Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Slide 58Slide 59Slide 60Slide 61Slide 62Slide 63Slide 64Slide 65Slide 66Slide 67Slide 68Slide 69Slide 70CSEP 590tv: Quantum ComputingDave BaconAug 10, 2005Today’s MenuEntanglementBell’s TheoremQuantum Communication ComplexityEntanglement and Quantum ComputationEntanglement as a resourceAdministriviaQuestions on final?Last lecture will be a non-technical summary of the sectionsof quantum information science we didn’t cover.Quantum cryptography, quantum error correction,quantum communication, physical implementations.Plus probably a little something on some far off topics likequantum information and the black hole information paradox.Date: Wed, 20 Jul 2005 13:49:33 -0700 (PDT)From: Nancy Bacon To: Dave BaconSubject: teachingParts/Attachments: 1 OK 1 lines Text 2 Shown 0 lines Text----------------------------------------from you earthly mortal mother, i learn best when someone gives me a broad andgeneral concept, then goes into details, and then returns to the basic concept, ofcourse the trick is to find just the very right details to explain the generalconcept. ha! good luck daversEmail From HomeBig PictureEntanglement has long been known to be one of thefundamentally strange things about quantum theory.For years, people worried about entanglement.(Einstein, Schrodinger, Bell,….)What happening in quantum computing is that peoplestopped learning to worry about entanglement, and beganto realize that if they just accepted it, it was a very valuableresource!Accept Entanglement!But what is entanglement? Why is it “mysterious”?Why is it important for quantum computation?Bipartite EntanglementAlice’s qubitBob’s qubitTwo qubits have a wave function which is eitherSeparable: we can express it as valid wave functionsEntangled: we cannot express it asTwo Qubits, SeparableExample:Two Qubits, EntangledExample:Eitherorbut this impliesbut this impliescontradictionsAssume:So is not a separable state. It is entangled.Special RelativityTo understand what makes entanglement so interesting inphysics we need to know a little special relativity.We don’t need to know how to calculate in special relativity,but we do need to understand the concept of “locality” whicharises in special relativity.Spacetimepositiontime“event”:(time,position)“spacetime path”:curve in spacetime“inertial frame”:constant velocitySpecial RelativitypositiontimeSpecial relativity: (1) physics is the same in all inertial reference frames(2) speed of light is same in all reference frames.positiontimeReference frame 1 Reference frame 2“spacetime paths”“light”SimultaneitypositiontimeIn special relativity, the idea of simultaneity is relative:Reference frame 1event Aevent BpositiontimeReference frame 2event Aevent B positiontimeReference frame 3event Aevent BDifferent events are seen as occurring in a different order,depending on the reference frame.Special RelativitypositiontimeReference frame 1event Aevent BpositiontimeReference frame 2event Aevent B positiontimeReference frame 3event Aevent BSignal: I do something at my location and time such that you can,conditional on what I do, act conditionally at your locationand time.If I try to send a signal from A to B, then in some reference frame,B acts before A sends the signal. Better not allow this!Special RelativitypositiontimeFor all inertial observers moving through the origin, there are threeregions of spacetime which are preserved between inertial frames: “light”“light”“past”“future”“elsewhere”“elsewhere”Special RelativitypositiontimeWhen Alice and Bob both live in each other’s “elsewhere” they cannot communicate with each other:AliceBobWe call such setups “space-like separated”Cellular Automata0 1 0 0 0 1 0 10 0 000 0 110 1 000 1 101 0 011 0 111 1 011 1 10Cellular automata (CA):State:CA Rules0 1 0 0 0 1 0 1T=0T=10 1 0 1 0 1T=20 1 0 1These rules are localEvolution of a CA:Cellular Automataa b ceCA rules are localTimePositionCA “Spacetime”Cellular Automataa b ceCA rules are localTimePositionCA “Spacetime”Changing this valuecan only every changethe blue states (the future)Finite speed of signalingCellular Automataa b ceCA rules are localTimePositionCA “Spacetime”This state is a function ofthe states in red (the past)Finite speed of signalingBipartite EntanglementAlice’s qubitBob’s qubitWe will be interested in situations where Alice and Bob arespacelike separated. It is in these setups, where they cannotcommunicate, that quantum theory becomes interesting.Separable States: PropertiesAlice’s qubitBob’s qubitIf Alice acts on her qubit, while Bob does nothing, the resultingunitary is of the formIf Bob acts on his qubit, while Alice does nothing, the resultingunitary is of the formBoth of these preserve separability:What about entangled states?Schmidt DecompositionAlice BobAll bipartite quantum systems can be expressed as For some choice of basis of the two parties, and Example:Schmidt DecompositionAlice BobAll bipartite quantum systems can be expressed as For some choice of basis of the two parties, and When only one is nonzero (and hence equal to one) thenthe wave function is separableThus we can use Schmidt form to test for entanglement.Entangled States: PropertiesAlice’s qubitBob’s qubitIf Alice acts on her qubit, while Bob does nothing, the resultingunitary is of the formIf Bob acts on his qubit, while Alice does nothing, the resultingunitary is of the formBoth of these preserve entanglement.Entangled States: PropertiesAlice’s qubitBob’s qubitJust a change of basis.Does not change Hence and preserve entanglementLocal UnitariesAlice’s qubitBob’s qubitABLocal unitariesseparable separable entangled entangledentangledseparableEntanglement GenerationAlice’s qubitBob’s qubitABTwo party unitary:separable separableseparable entangledentangled separableentangled entangled Depends on the details of the unitary and on the inputsEntanglement GenerationExample:separable
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