• Compass states, et cetera:another caveat to theuncertainty principle• N00N states revisited, alongwith their competition

29 Mar 2012(part I)

Not all entangled states are created equal(continued)

Background (A)I’ve already mentioned “N00N” states as the most sensitiveprobe of phase shifts, because of their narrow features inphase space.

Background (A)I’ve already mentioned “N00N” states as the most sensitiveprobe of phase shifts, because of their narrow features inphase space.

But these features are narrow in the azimuthal direction: these states are optimizing for measuring rotations about z.

Are they also suited to measuring arbitrary SU(2) rotations, or is it better to stick with coherent states,or perhaps use a completely different entangled state?

Background (B)Full process tomography can be done on the em fieldby preparing only coherent states and doing homodynemeasurements [Lobino et al., Science 322, 563 (2008).]

Easy to imagine that this would be a good way to characterize translation, squeezing, et cetera. Buthow sensitive is it to generic Fock-space trans-formations? And could it possibly detect small dephasing as well as, say, a N00N or “compass” state?

The x- and p-displacements δxand δp required to significantlylower the overlap with the initialstate may have δxδp << h!

[Zurek, Nature 412, 712 (2001)]

Full vs partial characterization

Complete Characterization –Process Tomography

Detection of a Single Parameter –e.g. degree of decoherencedue to “SU(2) blurring”

Send a complete set of states throughthe process, fully characterize eachstate at the output.

Sample the process with a singlestate, perform a measurement onthe state at the output

General, requires no prior information andyields a complete description of the process

Requires assumptions about theprocess, and then yields a only a singleparameter

Hard (many (~d4) measurements)but general

Easy (1 measurement) butrequires assumptions

VS

Process Tomography - Procedure• Perform process tomography using different sets of input

states• Each set generated from SU(2) rotations of one fiducial state

??????

• Experimentally test process tomography result–Use the result to predict the output for different states–Experimentally send those states through the process

.........

OR

OR...

The experiment:

• Prepares desired biphoton states with >95% fidelitysee Hofmann & Ono, Phys. Rev. A 76, 031806 (07)and Afek, Ambar, &Silberberg, Science 328, 879-881 (10)

Process Tomography - Results

• Results for several sets ofinput states and 2decoherence strengths

• When x≈0.15 the SU(2)orbit generates a 2-design

• N00N states x=0.5 andSU(2) operations cannotgenerate a complete set (Lines = simulations)

Using a single state as a detector ofdecoherence or SU(2)* rotations

• Prepare some initial state and add some decoherence by small randompolarisation rotations; project onto original state

* not arbitrary SU(2n) rotations, which would be process tomography

* not U(1) [phase] rotations, which is where N00N shines

What is the best input state to use?Coherent states? N00N states? Compass states?“2-design promotive states”? ......?

????????

2

2OR OROR...

OR...

Detecting DecoherenceX=0X=0.15X=0.5

theory: exp’t (so far):

(dash=theory)

(solid fromexpt fits)

Summary of Tomography/Metrologyusing Entangled States

• The sensitivities δx and δp of a state (“how much must I disturb it to see adifference”) obey no uncertainty principle such as δx δp ≥ h/2...

• Although coherent states alone can be used to do process tomography, theyare less sensitive than “2-design promotive” squeezed states.–N00N states actually do not even form a tomographically complete set

• N00N states are more sensitive to small, random SU(2) rotations than eithercoherent states or 2-design promotive states (even though the latter arebetter for full tomography).

• N00N states are the most sensitive to decoherence, yet they form the worstset of states for process tomography on decoherence

• The moral of the course (what is measurement?)• POVMs (generalized quantum measurements)• Discrimination of non-orthogonal states

• "Best guess" approach• Unambiguous discrimination• POVMs versus projective measurements• A linear-optical experiment• State-filtering (discrimination of mixed states)

29 Mar 2012[part II}

Generalized measurement...POVMs, non-orthogonal statediscrimination, and a few final

remarks

The moral of the courseMeasurement can be almost anything!

If two systems interact, such that depending on the initial state of system 1,system 2 may evolve differently, then by looking at system 2, I can learn aboutthe initial state of system 1.

The question of how I measure system 2 remains, but if we understand that firststep, the measurement interaction, we already know what has happened to system 1, and how much information is available in principle.

THIS IS NOT AN “APPROXIMATION” TO THE “REAL MEASUREMENT”DESCRIBED BY PROJECTION OPERATORS!

Rather, the theory of projective measurement is an approximate treatmentof one idealized class of such interactions.

Inaccurate measurementAll real measurements have finite accuracy.

Either this means we never perform “true” (projective) measurements, or itmeans that the theory of projective measurements is never an exact descriptionof “true” (experimental) measurements.

spin-1/2

A Stern-Gerlach setup:

There are exactly 2 orthogonalprojectors in a 2D space, so 2measurement outcomes.

Inaccurate measurementAll real measurements have finite accuracy.

Either this means we never perform “true” (projective) measurements, or itmeans that the theory of projective measurements is never an exact descriptionof “true” (experimental) measurements.

spin-1/2

A Stern-Gerlach setup:

There are exactly 2 orthogonalprojectors in a 2D space, so 2measurement outcomes.

Except really, the screen (orparticle position) is contin-uous; there are an infinitenumber of possible outcomeseach giving different infor-mation about the initial state.

No point means “definitely +”,but some mean “99% chanceof +, 1% of -”; while others mean“it’s really a toss-up.”

A real theory of measurement?The real theory of measurement should recognize that

(a) there’s no reason the measurement device should have exactly the same dimensionality as the system(b) the measurements may be uncertain(c) the system may change state when being measured (recall thedecaying atoms)

They need to be positive and sum to I, but they are not necessarily orthogonal projectors; therefore, there may be an arbitrary number of them.

POVMs (positive operator-valued measures), or “generalized measurements”--a set of operators Ei defining probabilites Pi of outcome i:

Note that the probability rule is the same as the familiar one.And the “update rule” is that upon finding result i, you know:

The example of projectors:

A destructive measurement:

If you detect a photon in mode i, you get result “i”; but you end up in the vacuum.The Ei “POVM elements” are the same as for projective measurements, but theMi’s also show the different action of this measurement.

Another simple example –an imperfect spin measurement:

But remember: there is no limitation on the number of operators in this sum.

(Continuous measurement)We never had a chance to discuss continuous measurement,beyond the case of the decaying atom, but it becomes trivialonce you use POVMs.

Each instant of time Δt yields possible outcomes, one of whichis always “no new information” (no photon emitted, forinstance). The limit of Δt -> 0 is well behaved, and one cannow ask about constructing different measurement series,even measuring X for a while, P for a while, et cetera.

(See the references on Jessen & Deutsch’s work on continuousmeasurement of cold atoms, for example, or Mabuchi’s workon adaptive phase estimation)

Can one distinguish betweennonorthogonal states?

• Single instances of non-orthogonal quantum states cannot bedistinguished with certainty. Obviously, ensembles can.

• This is one of the central features of quantum informationwhich leads to secure (eavesdrop-proof) communications.

• Crucial element: we must learn how to distinguish quantumstates as well as possible -- and we must know how well apotential eavesdropper could do.

H-polarized photon 45o-polarized photon

What's the best way to tell these apart?|a〉

|b〉θ

Error rate = (1- sin θ)/2

0 if <a|b> = 0 (ideal measurement)1/2 if <a|b> = 1 (pure guessing)15% for 45oBUT: can we ever tell for sure?

Some interaction would take input states |a> and |b>to "meter states" |"A"> and |"B">, which we could distinguish perfectly.

But unitary interactions preserve overlap:

|"A"〉

|"B"〉

(if they occur with equala priori probability)

Non-unitarity just means asuccess rate < 100% ...

|a〉

|b〉θ

Assuming, as always, equal probability of |a> or |b>, we choose inwhich basis to measure randomly.

The success probability is then:

|a〉

|b〉|"A"〉

|"B"〉

The only way to be sure"A" means a is to be sureit doesn't mean b...

(For 45o, you can succeed up to 25% of the time)

Theory: how to distinguish non-orthogonal states optimally

Step 1:Repeat the letters "POVM" over and over.

The view from the laboratory:A projective measurement of a two-state system can only yield two possible results.

If the measurement isn't guaranteed to succeed, thereare three possible results: (1), (2), and ("I don't know").

Therefore, to discriminate between two non-orth.states, we need three measurement outcomes –a POVM: or practically speaking, some interactionwith a higher-dimensional system.

Step 2:Ask your theorist friends for help.[or see, e.g., Y. Sun, J. Bergou, and M. Hillery, Phys. Rev. A 66, 032315 (2002).]

General bounds on the information...We need a nonunitary transformation to take non-orthogonal a and b to orthogonal "A" and "B".This can be accomplished by post-selection – i.e., by throwing out events.

But wait - 1-0.707= 29.3% > 25% ...is this upper bound unattainable?

Θ

The geometric picture

Two non-orthogonal vectorsThe same vectors rotated so theirprojections onto x-y are orthogonal

(The z-axis is “inconclusive”)

90o

Θ1

2

1

The POVM picture

We want one measurement operator to be |a><a| and one to be |b><b|.If a & b are not orthogonal, these don’t sum to I, and can’t form part of a projective-measurement basis.

But we can have three measurement operators

• α |a><a|• β |b><b|• I - α|a><a| - β|b><b|

which sum to the identity. The third one is the “inconclusive” result.We require α & β < 1 to maintain the positivity of the third operator.

|a〉

|b〉θ

see also:

attenuate the 45o component...

... if you choose the right attenuation,a -> V and b -> H. Really, the “attenuation” is a third outputport indicating “DK.”a = V + e•DKb = H + e•DK

POVM

von Neumannmeasurement

How do they compare?

At 0, the von Neumann strategy has a discontinuity-- only then can you succeed regardless of measurement choice.

At <a|b> = 0.707, the von Neumann strategy succeeds 25% of the time,while the optimum is 29.3%.

But a unitary transformation in a 4D space produces:

…and these states can be distinguished with certaintyup to 55% of the time

The advantage is higher in higher dim.Consider these three non-orthogonal states:

Projective measurements can distinguish these stateswith certainty no more than 1/3 of the time.

(No more than one member of an orthonormal basis is orthogonal to two of the above states, so only one pair may be ruled out.)

Experimental schematic

(ancilla)

A 14-path interferometer forarbitrary 2-qubit unitaries...

Success!

The correct state was identified 55% of the time--Much better than the 33% maximum for standard measurements.

"I don't know"

"Definitely 3"

"Definitely 2""Definitely 1"

Further interesting result: mixed states may also be discriminated, contraryto earlier wisdom.

STATE-DISCRIMINATIONSUMMARY

Non-orthogonal states may be distinguished with certainty("unambiguously") if a finite rate of "inconclusives" is tolerated.

The optimal (lowest) inconclusive rate is the absolute value of theoverlap between the states (in 2D), and cannot be achieved by anyprojective measurement.

POVMs, implementable by coupling to a larger Hilbert space,can achieve this optimum. In optics, they may be realized withoptical multiports (interferometers).

We successfully distinguish among 3 non-orthogonal states 55%of the time, where standard quantum measurements are limitedto 33%.

More recent observation: "state filtering" or discrimination ofmixed states is also possible.

If measurement affects things,it can be useful!

Examples: • Which-path information -- measurement destroys interference  • Zeno effect -- measurement modified dynamics. • Squeezing -- if I measure a previously unknown variable to some accuracy, then I know it tothat accuracy; if I measure N to better than sqrt(N), for instance, then I have a squeezed state. • Error correction / teleportation / et cetera -- by doing the right measurement, you can“project” all the possible (continuous) range of errors to a few discrete possibilities which can becompactly described and efficiently corrected. • KLM-style LOQC (linear-optical quantum computation) • Cluster-state computation • Entanglement...

- Recall that entanglement of two modes can just be related to squeezing of their sumsand differences. Polzik, for instance, has used this to entangle two atomic ensembles .

- If I have two independent excited ions, and I detect one photon the right way, I don’tknow which ion is still excited -- Monroe, for instance, has used this to entangle two separatedtrapped ions.

- If I can measure the sum of the photon number in two beams (by letting the sameprobe interact with both before observing it), then I entangle that: Munro & Nemoto have shownthat this “interaction” would be enough to build a whole quantum computer.

ReferencesSTATE DISCRIMINATIONC. W. Helstrom, Quantum Detection and Estimation

Theory (Academic Press, New York, 1976)I. D. Ivanovic, Phys. Lett. A \23} 257 (1987).A. Chefles and S. M. Barnett, J. Mod. Opt. 45, 1295 (1998)S. M. Barnett and E. Riis, J. Mod. Opt. 44, 1061 (1997)B. Huttner et al., Phys. Rev. A 54, 3783 (1996)R. B. M. Clarke et al., Phys Rev A 63, 040305 (2001)R. B. M. Clarke et al., Phys Rev A 64, 012303 (2001)T. Rudolph, R. W. Spekkens, and P. S. Turner, Phys. Rev.A 68, 0101301 (2003)M. Takeoka, M. Ban, and M. Sasaki, Phys. Rev. A 68,012307 (2003).

A. Chefles, Phys. Lett. A 239, 339 (1998)D. Dieks, Phys. Lett. A 126, 303 (1998)A. Peres, Phys. Lett. A 128, 19 (1988)A. Chefles and S. M. Barnett, Phys. Lett. A 250, 223

(1998)Y. Sun, M. Hillery, and J. A. Bergou, Phys. Rev. A 64,

022311 (2001)J. A. Bergou, M. Hillery, and Y. Sun, J. Mod. Opt. 47, 487

(2000)Y. Sun, J. A. Bergou, and M. Hillery, Phys. Rev. A 66,

032315 (2002)J. A. Bergou, U. Herzog, and M. Hillery, Phys. Rev. Lett.

90, 257901 (2003)

EXCITING TOPICS WE DIDN’T REACH(or not fully)

Compass states: Zurek, Nature 412, 712 (2001)Coherent-state process toography: Lobino et al, Science

322, 563 (2008)Reference frames & superselection: Bartlett et al,

quant-ph/0610030Phase estimation using entangled states: Krischek et al,

PRL 107, 080504 (2011)Adaptive homodyne measurement of optical phase:

Armen et al, PRL 89, 133602 (2002)Quantum State Reconstruction via Continuous

Measurement: Silberfarb et al, PRL 95, 030402(2005)

Continuous Weak Measurement and NonlinearDynamics: Smith et al, PRL 93, 163602 (2004)

Spin-squeezing of a large-spin system via QNDmeasurement: Sewell et al, quant-ph/1111.6969

Can the quantum state be interpreted statistically?:Pusey, Barrett, & Rudolph, quant-ph1111.3328

Mladen Pavicic (and Oliver Benson?), to appear: aclaim that all 4 Bell states can be efficientlydiscriminated using linear optics?!

Menzel, Puhlmann, Heuer, & Schleich: “Wave-particledualism...”, a suggestion that my presentation ofcomplementarity this year has been wrong? Toappear in PNAS