Docstoc

NMR Quantum Information Processing and Entanglement

Document Sample
NMR Quantum Information Processing and Entanglement Powered By Docstoc
					 NMR Quantum Information
Processing and Entanglement
       R.Laflamme, et al.
     presented by D. Motter
               Introduction
► Does  NMR entail true
  quantum computation?
► What about entanglement?
► Also:
   What is entanglement (really)?
   What is (liquid state) NMR?
► Why are quantum computers more powerful
 than classical computers
                   Outline
► Background
   States
   Entanglement
► Introductionto NMR
► NMR vs. Entanglement
► Conclusions and Discussion
   Background: Quantum States
► Pure   States
   |  > = 0|0000> + 1|0001> + … + n|1111>
► Density    Operator 
   Useful for quantum systems whose state is not known
     ► In   most cases we don’t know the exact state
   For pure states
     ►   = |  ><  |
   When acted on by unitary U
     ►    UU†
   When measured, probability of M = m
     ► P{   M = m } = tr(Mm†Mm )
   Background: Quantum States
► Ensemble       of pure states
    A quantum system is in one of a number of states | i>
        ►i  is an index
        ► System in | i> with probability pi

    {pi, | i>} is an ensemble
► Density     operator
     = Σ pi| i>< i|
► If   the quantum state is not known exactly
    Call it a mixed state
               Entanglement
► Seems  central to quantum computation
► For pure states:
   Entangled if can’t be written as product of states
   |  >  | 1>| 2>| n>
► For   mixed states:
   Entangled if cannot be written as a convex sum of
    bi-partite states
     Σ ai(1  2)
      Quantum Computation w/o
           Entanglement
► For   pure states:
     If there is no entanglement, the system can be
      simulated classically (efficiently)
      ►Essentially   will only have 2n degrees of freedom
►   For mixed states:
     Liquid State NMR at present does not show
      entanglement
     Yet is able to simulate quantum algorithms
  Power of Quantum Computing
► Why   are Quantum Computers more powerful than
  their classical counterparts?
► Several alternatives
   Hilbert space of size 2n, so inherently faster
     ► But   we can only measure one such state
   Entangled states during computation
     ► Forpure states, this holds. But what about mixed states?
     ► Some systems with entanglement can be simulated classically

   Universe splits  Parallel Universes
   All a consequence of superpositions
      Introduction to NMR QC
► Nuclei   possess a magnetic moment
   They respond to and can be detected by their
    magnetic fields
► Single   nuclei impossible to detect directly
   If many are available they can be observed as
    an ensemble
► Liquid   state NMR
  Nuclei belong to atoms forming a molecule
  Many molecules are dissolved in a liquid
           Introduction to NMR QC




►   Sample is placed in external magnetic field
     Each proton's spin aligns with the field
►   Can induce the spin direction to tip off-axis by RF pulses
     Then the static field causes precession of the proton spins
              Difficulties in NMR QC

► Standard    QC is based on pure states
     In NMR single spins are too weak to measure
    Must consider ensembles
• QC measurements are usually projective
    • In NMR get the average over all molecules
    • Suffices for QC
•   Tendency for spins to align with field is weak
    • Even at equilibrium, most spins are random
    • Overcome by method of pseudo-pure states
        Entanglement in NMR
► Today’s   NMR  no entanglement
   It is not believed that Liquid State NMR is a
    promising technology
       NMR experiments could show
► Future
 entanglement
   Solid state NMR
   Larger numbers of qubits in liquid state
       Quantifying Entanglement
► Measureentanglement by entropy
► Von Neumann entropy of a state

          S    tr log 2  
► Ifλi are the eigenvalues of ρ, use the
  equivalent definition:
            S     i log 2 i
                      i
      Quantifying Entanglement
► Basic   properties of Von Neumann’s entropy

     S    0 , equality if and only if in “pure state”.
                                               
   In a d-dimensional Hilbert space: S   log 2 d,
     the equality if and only if in a completely mixed
    state, i.e.
                   1 / d 0  0 
                    0 1/ d  0 
                I                 
               
                d            
                                  
                    0    0  1/ d 
     Quantifying Entanglement
► Entropy   is a measure of entanglement
   After partial measurement
    ►Randomizes the initial state
    ►Can compute reduced density matrix by partial trace

   Entropy of the resulting mixed state measures
    the amount of this randomization
    ►The larger the entropy
    The more randomized the state after measurement
    The more entangled the initial state was!
        Quantifying Entanglement
► Consider a pair of systems (X,Y)
► Mutual Information
   I(X, Y) = S(X) + S(Y) – S(X,Y)
   J(X, Y) = S(X) – S(X|Y)
   Follows from Bayes Rule:
     ►p(X=x|Y=y)  = p(X=x and Y=y)/p(Y=y)
     ►Then S(X|Y) = S(X,Y) – S(Y)

► For   classical systems, we always have I = J
        Quantifying Entanglement
► Quantum      Systems
   S(X), S(Y) come from treating individual subsystems
    independently
   S(X,Y) come from the joint system
   S(X|Y) = State of X given Y
       ► Ambiguous   until measurement operators are defined
       ► Let Pj be a projective measurement giving j with prob pj

   S(X|Y) = Σj pj S(X|PjY)
► Define discord (dependent on projectors)
   D = J(X,Y) – I(X,Y)
► In   NMR, reach states with nonzero discord
   Discord central to quantum computation?
                Conclusions
        over unitary evolution in NMR has
► Control
 allowed small algorithms to be implemented
   Some quantum features must be present
   Much further than many other QC realizations
► Importance   of synthesis realized
   Designing a RF pulse sequence which
    implements an algorithm
   Want to minimize imperfections, add error
    correction
                References
► NMR   Quantum Information Processing and
  Entanglement. R. Laflamme and D. Cory.
  Quantum Information and Computation, Vol 2. No
  2. (2002) 166-176
► Introduction to NMR Quantum Information
  Processing. R. Laflamme, et al. April 8, 2002.
  www.c3.lanl.gov/~knill/qip/nmrprhtml/
► Entropy in the Quantum World. Panagiotis
  Aleiferis, EECS 598-1 Fall 2001

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:4
posted:10/16/2012
language:Latin
pages:19