NMR Quantum Information Processing and Entanglement

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(Mm†Mm )

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 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

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 ai(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 2n, 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. 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 Introduction to NMR QC

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 λi are the eigenvalues of ρ, use the equivalent definition:

14. Quantifying Entanglement • 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.

15. 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!

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 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?

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/ • Entropy in the Quantum World. Panagiotis Aleiferis, EECS 598-1 Fall 2001