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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Qubit w otoczeniu termicznym\n",
      "============================\n",
      "Plan\n",
      "----\n",
      "\n",
      "1. Przedstawienia zakresu stoswalno\u015bci przybli\u017cenia Daviesa\n",
      "2. Definicja i w\u0142asnosci map Daviesa, konstrukcja numeryczna \n",
      "3. Mapa Daviesa w dzia\u0142aniu na qubit: wizualizacja wybranych w\u0142asno\u015bci \n",
      "4. Dynamika spl\u0105tania kwantowego a wp\u0142yw otoczenia termicznego.\n",
      "5. Zadania."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "__Przybli\u017cenie Davisa__ to znane w kwantowej fizyce statystycznej narz\u0119dzie s\u0142u\u017c\u0105ce do opisu dynamiki zredukowanej\n",
      "\n",
      "$\\rho(t)=\\mbox{Tr}_{E} R(t)$\n",
      "\n",
      "gdzie $R$ jest stanem uk\u0142adu kwantowego sk\u0142adaj\u0105cego si\u0119 z badanego poduk\u0142adu (w naszym przypadku to qubit lub para qubit\u00f3w)\n",
      "oraz termostatu (zwyczajowo modelowanego jako _pole_ bozonowe lub fermionowe), za\u015b \u015blad wykonuje si\u0119 jedynie po stopniach swobody termostatu. Powy\u017cszy \u015blad cz\u0119\u015bciowy oblicza si\u0119 przy wykorzystaniu szeregu przybli\u017ce\u0144. Spo\u015br\u00f3d najbardziej znanych wymie\u0144my przybli\u017cenia Borna i Markowa. Ich om\u00f3wieniu po\u015bwi\u0119cimy si\u0119 kiedy indziej. Celeowo\u015b\u0107 znajdowania dynamiki zredukowanej wi\u0105\u017ce si\u0119 z jej znaczeniem statystycznym: dowolne loklane obserwable obliczone dla dynamiki zredukowanej daj\u0105 wyniki zgodne z warto\u015bciami oczekiwanymi obliczonymi dla pe\u0142nej dynamiki. Co wi\u0119cej w wi\u0119kszo\u015bci przypadk\u00f3w \"otoczenie\" jest modelowane przez niesko\u0144czenie wymiarowe pole (bozonowe lub fermionowe) za\u015b dynamika zredukowana jest sko\u0144czenie wymiarowa (w przypadku qubit\u00f3w dwuwymiarowa), w przypadkach kwantowo-optycznych przeliczalnie wymiarowa. Dzi\u0119ki temu obliczenia numeryczne s\u0105 znacz\u0105co uproszoczone ze wzgl\u0119du na brak konieczno\u015bci wykonywania wiarygodnych obci\u0119\u0107 przestrzeni stan\u00f3w.  \n",
      "\n",
      "Skuteczne i wiarygodne modelowanie dynamiki zredukowanej wymaga przyjecia szeregu za\u0142o\u017ce\u0144 opisuj\u0105cych zar\u00f3wno otoczenie, jak i badany uk\u0142ad. Najpowszechniejsze przybli\u017cenia wymagaj\u0105 za\u0142o\u017cenia o dobrej separacji skal czasowych/energetycznych uk\u0142ad\u00f3w. Separacja ta przek\u0142ada si\u0119 na separacj\u0119 odpowiednich czas\u00f3w relaksacji otoczenia i uk\u0142adu wynikiem czego mo\u017ce by\u0107 uzasadnione przybli\u017cenie Markowa.  \n",
      "\n",
      "W przypadku przybli\u017cenia Daviesa mo\u017cemy m\u00f3wi\u0107 o wysoce posuni\u0119tej \u015bcis\u0142o\u015bci matematycznej. W oparciu o szczeg\u00f3\u0142ow\u0105 znajomo\u015b\u0107 parametr\u00f3w uk\u0142adu i termostatu otrzymuje si\u0119 dynamik\u0119 zredukowan\u0105 b\u0119d\u0105c\u0105 rozwi\u0105zaniem pewnego typu _r\u00f3wnania fundamentalnego_ (tzw. r\u00f3wnania Master):\n",
      "\n",
      "$\\frac{d}{dt}\\rho(t)=\\mathcal{L}[\\rho(t)]$\n",
      "\n",
      "gdzie informacja o w\u0142asno\u015bciach uk\u0142adu kryje si\u0119 w superoperatorze $\\mathcal{L}$. Nale\u017cy podkre\u015bli\u0107, \u017ce rozwi\u0105zanie r\u00f3wnania fundamentalnego jest w og\u00f3lnym przypadku trudne. Istot\u0105 wyprowadzenia dynamiki zredukowanej w przybli\u017ceniu Daviesa jest rachynek zaburze\u0144 przeprowadzony ze wzgl\u0119du na energi\u0119 sprz\u0119\u017cenia pomi\u0119dzy uk\u0142\u0105dem i otoczeniem.    \n",
      "\n",
      "W tych materia\u0142ach badamy, ograniczaj\u0105c si\u0119 do przypadku qubit\u00f3w,  rozwi\u0105zanie uproszczone dane  za pomoc\u0105 tzw.  __mapy Daviesa__ [W. Roga, M. Fannes, K. \u017byczkowski, RMP 66, 311 (2010)]. Opisuje ona rozwi\u0105zania r\u00f3wnania funadamentalnego w wybranych chwilach czasu.    "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Mapa Daviesa $D=D(p,A,G,\\omega,t)$ okre\u015blona na macierzach bazowych zbioru operator\u00f3w qubitowych (2 wymiary)\n",
      "ma nast\u0119puj\u0105c\u0105 posta\u0107\n",
      "\n",
      "$ D|1\\rangle\\langle 1|= [1-(1-p)(1-e^{-At})]|1\\rangle\\langle 1|+(1-p)(1-e^{-At})|0\\rangle\\langle 0| $\n",
      "\n",
      "$ D|1\\rangle\\langle 0|= e^{i\\omega t -Gt}|1\\rangle\\langle 0| $\n",
      "\n",
      "$ D|0\\rangle\\langle 1|= e^{-i\\omega t -Gt}|0\\rangle\\langle 1| $\n",
      "\n",
      "$ D|0\\rangle\\langle 0|= (1-e^{-At})|1\\rangle\\langle 1|+[1-(1-e^{-At})]|0\\rangle\\langle 0|$\n",
      "\n",
      "gdzie $p\\in[0,1/2]$ wi\u0105\u017ce si\u0119 z temperatur\u0105 (przy $k_B=1$) poprzez: \n",
      "\n",
      "$ p=\\exp(-\\omega/2T)/[\\exp(-\\omega/2T)+\\exp(\\omega/2T)].  $\n",
      "\n",
      "$\\omega$ opisuje odleg\u0142o\u015b\u0107 mi\u0119dzy poziomami energetycznymi qubitu:\n",
      "\n",
      "$H_Q=\\frac{\\omega}{2}(|1\\rangle\\langle 1|-|0\\rangle\\langle 0|)$\n",
      "\n",
      "Parametry $A = 1/\\tau_R$ oraz $G = 1/\\tau_D$  wi\u0105\u017c\u0105 si\u0119 z czasem relaksacji enregii $\\tau_R$ oraz czasem defazingu $\\tau_D$.\n",
      "\n",
      "Zauwa\u017cmy, \u017ce dla d\u0142ugich czas\u00f3w  mapa Daviesa daje r\u00f3wnowagowy stan Gibbsa:\n",
      "\n",
      "$ \\lim_{t\\rightarrow\\infty}D(p,A,G,\\omega,t)\\rho=p|1\\rangle\\langle 1|+(1-p)|0\\rangle\\langle 0|$\n",
      "\n",
      "__UWAGA__ Parametry mapy Daviesa wymagaj\u0105 spe\u0142nienia $ G \\ge A/2 $ dla uzyskania fizycznie sensownych w\u0142asno\u015bci mapy (takich jak zupe\u0142na dodatnio\u015b\u0107)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Przyst\u0105pmy do numerycznej konstrukcji mapy Daviesa. W pierwszym kroku importujemy odpowiednie narz\u0119dzia "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from qutip import *\n",
      "from pylab import *\n",
      "%pylab inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stderr",
       "text": [
        "WARNING: pylab import has clobbered these variables: ['power', 'linalg', 'draw_if_interactive', 'random', 'save', 'load', 'info', 'fft']\n",
        "`%pylab --no-import-all` prevents importing * from pylab and numpy\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "i wprowadzamy uproszczenia notacji dla bazy przestrzeni Hilberta qubitu"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "up=basis(2,0)\n",
      "dn=basis(2,1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Szczeg\u00f3ln\u0105 rol\u0119 odgrywaj\u0105 dalej macierze bazowe, za pomoc\u0105 kt\u00f3rych wyrazi\u0107 mo\u017cna dowoln\u0105 macierz dzia\u0142aj\u0105c\u0105 na przestrzeni Hilberta pojedynczego qubitu:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "rpp=up*up.dag()\n",
      "rpm=up*dn.dag()\n",
      "rmp=rpm.trans()\n",
      "rmm=dn*dn.dag()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Nast\u0119pnie definiujemy dzia\u0142anie mapy Daviesa na macierzach bazowych jako funkcj\u0119 parametr\u00f3w termostatu i czasu $t$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def dav_pp(A,G,p,W,t):\n",
      "    xx=(1-(1-p)*(1-exp(-A*t)))*rpp+(1-p)*(1-exp(-A*t))*rmm\n",
      "    return xx\n",
      "def dav_pm(A,G,p,W,t):\n",
      "    xx=exp(-1j*W*t)*exp(-G*t)*rpm\n",
      "    return xx\n",
      "def dav_mp(A,G,p,W,t):\n",
      "    xx=exp(1j*W*t)*exp(-G*t)*rmp\n",
      "    return xx\n",
      "def dav_mm(A,G,p,W,t):\n",
      "    xx=(1-((1-p)*(1-exp(-A*t)))*p/(1-p))*rmm+(1-p)*(1-exp(-A*t))*p/(1-p)*rpp\n",
      "    return xx"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Oczywi\u015bcie funkcj\u0119 tak\u0105 mo\u017cemy teraz okre\u015bli\u0107 bezpo\u015brednio dla stanu $\\rho$ uwzgl\u0119dniaj\u0105c fakt, \u017ce \n",
      "w zbiorze macierzy g\u0119sto\u015bci mo\u017cna wprowadzi\u0107 iloczyn skalarny Hilberta Schmidta \n",
      "\n",
      "$\\langle A,B\\rangle=\\mbox{Tr}(A^\\dagger B) $ \n",
      "\n",
      "pozwalaj\u0105cy na znalezienie sk\u0142adowych dowolnej macierzy w wybranej ortonormalnej bazie macierzowej:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def davis(rho,A,G,p,W,t):\n",
      "    #\n",
      "    dpp=dav_pp(A,G,p,W,t)\n",
      "    dpm=dav_pm(A,G,p,W,t)\n",
      "    dmp=dav_mp(A,G,p,W,t)\n",
      "    dmm=dav_mm(A,G,p,W,t)\n",
      "    #\n",
      "    xpp=(rpp*rho.dag()).tr()\n",
      "    xmp=(rmp*rho.dag()).tr()\n",
      "    xpm=(rpm*rho.dag()).tr()\n",
      "    xmm=(rmm*rho.dag()).tr()\n",
      "    out=xpp*dpp+xmm*dmm+xmp*dmp+xpm*dpm\n",
      "    return out\n",
      "    "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Teraz mo\u017cemy ustali\u0107 parametry:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A=1.0\n",
      "G=1.0\n",
      "p=1.0/2.0\n",
      "q=1.0/2.0\n",
      "W=0.0"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "i stan pocz\u0105tkowy:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "vv=(up+dn).unit()\n",
      "rho=ket2dm(vv)\n",
      "rho"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "\\begin{equation}\\text{Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isHerm = True}\\\\[1em]\\begin{pmatrix}0.5 & 0.5\\\\0.5 & 0.5\\\\\\end{pmatrix}\\end{equation}"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True\n",
        "Qobj data =\n",
        "[[ 0.5  0.5]\n",
        " [ 0.5  0.5]]"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Naszym pierwszym zadniem b\u0119dzie wizualizacja warto\u015bci entropii von Neumanna\n",
      " \n",
      "$S(\\rho)=-\\mbox{Tr}(\\rho\\log\\rho) $\n",
      "\n",
      "dla stanu $\\rho(t)=D\\rho$\n",
      "\n",
      "Na wst\u0119pie okre\u015blmy listy dla chwil czasu (tlist) oraz wynik\u00f3w (outlist): "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "tlist=linspace(0.0,3.0,20.0)\n",
      "outlist=zeros(len(tlist))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "W nast\u0119pnym kroku obliczamy, u\u017cywaj\u0105c funkcji QuTip entropy_vn(), entropi\u0119 dla ka\u017cdej chwili czasu. Wczae\u015bniej przy u\u017cyciu naszej funkcji davis() znajdujemy oczywi\u015bci $\\rho(t)$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "for i1 in range(len(tlist)):\n",
      "    t=tlist[i1]\n",
      "    out=davis(rho,A,G,p,W,t)\n",
      "    outlist[i1]=entropy_vn(out)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Uzyskane wyniki mo\u017cna nakre\u015bli\u0107:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(tlist,outlist,'-.r',label='xx', linewidth=4)\n",
      "xlabel('t',fontsize=20)\n",
      "ylabel('S_vN',fontsize=20)\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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dcsvkcU1z5gAffwxER4sxHjNmALfc4vjvExEpUOM/rOfOndvic8labaXRaGA0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\n/ToAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x3158c90>"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Rozwa\u017cmy teraz inny, nie mniej ciekawy, problem. Maj\u0105c dany maksymalnie spl\u0105tany stan Bella:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "BB0=(tensor(rpp,rmm)+tensor(rpm,rmp)+tensor(rmp,rpm)+tensor(rmm,rpp))/2.0"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "spr\u00f3bujmy okre\u015bli\u0107 i zwizualizowa\u0107 wp\u0142yw otoczenia termicznego na spl\u0105tanie tego stanu. \n",
      "Korzystaj\u0105c z faktu, \u017ce mapa Daviesa jest operacj\u0105 liniow\u0105 wystarczy okre\u015bli\u0107 jej dzia\u0142anie na macierzach bazowych przy ustalonych parametrach termostatu: "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A=1.0\n",
      "G=1.0\n",
      "p=1.0/2.0\n",
      "q=1.0/2.0\n",
      "W=0.0"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "t=0.1\n",
      "dpp=dav_pp(A,G,p,W,t)\n",
      "dpm=dav_pm(A,G,p,W,t)\n",
      "dmp=dav_mp(A,G,p,W,t)\n",
      "dmm=dav_mm(A,G,p,W,t)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Je\u015bli przyjmiemy, \u017ce tylko jeden z qubit\u00f3w tworz\u0105cych stan Bella jest w kontakcie z termostatem, w\u00f3wczas , w chwili $t=0.1$ stan BB0 przekszta\u0142aca si\u0119 do postaci:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Env=(tensor(dpp,rpp)+tensor(dpm,rpm)+tensor(dmp,rmp)+tensor(dmm,rmm))/2.0"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Spl\u0105tanie stanu Env mo\u017cna  zmierzy\u0107 i obliczy\u0107 wykorzystuj\u0105c funkcj\u0119 concurrence(). Przypomnijmy, \u017ce przyjmuje ona warto\u015b\u0107 1 w przypadku stan\u00f3w maksymalnie spl\u0105tanych, za\u015b 0 dla stan\u00f3w niespl\u0105tanych "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "concurrence(Env)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 15,
       "text": [
        "0.85725612705393783"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Teraz za\u015b wykonamy powy\u017csze zadanie dla ca\u0142ego przedzia\u0142u czasu:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "tlist=linspace(0.0,3.0,20.0)\n",
      "outlist=zeros(len(tlist))\n",
      "for i1 in range(len(tlist)):\n",
      "    t=tlist[i1]\n",
      "    #\n",
      "    dpp=dav_pp(A,G,p,W,t)\n",
      "    dpm=dav_pm(A,G,p,W,t)\n",
      "    dmp=dav_mp(A,G,p,W,t)\n",
      "    dmm=dav_mm(A,G,p,W,t)\n",
      "    #\n",
      "    out=(tensor(dpp,dpp)+tensor(dpm,dpm)+tensor(dmp,dmp)+tensor(dmm,dmm))/2.0\n",
      "    outlist[i1]=concurrence(out)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Uzyskane wyniki nakre\u015blimy:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(tlist,outlist,'-.r',label='xx', linewidth=4)\n",
      "xlabel('t',fontsize=20)\n",
      "ylabel('C',fontsize=20)\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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qKysLyL4sKxTdfehf7PHHH9eLL77YOoyru1tPFxcKAEBnHf+ILioq6vW+LCsUcXFx8ng8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\nolAAAHyiUAAAfKJQAAB8+j+PGvsRNhJaagAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x348ecd0>"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Zadania\n",
      "-------\n",
      "1. Przeanalizowa\u0107 w\u0142asno\u015bci entropii qubitu w przypadku innych stan\u00f3w pocz\u0105tkowych\n",
      "2. Przeanalizowa\u0107 w\u0142asno\u015bci entropii qubitu w przypadku $A=0$\n",
      "3. Wykre\u015bli\u0107 wp\u0142yw otoczenia termicznego na dymanik\u0119 spl\u0105tania innych stan\u00f3w Bella "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 17
    }
   ],
   "metadata": {}
  }
 ]
}