NMR Quantum Information Processing and Entanglement
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Transcript NMR Quantum Information Processing and Entanglement
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
► 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 )
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
=
► If
operator
Σ pi| i>< i|
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
► 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
► 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
► 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
S tr log 2
► If
λi 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.
0 0
1 / d
0 1/ d 0
I
d
0 1/ d
0
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
► 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