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!)

Some final words about quantum measurement...

Information and measurementAny 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 qubitinvolves 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 preservesthe 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 & TeleportationObservation: a pair of entangled photons has four orthogonalbasis states – the Bell states – but they can be connected byoperations on a single photon.

Thus sending that single photon to a partner who already possessesthe other entangled photon allows one to convey 2 classical bitsusing a single photon.

Bennett & Wiesner, PRL 69, 2881 (1992)

The Bell state basis:

flipphase

flippol.

single-photon operations:

Log2(3) bits in a single photon

To extract both bits, one wouldneed to distinguish all 4 Bellstates – this can't be done withlinear optics, but 3 of the 4 can.(Recall: a Hong-Ou-Mandelalready filters out the singlet)

Mattle et al., PRL 76, 4656 (1996)

Quantum Teleportation

(And the other three results just leave Bob with a unitary operation to do)

Bennett et al., Phys. Rev. Lett. 70, 1895 (1993)

BSM

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!

singletstates

S and I haveopposite polarisations

S I

Alice Bob

A (unknown state)

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 errorcould in principle move it anywhere else. Can we use measurementto 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 youcan similary measure whether or not the phase was flipped, and then correctthat.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

Encode: a|0> + b|1> a|000> + b|111>

Symptom State i1i2 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

In case of bit flips, use redundancy – it's unlikely that more than 1 bitwill flip at once, so we can use "majority rule"...

BUT: we must not actually measure the value of the bits!

And now just flip i1 back if you found that it was flipped –note that when you measure which of these four errorsyndromes occurred, you exhaust all the information in thetwo extra bits, and no record is left of the value of i1!

errors

i1

i2

i3

i1 i2

i1 i3

special |i >

a|0> + b|1> + c|2> a|0> + b|1> – c|2>

The dream of optical quantum computing

INPUT STATE

ANCILLA TRIGGER (postselection)

OUTPUT STATE

particular |f >

MAGIC MIRROR:Acts differently if there are 2 photons or only 1.In other words, can be a “transistor,” or “switch,”or “quantum logic gate”...

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 whichcomes out here musthave the opposite polarisation.

Two non-interacting photons became entangled, not only by meeting ata beam-splitter, but by being found on opposite sides (postselection).Choosing the state of one can determine which states of the other are allowedto be reflected (if we only pay attention to cases where coincidences occur.)

|1>

a|0> + b|1> + c|2> a'|0> + b'|1> + c'|2>

The germ of the KLM ideaINPUT STATE

ANCILLA TRIGGER (postselection)

OUTPUT STATE

|1>In particular: with a similar but somewhat more complicatedsetup, one can engineer

a |0> + b |1> + c |2> a |0> + b |1> – c |2> ;effectively a huge self-phase modulation ( 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?

The probability of 2 photons upconverting in a typicalnonlinear crystal is roughly 10 (as is the probabilityof 1 photon spontaneously down-converting).

(And if so, then why doesn't it exist in classical e&m?)

Quantum Interference

2-photon "Switch": experiment

(57% visibility)

Suppression/Enhancementof Spontaneous Down-Conversion

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 switchResch et al, Phys. Rev. Lett. 89, 037914 (2002)

Fringe data with and w/o postsel.

So: problem solved?

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 ornot it is present.

Preparation: weak coherent state with definite phasePost-selection: a "trigger" photon present in that stateResult: a strong nonlinearity mediated by that state,

between preparation and postselection.

Not entirely...

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 tellyou which one I received; but I can't span the space of allpossible states by projecting onto one of the four. (No teleportation.)

Detection is not all there is to measurement...

SUMMARY

(Cartoon stolen from Jonathan Dowling)

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