(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 1848178, 34854] NotebookOptionsPosition[ 1817138, 33933] NotebookOutlinePosition[ 1817630, 33952] CellTagsIndexPosition[ 1817587, 33949] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"2", "-", "2"}]], "Input", CellChangeTimes->{{3.529007641232791*^9, 3.529007641454632*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.529007642726614*^9, 3.529071381668146*^9, 3.529093756696607*^9, 3.529154400975184*^9, 3.5291746186997137`*^9, 3.529243947960511*^9, 3.529283523152419*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"Adjoint", "[", "x_", "]"}], ":=", RowBox[{"Transpose", "[", RowBox[{"Conjugate", "[", "x", "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.529098185117447*^9, 3.52909819005787*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ StyleBox["CompConj", FontColor->GrayLevel[0]], "[", "A_", "]"}], " ", ":=", " ", RowBox[{"A", " ", "/.", "\[InvisibleSpace]", RowBox[{ RowBox[{"Complex", "[", RowBox[{"0", ",", "n_"}], "]"}], "->", RowBox[{"-", RowBox[{"Complex", "[", RowBox[{"0", ",", "n"}], "]"}]}]}]}]}], "\[IndentingNewLine]"}]], "Input", CellChangeTimes->{3.5290076611006393`*^9, 3.529098202089488*^9, 3.529155947514908*^9}], Cell[CellGroupData[{ Cell["Quantum Walks on Graphs", "Title", CellChangeTimes->{{3.529007666732506*^9, 3.529007675661777*^9}}], Cell["\<\ Nicholas Wheeler 30 October 2011\ \>", "Text", CellChangeTimes->{{3.529007683912573*^9, 3.529007700791431*^9}}, FontSize->10], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.529007708280631*^9}], Cell[CellGroupData[{ Cell["Introduction", "Subsection", CellChangeTimes->{{3.529007926931229*^9, 3.529007932110867*^9}}], Cell["\<\ Matthew Jemielita (October 2008) announced to me\[LongDash]his designated \ thesis advisor\[LongDash]that he proposed to explore aspects of the quantum \ theory of random walks. I was, at that time, only dimly aware that such a \ topic even existed; I had studied none of the literature, and had no idea why \ that subject was of interest, and certainly not of why it had recently \ acquired urgent interest, though I was aware that it had engaged the \ attention of two applicants (Andrew Childs and Cesar Rodriguez-Rosario) for \ positions on the 糖心视频 physics faculty. I return to the subject now because it involves many of the ideas and methods \ that play roles in the subject of primary interest to me at the moment\ \[LongDash]the quantum theory of open systems, to which the quantum theory of \ walks provides a kind of introduction. The literature in these areas has become immense (ask Google). Basic \ references can be found in Matt's bibliography: \"Quantum Walks on Graphs,\" \ 糖心视频 thesis (2009). I, however, will be content to follow my own \ nose, drawing upon and citing literature only when it bears upon a specific \ problem or issue, serves a specific need. \ \>", "Text", CellChangeTimes->{{3.5290079954856577`*^9, 3.529007997244752*^9}, { 3.529008051558297*^9, 3.529008082869542*^9}, {3.529008134761869*^9, 3.529008197641864*^9}, {3.529008240768055*^9, 3.5290083723209267`*^9}, { 3.529008403773985*^9, 3.5290084938246403`*^9}, {3.529008574121305*^9, 3.529008579944439*^9}, {3.529008613772225*^9, 3.529008633710526*^9}, 3.5290086840122623`*^9, {3.5290087320717583`*^9, 3.529008732319058*^9}, { 3.5290087824596567`*^9, 3.529009275605845*^9}, 3.5290709048207817`*^9, 3.529093830155149*^9}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.5290092802422123`*^9}] }, Open ]], Cell[CellGroupData[{ Cell["Representatives of quantum state", "Subsection", CellChangeTimes->{{3.529009286786727*^9, 3.5290093046158447`*^9}}], Cell[CellGroupData[{ Cell["State vectors", "Subsubsection", CellChangeTimes->{{3.5290109603334103`*^9, 3.5290109638624563`*^9}}], Cell[TextData[{ "The state of a quantum walker on a finite graph can, in the simplest \ instance, be described by a \"complex stochastic vector\"\[LongDash]a complex \ vector (dimension = number of graph nodes) with unit norm. 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It descibes a \"mixed state\" of the quantum system\ \[LongDash]a circumstance usually typified\ \>", "Text", CellChangeTimes->{{3.529078373023075*^9, 3.529078392125822*^9}, { 3.529078559135302*^9, 3.5290786111768217`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Chop", "[", RowBox[{"Tr", "[", RowBox[{"\[Rho]", ".", "\[Rho]"}], "]"}], "]"}], "<", RowBox[{"Tr", "[", "\[Rho]", "]"}], "\[Equal]", "1"}]], "Input", CellChangeTimes->{{3.529078435210783*^9, 3.529078463868556*^9}, { 3.529078494286541*^9, 3.529078528738453*^9}}], Cell[BoxData["True"], "Output", CellChangeTimes->{{3.529078455862228*^9, 3.5290784651893597`*^9}, { 3.529078512767695*^9, 3.529078534188492*^9}}] }, Open ]], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.529079128022784*^9}], Cell[TextData[{ "I describe now a ", StyleBox["more efficient way to construct random density matrices", FontVariations->{"Underline"->True}], "." }], "Text", CellChangeTimes->{{3.529079137162208*^9, 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Note that the eigenvalues\ \>", "Text", CellChangeTimes->{{3.5290847741463947`*^9, 3.529084788201111*^9}, { 3.529084830478941*^9, 3.5290848484535646`*^9}, 3.529094544956263*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Eigenvalues", "[", "\[DoubleStruckR]", "]"}], "//", "Chop"}]], "Input", CellChangeTimes->{{3.529078973101688*^9, 3.52907897972646*^9}, { 3.529079041726652*^9, 3.529079043761314*^9}, {3.5290795168826857`*^9, 3.529079524965506*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ "0.860892769916727`", ",", "0.08926807043633139`", ",", "0.03385997184534731`", ",", "0.012734075282516907`", ",", "0.0032451125190781638`"}], "}"}]], "Output", CellChangeTimes->{ 3.5290789814971027`*^9, {3.529079033621545*^9, 3.529079045484807*^9}, 3.529079526841955*^9, 3.529094550345212*^9}] }, Open ]], Cell[" are non-negative, and sum to unity:", "Text", CellChangeTimes->{3.5290945586696672`*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Total", "[", "%", "]"}]], "Input", 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I omit discussion \ of the probability of finding the particle at the ", Cell[BoxData[ FormBox[ SuperscriptBox["n", "th"], TraditionalForm]], FormatType->"TraditionalForm"], " node when the system is in an evolving mixed state." }], "Text", CellChangeTimes->{{3.5291695454291897`*^9, 3.5291697244504766`*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.52916973671736*^9}] }, Open ]], Cell[CellGroupData[{ Cell["Continuous-time quantum walks", "Subsection", CellChangeTimes->{{3.5291697657224216`*^9, 3.529169777039427*^9}}], Cell[TextData[{ "It is difficult\[LongDash]not on mathematical grounds, but on ", StyleBox["physical", FontSlant->"Italic"], " grounds\[LongDash]to imagine a quantum dynamical process that proceeds by \ discrete iterative jumps, so I concentrate here on ", StyleBox["continuous-time", FontVariations->{"Underline"->True}], " quantum walks.\nCuriously, on the classical side of the street the \ situation is reversed: the random walk concept contains within its name an \ allusion to step-wise advance, and it is the notion of a \"continuous walk\" \ that calls for a leap of imagination (except for snakes and other slithery \ creatures?)." }], "Text", CellChangeTimes->{{3.529244557419642*^9, 3.529244750421589*^9}, { 3.5292448127807703`*^9, 3.529244813201848*^9}, {3.5292448584178047`*^9, 3.529244864951274*^9}, {3.5292449257392883`*^9, 3.529244974752143*^9}, { 3.52924501809414*^9, 3.529245060427328*^9}}], Cell[TextData[{ "\"Continuous-time quantum walk theory\" is got by specializing the general \ theory reviewed in the preceding section. 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Open ]], Cell[TextData[{ "But those matrices are ", StyleBox["complex", FontSlant->"Italic"], ", so at non-integral times Markov's construction" }], "Text", CellChangeTimes->{{3.5292558736576*^9, 3.529255908593577*^9}}], Cell[BoxData[ RowBox[{ SubscriptBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { SubscriptBox["p", "1"]}, { SubscriptBox["p", "2"]}, {"\[VerticalEllipsis]"}, { SubscriptBox["p", "N"]} }], "\[NoBreak]", ")"}], "t"], "=", RowBox[{ RowBox[{"Power", "[", RowBox[{"\[DoubleStruckCapitalM]", ",", "t"}], "]"}], ".", SubscriptBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { SubscriptBox["p", "1"]}, { SubscriptBox["p", "2"]}, {"\[VerticalEllipsis]"}, { SubscriptBox["p", "N"]} }], "\[NoBreak]", ")"}], "0"]}]}]], "DisplayFormula"], Cell["\<\ leads to \"stochastic vectors\" which\[LongDash]since complex\[LongDash]are \ uninterpretable. 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More surprising \ is the evidence that the classical entropy always exceeds the quantum \ entropy, and that temporal variation of the latter is ocassionally so broad.\ \>", "Text", CellChangeTimes->{{3.529346822901692*^9, 3.5293469184317827`*^9}, { 3.529347059482881*^9, 3.529347073990715*^9}, {3.529347120780424*^9, 3.529347138234825*^9}, {3.529347194843944*^9, 3.5293472129505253`*^9}}], Cell[TextData[{ "I will return later to consideration of this ", StyleBox["BASIC QUESTION", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], ": ", StyleBox["Does the preceding \"quantum entropy\" notation make physical \ sense? 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But none of those devices are \ available to the quantum theory of walks on graphs (or, more generally, to \ any form of finite-dimensional quantum mechanics, which includes the quantum \ theory of angular momentum (spin), the pure/applied theory of qubits). \ Formally, the question is very simple: \"How to get rid of the ", StyleBox["\[ImaginaryI]", "Output"], "?\" But on what compelling ", StyleBox["physical", FontSlant->"Italic"], " grounds can one motivate the adjustment" }], "Text", CellChangeTimes->{{3.529361143232815*^9, 3.529361309146124*^9}, { 3.529361339682413*^9, 3.529361491740388*^9}, {3.529363470639195*^9, 3.529363513006178*^9}, {3.529363883427587*^9, 3.529363966977255*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"MatrixExp", "[", RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], " ", "t", " ", "\[DoubleStruckCapitalL]ap"}], "]"}], "\[DoubleLongRightArrow]", RowBox[{"MatrixExp", "[", RowBox[{"t", " ", "\[DoubleStruckCapitalL]ap"}], "]"}]}]], "DisplayFormula",\ CellChangeTimes->{{3.529363985478154*^9, 3.529364036719326*^9}, 3.529364300362705*^9}], Cell[TextData[{ StyleBox["SEE IN THIS CONNECTION THE ", FontColor->RGBColor[0, 0, 1]], StyleBox["AFTERWORD", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[" AT THE END OF THIS NOTEBOOK", FontColor->RGBColor[0, 0, 1]] }], "Text", CellChangeTimes->{{3.529439394399416*^9, 3.52943942066547*^9}, { 3.529442080133747*^9, 3.5294421007097473`*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.529364051087274*^9}] }, Open ]], Cell[CellGroupData[{ Cell["The reversibility-irreversibility distinction", "Subsection", CellChangeTimes->{{3.529364056805608*^9, 3.529364076085578*^9}}], Cell["Quantum dynamical processes mediated by ", "Text", CellChangeTimes->{{3.529364092712935*^9, 3.5293641334832687`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"\[DoubleStruckCapitalU]", "[", "t", "]"}], "=", RowBox[{"MatrixExp", "[", RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], " ", "t", " ", "\[DoubleStruckCapitalL]ap"}], "]"}]}]], "DisplayFormula", CellChangeTimes->{{3.529364152789896*^9, 3.529364172377125*^9}}], Cell["\<\ make\[LongDash]in the absence of measurements\[LongDash]unobjectionable good \ sense when inverted. But\ \>", "Text", CellChangeTimes->{{3.5293642005454483`*^9, 3.529364254943809*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"\[DoubleStruckCapitalM]", "[", "t", "]"}], "=", RowBox[{"MatrixExp", "[", " ", RowBox[{"t", " ", "\[DoubleStruckCapitalL]ap"}], "]"}]}]], "DisplayFormula",\ CellChangeTimes->{{3.52936425958255*^9, 3.52936429201171*^9}}], Cell[TextData[{ StyleBox["becomes non-Markovian", FontVariations->{"Underline"->True}], " when inverted. This is a profoundly consequential distinction, and serves \ to further complicate the problem of constructing a plausible classical limit \ of the quantum theory of walks on graphs." }], "Text", CellChangeTimes->{{3.5293643262157583`*^9, 3.5293643986216583`*^9}, { 3.529364458523657*^9, 3.529364569304274*^9}}], Cell[BoxData[""], "Input", CellChangeTimes->{{3.5293645820911922`*^9, 3.529364608583352*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ The especially simple quantum walks contemplated in the quantum walk \ literature\ \>", "Subsection", CellChangeTimes->{{3.529364641807819*^9, 3.529364683818529*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Needs", "[", "\"\\"", "]"}], ";"}]], "Input"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.529421560942494*^9, 3.5294215611848497`*^9}}], Cell["\<\ Matt Jemalita, and the authors upon whose work he drew\[LongDash]see\ \>", "Text", CellChangeTimes->{{3.529364730831641*^9, 3.5293647886366158`*^9}, { 3.529416256180011*^9, 3.529416257171476*^9}, {3.529421577127573*^9, 3.529421578094035*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Hyperlink", "[", RowBox[{ "\"\\"", ",", "\"\\""}], "]"}]], "Input", CellChangeTimes->{{3.529375112724073*^9, 3.529375118606497*^9}, { 3.529418409129434*^9, 3.529418441354246*^9}}], Cell[BoxData[ TagBox[ ButtonBox[ PaneSelectorBox[{False->"\<\"Andrew Childs et al\"\>", True-> StyleBox["\<\"Andrew Childs et al\"\>", "HyperlinkActive"]}, Dynamic[ CurrentValue["MouseOver"]], BaseStyle->{"Hyperlink"}, BaselinePosition->Baseline, FrameMargins->0, ImageSize->Automatic], BaseStyle->"Hyperlink", ButtonData->{ URL["http://arxiv.org/abs/quant-ph/0209131"], None}, ButtonNote->"http://arxiv.org/abs/quant-ph/0209131"], Annotation[#, "http://arxiv.org/abs/quant-ph/0209131", "Hyperlink"]& ]], "Output", CellChangeTimes->{3.5294184487103367`*^9}] }, Open ]], Cell[TextData[{ "\[LongDash]restricted their attention to walks with generators that are \ \"proportional to the adjacency matrix,\" or so they claim. 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Our interest in \[DoubleStruckCapitalK] derives from the fact \ that it ", StyleBox["generates a symmetric Markovian matrix", FontWeight->"Bold"], ":" }], "Text", CellChangeTimes->{{3.529422607139269*^9, 3.529422634359407*^9}, { 3.5294226680930843`*^9, 3.529422718097981*^9}, 3.5294227587678022`*^9, { 3.5294227896538*^9, 3.529422844874795*^9}, {3.529422909954681*^9, 3.529422912983347*^9}, {3.529434941305835*^9, 3.529434942407228*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"\[DoubleStruckCapitalM]", "=", RowBox[{"MatrixExp", "[", "\[DoubleStruckCapitalK]", "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"\[DoubleStruckCapitalM]", "//", "MatrixForm"}], "\[IndentingNewLine]", RowBox[{"Table", "[", RowBox[{ RowBox[{"Total", "[", RowBox[{ "\[DoubleStruckCapitalM]", "\[LeftDoubleBracket]", "k", "\[RightDoubleBracket]"}], "]"}], ",", RowBox[{"{", RowBox[{"k", ",", "1", ",", "8"}], "}"}]}], "]"}]}], "Input", CellChangeTimes->{{3.5294208132117367`*^9, 3.529420823701372*^9}, { 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The associated Laplacian matrix is \ \>", "Text", CellChangeTimes->{{3.529424909954103*^9, 3.5294250146807756`*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"\[DoubleStruckCapitalL]", "=", RowBox[{ RowBox[{"3", " ", RowBox[{"IdentityMatrix", "[", "8", "]"}]}], "-", "\[DoubleStruckCapitalA]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"\[DoubleStruckCapitalL]", "//", "MatrixForm"}]}], "Input", CellChangeTimes->{{3.5294250471382313`*^9, 3.5294251362286*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"3", RowBox[{"-", "1"}], "0", RowBox[{"-", "1"}], "0", RowBox[{"-", "1"}], "0", "0"}, { RowBox[{"-", "1"}], "3", RowBox[{"-", "1"}], "0", RowBox[{"-", "1"}], "0", "0", "0"}, {"0", RowBox[{"-", "1"}], "3", RowBox[{"-", "1"}], "0", "0", "0", RowBox[{"-", "1"}]}, { RowBox[{"-", "1"}], "0", RowBox[{"-", "1"}], "3", "0", "0", RowBox[{"-", "1"}], "0"}, {"0", RowBox[{"-", "1"}], "0", "0", "3", RowBox[{"-", "1"}], "0", RowBox[{"-", "1"}]}, { RowBox[{"-", "1"}], "0", 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But when raised to high powers \ (as one would do if looking to the asymptotics of classic walks on the cube) \ it assumes a form\ \>", "Text", CellChangeTimes->{{3.529426033564107*^9, 3.529426156110281*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"MatrixPower", "[", RowBox[{"\[DoubleStruckCapitalM]", ",", "50"}], "]"}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.529425485320816*^9, 3.529425497790522*^9}, { 3.5294255833665133`*^9, 3.529425603104809*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0.12500002166655827`", "0.12500000722218532`", "0.12499999277781403`", "0.12500000722218535`", "0.12499999277781403`", "0.12500000722218535`", "0.12499999277781405`", "0.12499997833344441`"}, {"0.12500000722218527`", "0.12500002166655827`", "0.1250000072221853`", "0.12499999277781403`", "0.1250000072221853`", "0.12499999277781403`", "0.12499997833344441`", "0.12499999277781403`"}, {"0.12499999277781404`", "0.1250000072221853`", "0.1250000216665583`", 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CellChangeTimes->{{3.529441992335765*^9, 3.529441995791421*^9}}], Cell["Earlier on, I was led to pose this", "Text", CellChangeTimes->{{3.529439525036972*^9, 3.529439542445731*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ OPEN QUESTION: Does the theory of quantum walks possess a classical limit?\ \>", "Subsection", CellChangeTimes->{{3.529360610141438*^9, 3.529360613571744*^9}, { 3.529360869495079*^9, 3.5293608829842167`*^9}, {3.5293609149954653`*^9, 3.529360963295999*^9}}], Cell["\<\ After bringing this notebook to its present state of completion, I chanced \ (an hour ago) to be reading \ \>", "Text", CellChangeTimes->{{3.5294397796646643`*^9, 3.529439780619227*^9}, { 3.529439835232893*^9, 3.529439877047222*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Hyperlink", "[", RowBox[{ "\"\\"", ",", "\"\\""}], "]"}]], "Input", CellChangeTimes->{{3.529439898339073*^9, 3.529439958271474*^9}}], Cell[BoxData[ TagBox[ ButtonBox[ PaneSelectorBox[{ False->"\<\"Decoherence in quantum walks\[LongDash]a review\"\>", True-> StyleBox["\<\"Decoherence in quantum walks\[LongDash]a review\"\>", "HyperlinkActive"]}, Dynamic[ CurrentValue["MouseOver"]], BaseStyle->{"Hyperlink"}, BaselinePosition->Baseline, FrameMargins->0, ImageSize->Automatic], BaseStyle->"Hyperlink", ButtonData->{ URL["http://arxiv.org/abs/quant-ph/0606016"], None}, ButtonNote->"http://arxiv.org/abs/quant-ph/0606016"], Annotation[#, "http://arxiv.org/abs/quant-ph/0606016", "Hyperlink"]& ]], "Output", CellChangeTimes->{3.5294399639881287`*^9}] }, Open ]], Cell["\<\ by one Viv Kendon (2006), of the School of Physics & Astronomy, University of \ Leeds. Near the beginning of his \[Section]5 (page 19) Kendon remarks that \ \"One way to justify a particular quantum dynamics as being a \"quantum \ walk\" is to see if it turns into a classical random walk when decohered. \ \[Ellipsis] Many early studies of quantum walks took the trouble to show \ numerically that, for specific cases, their quantum walks decohered into \ classical random walks\[Ellipsis] A more systematic treatment of the quantum \ to classical transition in a general quantum walk appears in \ \>", "Text", CellChangeTimes->{{3.529439976047453*^9, 3.5294399829292088`*^9}, { 3.529440045663829*^9, 3.52944013755293*^9}, 3.529440215272187*^9, { 3.5294402854718113`*^9, 3.529440402507296*^9}, {3.529440632867858*^9, 3.529440687969349*^9}, {3.529440722940549*^9, 3.529440796972001*^9}, 3.529440986952661*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Hyperlink", "[", RowBox[{ "\"\\"", ",", "\"\\""}], "]"}]], "Input", CellChangeTimes->{{3.529440994529047*^9, 3.529441043980147*^9}}], Cell[BoxData[ TagBox[ ButtonBox[ PaneSelectorBox[{False->"\<\"Complementarity and quantum walks\"\>", True-> StyleBox["\<\"Complementarity and quantum walks\"\>", "HyperlinkActive"]}, Dynamic[ CurrentValue["MouseOver"]], BaseStyle->{"Hyperlink"}, BaselinePosition->Baseline, FrameMargins->0, ImageSize->Automatic], BaseStyle->"Hyperlink", ButtonData->{ URL["http://arxiv.org/abs/quant-ph/0404043"], None}, ButtonNote->"http://arxiv.org/abs/quant-ph/0404043"], Annotation[#, "http://arxiv.org/abs/quant-ph/0404043", "Hyperlink"]& ]], "Output", CellChangeTimes->{3.529441048858528*^9}] }, Open ]], Cell[TextData[{ "where the authors (Kendon & B. C. Sanders, 2005) emphasize the importance \ of demonstrating that quantum walks exhibit both wave (pure quantum) and \ particle (decohered) dynamics and, for a non-unitary quantum walk, being able \ to interpolate between these two modes of behavior by turning the decoherence \ up or down.\"\n\nThat reference to ", StyleBox["non-unitary quantum motion", FontWeight->"Bold"], "\[LongDash]means precisely what?\[LongDash]puts me in mind of the quantum \ mechanics of open systems (Kossakowski-Lindblad equation)" }], "Text", CellChangeTimes->{{3.529441087541223*^9, 3.529441262600978*^9}, { 3.529441303028392*^9, 3.5294416562258873`*^9}, {3.5294418898336067`*^9, 3.5294419104824677`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Hyperlink", "[", RowBox[{ "\"\\"", ",", "\"\\""}], "]"}]], "Input",\ CellChangeTimes->{{3.529441792569663*^9, 3.5294418225758944`*^9}}], Cell[BoxData[ TagBox[ ButtonBox[ PaneSelectorBox[{False->"\<\"Lindblad equation\"\>", True-> StyleBox["\<\"Lindblad equation\"\>", "HyperlinkActive"]}, Dynamic[ CurrentValue["MouseOver"]], BaseStyle->{"Hyperlink"}, BaselinePosition->Baseline, FrameMargins->0, ImageSize->Automatic], BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Lindblad_equation"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Lindblad_equation"], Annotation[#, "http://en.wikipedia.org/wiki/Lindblad_equation", "Hyperlink"]& ]], "Output", CellChangeTimes->{3.529441827400607*^9}] }, Open ]], Cell["\<\ [Google supplies many other helpful links] \[Ellipsis]which, after all, was \ my original destination when I took up this review of elementary quantum walk \ theory. So that is the subject to which I now\[LongDash]finally\[LongDash](re)turn. \ \>", "Text", CellChangeTimes->{{3.529441850361659*^9, 3.5294418796977987`*^9}, { 3.52944195355951*^9, 3.529441953687254*^9}, {3.529442144062756*^9, 3.529442177355474*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.5295023442080193`*^9}], Cell[TextData[StyleBox["ANOTHER AFTERWORD", "Subsection", FontSize->24, FontColor->RGBColor[1, 0, 0]]], "Text", CellChangeTimes->{{3.529441992335765*^9, 3.529441995791421*^9}, { 3.5295023857153683`*^9, 3.529502393279401*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Distinction between discrete & continuous classical walks on a cube\ \>", "Subsection", CellChangeTimes->{{3.5295024322497396`*^9, 3.5295024577299557`*^9}}], Cell["\<\ Continous walks, if sampled at regular intervals, become discrete walks. I \ had on this ground always assumed that the continuous theory was primary, and \ gave back the discrete theory as a trivial special case\[Ellipsis]and \ therefore wondered how Frederick Strauch (recently an applicant for a \ position on the 糖心视频 physics faculty) found anything to write about in\ \>", "Text", CellChangeTimes->{{3.5295024970943413`*^9, 3.529502691425515*^9}, { 3.5295028710667553`*^9, 3.5295029191447678`*^9}, {3.529503138488414*^9, 3.529503150567738*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Hyperlink", "[", RowBox[{ "\"\\"", ",", "\"\\""}], "]"}]], "Input", CellChangeTimes->{{3.529503153892152*^9, 3.529503206517168*^9}}], Cell[BoxData[ TagBox[ ButtonBox[ PaneSelectorBox[{ False->"\<\"Connecting the discrete- and continuous-time quantum \ walks\"\>", True-> StyleBox["\<\"Connecting the discrete- and continuous-time quantum \ walks\"\>", "HyperlinkActive"]}, Dynamic[ CurrentValue["MouseOver"]], BaseStyle->{"Hyperlink"}, BaselinePosition->Baseline, FrameMargins->0, ImageSize->Automatic], BaseStyle->"Hyperlink", ButtonData->{ URL["http://arxiv.org/pdf/quant-ph/0606050v1"], None}, ButtonNote->"http://arxiv.org/pdf/quant-ph/0606050v1"], Annotation[#, "http://arxiv.org/pdf/quant-ph/0606050v1", "Hyperlink"]& ]], "Output", CellChangeTimes->{3.5295032118183537`*^9}] }, Open ]], Cell["\<\ But then I recalled that, in recent discussion of the discrete-time classical \ walk on the cube I used the transition matrix\ \>", "Text", CellChangeTimes->{{3.529503266705316*^9, 3.529503322753997*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"\[DoubleStruckCapitalT]", "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", FractionBox["1", "3"], "0", FractionBox["1", "3"], "0", FractionBox["1", "3"], "0", "0"}, { FractionBox["1", "3"], "0", FractionBox["1", "3"], "0", FractionBox["1", "3"], "0", "0", "0"}, {"0", FractionBox["1", "3"], "0", FractionBox["1", "3"], "0", "0", "0", FractionBox["1", "3"]}, { FractionBox["1", "3"], "0", FractionBox["1", "3"], "0", "0", 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