Some final words about quantum measurement...

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Some final words about quantum measurement...

• Some remarks on Bell-state measurement, its
  uses, and on quantum error correction
• Measurement as action gloss of KLM
• Related pre- and postselected effects quantum
  -interferometric effective nonlinearities
• importance of both preparation projection
• difference between detection measurement
• Summary
• Measurement is more general than projection
• One can measure postselected "subensembles"
• Measurement has important physical effects

16 Dec 2003 (and Happy Holidays!)
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Information and measurement
Any measurement on a qubit (two-level system)
yields at most 1 bit of info. On the other hand,
a full specification of the state (density
matrix) of a qubit involves 3 independent real
parameters (coordinates on Bloch/Poincaré
sphere) this is in principle an infinite amount
of information. How much information can be
stored or transferred using qubits? Measure
reproduce only one classical bit results from
the measurement, and this is all which can be
reproduced. "No cloning" cannot make faithful
copies of unknown, non-orthogonal quantum
states, because ltaltabgtbgt ltabgt2 and
unitary evolution preserves the inner product.
Wooters Zurek, Nature 299, 802 (1982).
(N.B. Applies to unitary evolution. With
projection, one can for instance distinguish 0
from 45 sometimes, and then reproduce the exact
state but notice, still only one classical
bit's worth of information.)
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Dense coding Teleportation
Bennett Wiesner, PRL 69, 2881 (1992)
Observation a pair of entangled photons has four
orthogonal basis states the Bell states but
they can be connected by operations on a single
photon. Thus sending that single photon to a
partner who already possesses the other entangled
photon allows one to convey 2 classical
bits using a single photon.
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Log2(3) bits in a single photon
To extract both bits, one would need to
distinguish all 4 Bell states this can't be
done with linear optics, but 3 of the 4
can. (Recall a Hong-Ou-Mandel already filters
out the singlet)
Mattle et al., PRL 76, 4656 (1996)
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Quantum Teleportation
Bennett et al., Phys. Rev. Lett. 70, 1895 (1993)
(And the other three results just leave Bob with
a unitary operation to do)
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Quantum Teleportation (expt)
Bouwmeester et al., Nature 390, 575 (1997)
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One striking aspect of teleportation

• Alice's photon and Bob's have no initial
  relationship Bob's could be in any of an
  infinite positions on the Poincaré sphere.
• The Bell-state measurement collapses photon S
  (and hence Bob's photon I) into one of four
  particular states states with well-defined
  relationships to Alice's initial photon.
• Thus this measurement transforms a continuous,
  infinite range of possibilities (which we
  couldn't detect, let alone communicate to Bob)
  into a small discrete set.
• All possible states can be teleported, by
  projecting the continuum onto this complete set.

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Quantum Error Correction
In classical computers, small errors are
continuously corrected built-in dissipation
pulls everything back towards a "1" or a
"0". Recall that quantum computers must avoid
dissipation and irreversibility. How, then, can
errors be avoided?
A bit could be anywhere on the Poincaré sphere
and an error could in principle move it anywhere
else. Can we use measurement to reduce the error
symptoms to a discrete set, à la teleportation?
Yes if you measure whether or not a bit flipped,
you get either a "YES" or a "NO", and can correct
it in the case of "YES". As in dense coding, the
phase degree of freedom is also important, but
you can similary measure whether or not the phase
was flipped, and then correct that. Any possible
error can be collapsed onto a "YES" or "NO" for
each of these.
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The four linearly independent errors
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Q. error correction Shor's 3-bit code
In case of bit flips, use redundancy it's
unlikely that more than 1 bit will flip at once,
so we can use "majority rule"... BUT we must
not actually measure the value of the bits!
Encode a0gt b1gt ? a000gt b111gt
Symptom State i1?i2 i1 ?i3
Nothing happens a000gt b111gt 0 0
i1 flips a100gt b011gt 1 1
i2 flips a010gt b101gt 1 0
i3 flips a001gt b110gt 0 1
And now just flip i1 back if you found that it
was flipped note that when you measure which of
these four error syndromes occurred, you exhaust
all the information in the two extra bits, and no
record is left of the value of i1!
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The dream of optical quantum computing
INPUT STATE
OUTPUT STATE
a0gt b1gt c2gt
a0gt b1gt c2gt
TRIGGER (postselection)
ANCILLA
special ?i gt
particular ?f gt
But real nonlinear interactions are typically
1010 times too weak to do this! What can one do
with purely "linear" optics?
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Hong-Ou-Mandel as interaction?
If I detect a "trigger" photon here...
...then anything which comes out here must have
the opposite polarisation.
Two non-interacting photons became entangled, not
only by meeting at a beam-splitter, but by being
found on opposite sides (postselection). Choosing
the state of one can determine which states of
the other are allowed to be reflected (if we only
pay attention to cases where coincidences occur.)
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The germ of the KLM idea
INPUT STATE
OUTPUT STATE
a0gt b1gt c2gt
a'0gt b'1gt c'2gt
TRIGGER (postselection)
ANCILLA
1gt
1gt
In particular with a similar but somewhat more
complicated setup, one can engineer a 0gt b
1gt c 2gt ??a 0gt b 1gt c 2gt
effectively a huge self-phase modulation (p per
photon). More surprisingly, one can efficiently
use this for scalable QC.
KLM Nature 409, 46, (2001) Cf. Kaoru's
experiment Sara's experiment experiments by
Franson et al., White et al., ...
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(What you really have to do)
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The mad, mad idea of Jim Franson
J.D. Franson, Phys. Rev. Lett 78, 3852 (1997)
Nonlinear coefficients scale linearly with the
number of atoms. Could the different atoms'
effects be made to add coherently, providing an
N2 enhancement (where N might be 1013)?
Appears to violate local energy conservation...
but consists of perfectly reasonable Feynman
diagrams, with energy conserved in final
state. Controversy regarding some magic
cancellations.... Each of N(N-1)/2 pairs of
atoms should contribute. Franson proposes that
this can lead to immense nonlinearities. No
conclusive data.
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John Sipe's suggestion
Franson's proposal to harness photon-exchange
terms investigates the effect on the real index
of refraction (virtual intermediate state). Why
not first search for such effects on real
intermediate states (absorption)?
Conclusion exchange effects do matter
Probability of two-photon absorption may be
larger than product of single-photon abs.
prob's. Caveat the effect indeed goes as N2,
... but N is the photon number (2)
and not the atom number (1013) !
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Ugly data,but it works.
Resch et al. quant-ph/0306198
Roughly a 4 drop observed in 2-photon
transmission when the photons are delayed
relative to one another.
Complicated by other effects due to
straightforward frequency correlations between
photons (cf. Wong, Sergienko, Walmsley,...), as
well as correlations between spatial and spectral
mode.
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What was the setup?
Type-II SPDC birefringent delay 45o polarizer
produces delayed pairs. Use a reflective notch
filter as absorbing medium, and detect remaining
pairs.

• This is just a Hong-Ou-Mandel interferometer,
  with detection in a complementary mode.
• Although the filter is placed after the output,
  this is irrelevant for a linear system.
• Interpretations
• Our "suppressed" two-photon reflection is
  merely the ratio of two different interference
  patterns the modified spectrum broadens the
  pattern.
• Yet photons which reach the filter in pairs
  really do not behave independently. The
  HOM interference pattern is itself a
  manifestation of photon exchange effects.

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Another approach to 2-photon interactions...Ask
Is SPDC really the time-reverse of SHG?
(And if so, then why doesn't it exist in
classical em?)
The probability of 2 photons upconverting in a
typical nonlinear crystal is roughly 10-10 (as
is the probability of 1 photon spontaneously
down-converting).
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Quantum Interference
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2-photon "Switch" experiment
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Suppression/Enhancementof Spontaneous
Down-Conversion
(57 visibility)
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Photon-photon transmission switch
On average, less than one photon per pulse. One
photon present in a given pulse is sufficient to
switch off transmission. The photons upconvert
with near-unit eff. (Peak power approx.
mW/cm2). The blue pump serves as a catalyst,
enhancing the interaction by 1010.
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Switchiness ("Nonlinearity")
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Controlled-phase switch
Resch et al, Phys. Rev. Lett. 89, 037914 (2002)
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Fringe data with and w/o postsel.
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...but it actually is true
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So problem solved?
Not entirely...
This switch relies on interference the input
state must have a specific phase. Single photons
don't have well-defined phase the switch does
not work on Fock states. The phase shifts if and
only if a control photon is present-- so long as
we make sure not to know in advance whether
or not it is present. Preparation weak
coherent state with definite phase Post-selection
a "trigger" photon present in that
state Result a strong nonlinearity mediated by
that state, between preparation and
postselection.
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Detection is not all there is to measurement...
We have shown theoretically that a polarisation
version could be used for Bell-state
determination (and, e.g., dense coding) a task
known to be impossible with LO. Resch et al.,
quant-ph/0204034 Like the case of
non-orthogonal state discrimination,
however, this is not the same as projective
measurement. If you promise me to give me one of
the Bell states, I can tell you which one I
received but I can't span the space of
all possible states by projecting onto one of the
four. (No teleportation.)
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SUMMARY

• Measurement is more general than projection
• One can measure postselected "subensembles"
• Measurement has important physical effects

(Cartoon stolen from Jonathan Dowling)