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# NMR Quantum Information Processing and Entanglement - PowerPoint PPT Presentation

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?

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### NMR Quantum Information Processing and Entanglement

R.Laflamme, et al.

presented by D. Motter

• Does NMR entail true

quantum computation?

• Also:

• What is entanglement (really)?

• What is (liquid state) NMR?

• Why are quantum computers more powerful than classical computers

• Background

• States

• Entanglement

• Introduction to NMR

• NMR vs. Entanglement

• Conclusions and Discussion

• 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 )

• 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

• 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)

• 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

• 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

• 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

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

• 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

• 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

• Measure entanglement by entropy

• Von Neumann entropy of a state

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

• 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.

• 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!

• 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

• 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?

• 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

• 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