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NMR Quantum Information Processing and EntanglementPowerPoint Presentation

NMR Quantum Information Processing and Entanglement

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Presentation Transcript

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

- Background
- States
- Entanglement

- Introduction to NMR
- NMR vs. Entanglement
- Conclusions and Discussion

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
- UU†

- When measured, probability of M = m
- P{ M = m } = tr(Mm†Mm )

- Useful for quantum systems whose state is not known

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

- A quantum system is in one of a number of states | i>
- Density operator
- = Σ pi| i>< i|

- If the quantum state is not known exactly
- Call it a mixed state

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

- For pure states:
- If there is no entanglement, the system can be simulated classically (efficiently)
- Essentially will only have 2n degrees of freedom

- If there is no entanglement, the system can be simulated classically (efficiently)
- For mixed states:
- Liquid State NMR at present does not show entanglement
- Yet is able to simulate quantum algorithms

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

- Hilbert space of size 2n, so inherently faster

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

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 QCDifficulties 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

- 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

- Measure entanglement by entropy
- Von Neumann entropy of a state
- If λi are the eigenvalues of ρ, use the equivalent definition:

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 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!

- After partial measurement

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 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

- 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

- 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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