Quantum State and Process Tomography: measuring mixed states

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Quantum State and Process Tomography measuring
mixed states

• Now for something different
• Instead of "weak" measurements, let's try to
  measure everything.
• Instead of a state overdetermined by preparation
  and postselection, let's consider states which
  may be incompletely defined.
• Description of mixed states
• Density matrices
• Wigner functions
• Superoperators
• Brief nod to "traditional" (!) quantum state
  tomography
• Two-photon state process tomography
• Optical-latice state process tomography

2 Dec 2003
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Recall mixed states are described by density
matrices, not by wave functions.
Can't determine the coefficients by measuring a
single system - to extract all this information,
we must study a large ensemble of identical
particles.
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How to extract coefficients?
Measure the expectation value of various
operators A each one provides a given linear
combination of the matrix elements. To measure
all n2 elements for an n-dimensional system,
should make n2 different (linearly independent)
measurements.
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What about continuous variables?
It may be negative, but P(x) and P(p) never
are... in fact, W is almost always negative
somewhere.
(In principle, an infinite number of observables
must be measured to extract all the infinite
number of points in W(x,p).)
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How does one measure these things?
Heterodyning allows one to measure Re (Es
eif??for various f? thus we can extract integrals
along every possible angle...
Much work over the last 10-20 years applying
algorithms from medical imaging to extract Wigner
functions, e.g., of light...
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E.g. Wigner function of a photon
Lvovsky et al., Physical Review Letters 87 ,
050402 (2001)
Circularly symmetric no phase-dependence when
you homodyne. Dip in each marginal at 0 -- the
only way this can be is negative
quasiprobability at E1E20. Dip at middle is
related to the Hong-Ou-Mandel dip, and its
high- photon-number analog put another way, it's
our old discussion of interference of number
states, and how the photons tend to bunch.
But with quantum information in mind, let's think
about something different polarisation states of
photon pairs, and how they evolve...
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Density matrices and superoperators
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Two-photon Process Tomography
Two waveplates per photon for state preparation
Detector A
HWP
HWP
PBS
QWP
QWP
SPDC source
QWP
QWP
PBS
HWP
HWP
Detector B
Argon Ion Laser
Two waveplates per photon for state analysis
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Hong-Ou-Mandel Interference
How often will both detectors fire together?
r2t2 0 total destructive interference.
...iff the processes ( thus photons)
indistinguishable.
If the photons have same polarisation, no
coincidences.
Only in the singlet state HVgt VHgt are the
two photons guaranteed to be orthogonal. This
interferometer is a "Bell-state filter,"
needed for quantum teleportation and other
applications.
Our Goal use process tomography to test this
filter.
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Measuring the superoperator
Coincidencences
Output DM Input

HH



16 input states
HV
etc.
VV
16 analyzer settings
VH
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Measuring the superoperator
Superoperator
Input Output DM
HH
HV
VV
VH
Output
Input
etc.
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Testing the superoperator
LL input state
Predicted
Nphotons 297 14
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Testing the superoperator
LL input state
Predicted
Nphotons 297 14
Observed
Nphotons 314
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So, How's Our Singlet State Filter?
Bell singlet state ?? (HV-VH)/v2
Observed ? ??, but a different maximally
entangled state
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Model of real-world beamsplitter
Singlet filter
multi-layer dielectric
AR coating
45 unpolarized 50/50 dielectric beamsplitter
at 702 nm (CVI Laser)
birefringent element singlet-state
filter birefringent element
Necessary correction determined from leading
Kraus operator...
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Superoperator provides informationneeded to
correct diagnose operation
FUTURE more efficient extraction of information
for better correction of errors iterative search
for optimal encodings in presence of collective
noise...
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Tomography in Optical Lattices
Complete characterisation of process on arbitrary
inputs?
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Lattice experimental setup
Setup for lattice with adjustable position
velocity
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First task measuring state populations
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Time-resolved quantum states
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Quantum state reconstruction
Now, we can also perform translation directly in
both x and p
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Recapturing atoms after setting them into
oscillation...
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...or failing to recapture themif you're too
impatient
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Oscillations in lattice wells
essentially a measure of Q(r,?) at fixed
r-- recall, r is set by size of shift and ? by
length of delay
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Extracted phase-space distributions(Q rather
than W in this case)
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Data"W-like" Pg-Pe(x,p) for a mostly-excited
incoherent mixture
(For 2-level subspace, can also choose 4
particular measurements and directly extract
density matrix)
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Theory W(x,p) for 80 excitation
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Atomic state measurement(for a 2-state lattice,
with c00gt c11gt)
initial state
displaced
delayed displaced
left in ground band

tunnels out during adiabatic lowering
(escaped during preparation)
c0 i c1 2
c02
c0 c1 2
c12
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Time-evolution of some states
input density matrices
output density matrices
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Extracting a superoperatorprepare a complete
set of input states and measure each output
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Atom superoperators
sitting in lattice, quietly decohering
being shaken back and forth resonantly
Initial Bloch sphere
CURRENT PROJECTS On atoms, incorporate
"bang-bang" (pulse echo) to preserve coherence
measure homog. linewidth. With photons, study
"tailored" quantum error correction (adaptive
encodings for collective noise).
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Superoperator for resonant drive
Operation ?x (resonantly couple 0 and 1 by
modulating lattice periodically) Measure
superoperator to diagnose single-qubit
operation (and in future, to correct for errors
and decoherence)
Observed Bloch sphere
Upcoming goals generate tailored pulse sequences
to preserve coherence determine whether
decoherence is Markovian et cetera.
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SUMMARY
Any pure or mixed state may be represented by a
density matrix or phase-space distribution (e.g.,
Wigner function). These can be reconstructed by
making repeated measurements in various bases
(n2 measurements for a density matrix). A
superoperator determines the time-evolution of a
density matrix (including decoherence), and
requires n4 measurements. Elements in
quantum-information systems can be
characterized by performing such
measurements. More work needs to be done on (a)
optimizing the extraction of useful
information (b) determining how to use the
resulting superoperators.
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References
Your favorite quantum optics text -- Loudon,
Walls/Milburn, Meystre/Sargent, Milonni,
Scully/Zubairy, etc. -- for introduction to
quantum optics phase-space methods.
Schleich's Quantum Optics in Phase Space.
Leonhardt's Measuring the Quantum State of Light.
Single-photon process tomography White et al.,
PRA 65, 012301 (2002) James et al., PRA 65,
052312 (2001) Ancilla-assisted
photon-polarisation tomography Altepeter et
al., PRL 90, 193601 (2003) Phase-space
tomography on single-photon fields Lvovsky et
al., PRL 87, 050402 (2001) Two-photon process
tomography Mitchell et al., PRL. 91, 120402
(2003) Applications of process
tomography Weinstein et al., PRL 86, 1889
(2001) (in NMR experiment) Boulant et al.,
quant-ph/0211046 (interpreting
superoperators) White et al., quant-ph/0308115
(for 2-photon gates)
Theory Wigner, Phys. Rev. 40, 749
(1932) Hillery et al., Phys. Rep. 106, 121
(1984) Early tomography experiments Smithey
et al, PRL 70, 1244 (1993) (light modes) Dunn
et al., Phys. Rev. Lett. 74, 884
(1995) (molecules) Measurement of negative
Wigner functions Nogues et al, Phys. Rev. A 62,
054101 (2000) (cavity QED) Leibfried et al,
PRL 77, 4281 (1996) (trapped ion)