QM Postulate: The Time Evolution Of A State ψ Of A .

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Dynamics of a Quantum System:QM postulate: The time evolution of a state ψ of a closed quantum system is describedby the Schrödinger equationwhere H is the hermitian operator known as the Hamiltonian describing the closed system.a closed quantum system does not interact with any other systemgeneral solution:-the Hamiltonian:H is hermitian and has a spectral decompositionwith eigenvalues Eand eigenvectors E smallest value of E is the ground state energy withthe eigenstate E QSIT07.2 Page 1example:e.g. electron spin in a field:on the Bloch sphere:this is a rotation around the equator with Larmorprecession frequency ωQSIT07.2 Page 2

Rotation operators:when exponentiated the Pauli matrices give rise to rotation matrices around the threeorthogonal axis in 3-dimensional space.If the Pauli matrices X, Y or Z are present in theHamiltonian of a system they will give rise to rotationsof the qubit state vector around the respective axis.exercise: convince yourself that the operators Rx,y,z do perform rotations on the qubit statewritten in the Bloch sphere representation.QSIT07.2 Page 3Control of Single Qubit Statesby resonant irradiation:qubit Hamiltonian with ac-drive:ac-fields applied alongthe x and y componentsof the qubit stateQSIT07.2 Page 4

Rotating Wave Approximation (RWA)unitary transform:result:drop fast rotating terms (RWA):with detuning:I.e. irradiating the qubit with an ac-field with controlled amplitude and phase allows torealize arbitrary single qubit rotations.QSIT07.2 Page 5preparation of qubit states:initial state 0 :prepare excited state by rotating around x or y axis:Xπ pulse:Yπ pulse:preparation of a superposition state:Xπ/2 pulse:Yπ/2 pulse:in fact such a pulse of chosen length and phase can prepare any single qubit state, i.e. anypoint on the Bloch sphere can be reachedQSIT07.2 Page 6

Quantum MeasurementOne way to determine the state of a qubit is to measure the projection of its state vector alonga given axis, say the z-axis.On the Bloch sphere this corresponds to the following operation:After a projective measurement is completed the qubit will bein either one of its computational basis states.In a repeated measurement the projected state will be measuredwith certainty.QSIT07.2 Page 7QM postulate: quantum measurement is described by a set of operators {Mm} acting onthe state space of the system. The probability p of a measurement result m occurringwhen the state ψ is measured isthe state of the system after the measurement iscompleteness: the sum over all measurement outcomes has to be unityexample: projective measurement of a qubit in state ψ in its computational basisQSIT07.2 Page 8

measurement operators:measurement probabilities:state after measurement:measuring the state again after a first measurement yields the same state as the initialmeasurement with unit probabilityQSIT07.2 Page 9information content in a single qubit:-infinite number of qubit statesbut single measurement reveals only 0 or 1 with probabilitiesormeasurement will collapse state vector on basis stateto determine and an infinite number of measurements has to be madeBut, if not measured qubit contains 'hidden' information aboutQSIT07.2 Page 10and

A Few Physical Realizations of Qubitsenergy scales:nuclear spins in molecules:- nuclear magnetic moment in external magnetic field2F- solution of largenumber of moleculeswith nuclear spin612C41F13C3713CFF12CFeC5H55F4 2(CO)2531- distinct energies ofdifferent nuclei469.98figures from MIT group (www.mit.edu/ ichuang/)QSIT07.2 Page 11chain of ions in an ion trap:qubit states are implemented as long livedelectronic states of atomsfigures from Innsbruck group(http://heart-c704.uibk.ac.at/)QSIT07.2 Page 12470.00470.02 [MHz]

electrons in quantum dots:IDOT- double quantum dot- control individual electronsIQPCfigures from Delft group(http://qt.tn.tudelft.nl/)IQPCLPLMPRR200 nmGaAs/AlGaAs heterostructure2DEG 90 nm deepns 2.9 x 1011 cm-2- spin states of electrons as qubit states- interaction with external magnetic field BB 0B 0 gμBB QSIT07.2 Page 13superconducting circuits:- qubits made from circuit elements- circulating currents are qubit states µmDelftDelft- made from sub-micron scalesuperconducting inductors andcapacitorsQSIT07.2 Page 14IPHT

polarization states of photons:- qubit states corresponding to differentpolarizations of a single photon (in the visiblefrequency range)- are used in quantumcryptography and forquantumcommunication- photons are also used inthe one-way quantumcomputerQSIT07.2 Page 15Two Qubits:2 classical bits with states:- 2n different states (here n 2)- but only one is realized atany given time2 qubits with quantum states:- 2n basis states (n 2)- can be realized simultaneously- quantum parallelism2n complex coefficients describe quantum statenormalization conditionQSIT07.2 Page 16

Composite quantum systemsQM postulate: The state space of a composite systems is the tensor product of the state spacesof the component physical systems. If the component systems have states ψi the compositesystem state isThis is a product state of the individual systems.example:exercise: Write down the state vector (matrix representation) of two qubits, i.e. the tensorproduct, in the computational basis. Write down the basis vectors of the composite system.there is no generalization of Bloch sphere picture to many qubitsQSIT07.2 Page 17Information content in multiple qubits- 2n complex coefficients describe state of a composite quantum system with n qubits!- Imagine to have 500 qubits, then 2500 complex coefficients describe their state.- How to store this state. 2500 is larger than the number of atoms in the universe. It is impossiblein classical bits. This is also why it is hard to simulate quantum systems on classicalcomputers.- A quantum computer would be much more efficient than a classical computer at simulatingquantum systems.- Make use of the information that can be stored in qubits for quantum information processing!QSIT07.2 Page 18

Operators on composite systems:Let A and B be operators on the component systems described by state vectors a and b .Then the operator acting on the composite system is written astensor product in matrix representation (example for 2D Hilbert spaces):QSIT07.2 Page 19Entanglement:Definition: An entangled state of a composite system is a state that cannot be written as aproduct state of the component systems.example: an entangled 2-qubit state (one of the Bell states)What is special about this state? Try to write it as a product state!It is not possible! This state is special, it is entangled!QSIT07.2 Page 20

Measurement of single qubits in an entangled state:measurement of first qubit:post measurement state:measurement of qubit two will then result with certainty in the same result:The two measurement results are correlated! Correlations in quantum systems can bestronger than correlations in classical systems. This can be generally proven using theBell inequalities which will be discussed later. Make use of such correlations as a resourcefor information processing, for example in super dense coding and teleportation.QSIT07.2 Page 21Two Qubit Quantum Logic GatesThe controlled NOT gate (CNOT):function:addition mod 2 of basis statesCNOT circuit:control qubittarget qubitcomparison with classical gates:- XOR is not reversible- CNOT is reversible (unitary)Universality of controlled NOT:Any multi qubit logic gate can be composed of CNOT gates and single qubit gates X,Y,Z.QSIT07.2 Page 22

application of CNOT: generation of entangled states (Bell states):exercise: Write down the unitary matrix representations of the CNOT in the computationalbasis with qubit 1 being the control qubit. Write down the matrix in the same basis withqubit 2 being the control bit.QSIT07.2 Page 23Implementation of CNOT:Ising interaction:pair wise spin interactiongeneric two-qubit interaction:J 0: ferromagnetic couplingJ 0: anti-ferrom. coupling2-qubit unitary evolution:BUT this does not realize a CNOT gate yet. Additionally, single qubit operations on each ofthe qubits are required to realize a CNOT gate.QSIT07.2 Page 24

CNOT realization with the Ising-type interaction:CNOT - unitary:circuit representation:Any physical two-qubit interaction that can produce entanglement can be turned into auniversal two-qubit gate (such as the CNOT gate) when it is augmented by arbitrarysingle qubit operations. [Bremner et al., PRL 89, 247902 (2002)]QSIT07.2 Page 25Quantum Teleportation:Task: Alice wants to transfer an unknown quantum state ψ to Bob only using oneentangled pair of qubits and classical information as a resource.note:- Alice does not know the state to be transmitted- Even if she knew it the classical amount of information that she would need to send would beinfinite.The teleportation circuit:original article:Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channelsCharles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. WoottersPhys. Rev. Lett. 70, 1895 (1993) [PROLA Link]QSIT07.2 Page 26

How does it work?CNOT between qubit to be teleported and one bit of the entangled pair:Hadamard on qubit to be teleported:measurement of qubit 1 and 2, classical information transfer and single bitmanipulation on target qubit 3:QSIT07.2 Page 27(One) Experimental Realization of Teleportation using Photon Polarization:- parametric down conversion (PDC)source of entangled photons- qubits are polarization encodedExperimental quantum teleportationDik Bouwmeester, Jian-Wei Pan, Klaus Mattle, Manfred Eibl, Harald Weinfurter, Anton ZeilingerNature 390, 575 - 579 (11 Dec 1997) ArticleAbstract Full Text PDF Rights and permissions Save this linkQSIT07.2 Page 28

Experimental Implementationstart with statescombine photon to be teleported (1) andone photon of entangled pair (2) on a50/50 beam splitter (BS) and measure(at Alice) resulting state in Bell basis.- polarizing beam splitters(PBS) as detectors ofteleported statesanalyze resulting teleported state ofphoton (3) using polarizing beamsplitters (PBS) single photon detectorsQSIT07.2 Page 29teleportation papers for you to present:Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein-Podolsky-Rosen ChannelsD. Boschi, S. Branca, F. De Martini, L. Hardy, and S. PopescuPhys. Rev. Lett. 80, 1121 (1998) [PROLA Link]Unconditional Quantum TeleportationA. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. PolzikScience 23 October 1998 282: 706-709 [DOI: 10.1126/science.282.5389.706] (in Research Articles)Abstract » Full Text » PDF »Complete quantum teleportation using nuclear magnetic resonanceM. A. Nielsen, E. Knill, R. LaflammeNature 396, 52 - 55 (05 Nov 1998) Letters to EditorAbstract Full Text PDF Rights and permissions Save this linkDeterministic quantum teleportation of atomic qubitsM. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, D. J. WinelandNature 429, 737 - 739 (17 Jun 2004) Letters to EditorAbstract Full Text PDF Rights and permissions Save this linkDeterministic quantum teleportation with atomsM. Riebe, H. Hà ffner, C. F. Roos, W. Hà nsel, J. Benhelm, G. P. T. Lancaster, T. W. Körber, C. Becher, F. Schmidt-Kaler, D. F. V. James, R. BlattNature 429, 734 - 737 (17 Jun 2004) Letters to EditorAbstract Full Text PDF Rights and permissions Save this linkQuantum teleportation between light and matterJacob F. Sherson, Hanna Krauter, Rasmus K. Olsson, Brian Julsgaard, Klemens Hammerer, Ignacio Cirac, Eugene S. PolzikNature 443, 557 - 560 (05 Oct 2006) Letters to EditorFull Text PDF Rights and permissions Save this linkQSIT07.2 Page 30

- spin states of electrons as qubit states - interaction with external magnetic field B B 0 B 0 gμBB QSIT07.2 Page 13 superconducting circuits: µm Delft Delft IPHT - circulating currents are qubit states - made from sub-micron scale superconducting inductors and capaci

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