{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Entanglement Distillation -- DEJMPS Protocol\n", "\n", "\n", " Copyright (c) 2021 Institute for Quantum Computing, Baidu Inc. All Rights Reserved. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Overview\n", "\n", "Before reading this tutorial, we highly recommend you to read the [BBPSSW protocol](./EntanglementDistillation_BBPSSW_EN.ipynb) first if you are not familiar with entanglement distillation. The DEJMPS protocol, introduced by Deutsch et al. [1], is similar to the BBPSSW protocol. The main difference between these two protocols is the state for distillation: the DEJMPS protocol can distill Bell-diagonal states, while the BBPSSW protocol could distill isotropic states. In entanglement distillation, the main purpose is to generate a **maximally entangled state** $|\\Phi^+\\rangle$ from many copies of imperfect entangled states, using only LOCC operations. Recall the four Bell states,\n", "\n", "$$ \n", "\\begin{align*}\n", "|\\Phi^{\\pm}\\rangle_{AB} &= \\frac{1}{\\sqrt{2}}(|0\\rangle_A\\otimes|0\\rangle_B \\pm |1\\rangle_A\\otimes|1\\rangle_B), \\\\\n", "|\\Psi^{\\pm}\\rangle_{AB} &= \\frac{1}{\\sqrt{2}}(|0\\rangle_A\\otimes|1\\rangle_B \\pm |1\\rangle_A\\otimes|0\\rangle_B). \n", "\\tag{1}\n", "\\end{align*}\n", "$$\n", "\n", "where $A$ and $B$ represent the bi-party Alice and Bob. The Bell-diagonal state, by definition, is diagonal in the Bell basis that can be expressed as\n", "\n", "$$\n", "\\rho_{\\text{diag}} = p_1 | \\Phi^+\\rangle \\langle \\Phi^+ | + p_2 | \\Psi^+\\rangle \\langle \\Psi^+ | + \n", "p_3 | \\Phi^-\\rangle \\langle \\Phi^- | + p_4 | \\Psi^-\\rangle \\langle \\Psi^- |,\n", "\\tag{2}\n", "$$\n", "\n", "with $p_1 > p_2 \\geq p_3 \\geq p_4$ and $p_1 + p_2+ p_3+ p_4 = 1$. Then the entanglement quantification of a Bell-diagonal state can be described as:\n", "\n", "* State fidelity $F = \\langle \\Phi^+|\\rho_{\\text{diag}}|\\Phi^+\\rangle = p_1$\n", "* Negativity $\\mathcal{N}(\\rho_{\\text{diag}}) = p_1 - 1/2$\n", "\n", "**Note:** The Bell-diagonal state is distillable when $p_1 > 1/2$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## DEJMPS protocol\n", "\n", "Suppose that two parties, namely Alice($A$) and Bob($B$), possess two copies of entangled qubit: $\\{ A_0, B_0 \\}, \\{ A_1, B_1 \\}$. If these two pairs are all in the same Bell-diagonal state $\\rho_{\\text{diag}}$, with $p_1 > 0.5$. We can apply the following workflow to purify the input states and leads to an output state has fidelity closer to $|\\Phi^+\\rangle$:\n", "1. Alice and Bob firstly choose the qubit pair **they want to keep as the memory qubit pair after distillation**. Here, they choose $A_0$ and $B_0$. \n", "2. Alice performs $R_x(\\pi/2)$ gates on both qubits, and Bob performs $R_x(-\\pi/2)$ gates on both qubits.\n", "3. Then, Alice and Bob both apply a CNOT gate on their qubits. Here, they choose $A_0,B_0$ as the control qubits and $A_1,B_1$ as the target qubits.\n", "4. Two remote parties measure the target qubits and use a classical communication channel to exchange their measurement results $m_{A_1}, m_{B_1}$.\n", "5. If the measurement results of Alice and Bob are the same (00 or 11), the distillation is successful, and the qubit pair $A_0, B_0$ is stored as state $\\rho_{out}$; If the measurement results are different (01 or 10), they claim the distillation failed and the qubit pair $A_0, B_0$ will be discarded.\n", "\n", "