1. NMR Quantum Information Processing and Entanglement R.Laflamme, et al. presented by D. Motter

2. 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

3. Outline ► Background  States  Entanglement ► Introduction to NMR ► NMR vs. Entanglement ► Conclusions and Discussion

4. 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(M m † M m  )

5. 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 p i  {p i, |  i >} is an ensemble ► Density operator   =  Σ p i |  i > <  i | ► If the quantum state is not known exactly  Call it a mixed state

6. 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 a i (  1   2 )

7. 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

8. Power of Quantum Computing ► Why are Quantum Computers more powerful than their classical counterparts? ► Several alternatives  Hilbert space of size 2 n, so inherently faster ► But we can only measure one such state  Entangled states during computation ► For pure 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

9. 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

10. 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

11. 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

12. Entanglement in NMR ► Today’s NMR  no entanglement  It is not believed that Liquid State NMR is a promising technology ► Future NMR experiments could show entanglement  Solid state NMR  Larger numbers of qubits in liquid state

13. Quantifying Entanglement ► Measure entanglement by entropy ► Von Neumann entropy of a state ► If λ i are the eigenvalues of ρ, use the equivalent definition:

14. ► Basic properties of Von Neumann’s entropy , equality if and only if in “pure state”.  In a d-dimensional Hilbert space:, the equality if and only if in a completely mixed state, i.e. the equality if and only if in a completely mixed state, i.e. Quantifying Entanglement

15. ► 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!

16. 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

17. 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 p j  S(X|Y) = Σ j p j 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?

18. Conclusions ► Control over unitary evolution in NMR has 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

19. 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/ www.c3.lanl.gov/~knill/qip/nmrprhtml/ ► Entropy in the Quantum World. Panagiotis Aleiferis, EECS 598-1 Fall 2001