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
• ? gt ?00000gt ?10001gt ?n1111gt
• Density Operator ?
• Useful for quantum systems whose state is not
  known
• In most cases we dont know the exact state
• For pure states
• ? ? gtlt ?
• When acted on by unitary U
• ? ? U?U
• When measured, probability of M m
• P M m tr(MmMm ?)

5
Background Quantum States

• Ensemble of pure states
• A quantum system is in one of a number of states
  ?igt
• i is an index
• System in ?igt with probability pi
• pi, ?igt is an ensemble
• Density operator
• ? ?S pi ?igtlt ?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 cant be written as product of
  states
• ? gt ? ?1gt ?2gt? ?ngt
• For mixed states
• Entangled if cannot be written as a convex sum of
  bi-partite states
• ? ? ?S 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
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

• Todays 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 ?i are the eigenvalues of ?, use the
  equivalent definition

14
Quantifying Entanglement

• Basic properties of Von Neumanns 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(XY)
• Follows from Bayes Rule
• p(XxYy) p(Xx and Yy)/p(Yy)
• Then S(XY) 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(XY) State of X given Y
• Ambiguous until measurement operators are defined
• Let Pj be a projective measurement giving j with
  prob pj
• S(XY) Sj pj S(?XPjY)
• 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