Nmr quantum information processing and entanglement
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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

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Nmr quantum information processing and entanglement

NMR Quantum Information Processing and Entanglement

R.Laflamme, et al.

presented by D. Motter


Introduction

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

Outline

  • Background

    • States

    • Entanglement

  • Introduction to NMR

  • NMR vs. Entanglement

  • Conclusions and Discussion


Background quantum states

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 states1

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

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

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

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


Introduction to nmr qc

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 qc1

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


Difficulties in nmr qc

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

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


Quantifying entanglement

Quantifying Entanglement

  • Measure entanglement by entropy

  • Von Neumann entropy of a state

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


Quantifying entanglement1

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.


Quantifying entanglement2

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 entanglement3

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 entanglement4

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

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


References

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


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