labels, we can define the loss function as the probability of guessing wrong label,\n", "\n", "$$\n", "L = p_{\\lvert\\psi\\rangle\\_01}+p_{\\lvert\\psi\\rangle\\_11}+p_{\\lvert\\phi\\rangle\\_10}+p_{\\lvert\\phi\\rangle\\_00}.\n", "\\tag{1}\n", "$$\n", "\n", "where $p_{\\lvert\\psi\\rangle\\_01}$ stands for the probability of measuring 01 when the input state is $\\lvert\\psi\\rangle$. Then we can begin the training stage to minimize the loss function.\n", "\n", "\n", "![qsd](figures/discrimination-fig-circuit.png \"Figure 1: Schematic diagram of state discrimination with LOCCNet.\")\n", "