Some final words about quantum
measurement...
• Some remarks on Bell-state measurement, its uses,
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
and
16 Dec 2003
(and Happy Holidays!)
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 {<a|<a|}{|b>|b>} = {<a|b>}2 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.)
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.
single-photon
operations:
The Bell state basis:
flip
flip
phase pol.
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)
Quantum Teleportation
Bennett et al., Phys. Rev. Lett. 70, 1895 (1993)
singlet
states
Alice
A (unknown
state)
BSM
S
Bob
I
S and I have
opposite polarisations
If BSM finds A & S in a singlet state,
then we know they have opposite polarisation.
Let Bob know the result.
If S and I were opposite,
and A and S were opposite,
then I = A!
(And the other three results just leave Bob with a unitary operation to do)
Quantum Teleportation (expt)
Bouwmeester et al., Nature 390, 575 (1997)
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.
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.
The four linearly independent errors
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: a|0> + b|1>  a|000> + b|111>
i1
i2
i1 i2
i3
i1 i3
Symptom
State
i1i2
i1 i3
Nothing
happens
a|000> +
b|111>
0
0
i1 flips
a|100> +
b|011>
1
1
i2 flips
a|010> +
b|101>
1
0
i3 flips
a|001> +
b|110>
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!
The dream of optical quantum computing
INPUT STATE
a|0> + b|1> + c|2>
OUTPUT STATE
a|0> + b|1> – c|2>
MAGIC MIRROR:
Acts differently if there are 2 photons or only 1.
TRIGGER (postselection)
ANCILLA
In other words, can be a “transistor,” or “switch,”
or “quantum
special
|i > logic gate”...
particular | >
f
But real nonlinear interactions are typically 1010 times too weak to do this!
What can one do with purely "linear" optics?
Hong-Ou-Mandel as interaction?
|H>
a|H>+b|V>
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.)
The germ of the KLM idea
INPUT STATE
a|0> + b|1> + c|2>
ANCILLA
|1>
OUTPUT STATE
a'|0> + b'|1> + c'|2>
TRIGGER (postselection)
|1>
In particular: with a similar but somewhat more complicated
setup, one can engineer
a |0> + b |1> + c |2> a |0> + b |1> – c |2> ;
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., ...
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 e&m?)
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).
Quantum Interference
2-photon "Switch": experiment
Suppression/Enhancement
of Spontaneous Down-Conversion
(57% visibility)
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.
Switchiness ("Nonlinearity")
Controlled-phase switch
Resch et al, Phys. Rev. Lett. 89, 037914 (2002)
Fringe data with and w/o postsel.
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.
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.)
SUMMARY
• Measurement is more general than projection
• One can measure postselected "subensembles"
• Measurement has important physical effects
(Cartoon stolen from Jonathan Dowling)