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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? ► Why are quantum computers more powerful than classical computers Outline ► Background States Entanglement ► Introductionto 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 ) 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 ► 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 ► 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 ► Forpure 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 ► 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 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 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 ► Today’s NMR no entanglement It is not believed that Liquid State NMR is a promising technology NMR experiments could show ► Future entanglement Solid state NMR Larger numbers of qubits in liquid state Quantifying Entanglement ► Measureentanglement by entropy ► Von Neumann entropy of a state S tr log 2 ► Ifλi are the eigenvalues of ρ, use the equivalent definition: S i log 2 i i Quantifying Entanglement ► Basic properties of Von Neumann’s entropy S 0 , equality if and only if in “pure state”. In a d-dimensional Hilbert space: S log 2 d, the equality if and only if in a completely mixed state, i.e. 1 / d 0 0 0 1/ d 0 I d 0 0 1/ d 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 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 over unitary evolution in NMR has ► Control 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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posted: | 10/16/2012 |

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