Real-World Quantum Measurements: Fun With Photons and Atoms
description
Transcript of Real-World Quantum Measurements: Fun With Photons and Atoms
![Page 1: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/1.jpg)
Aephraim M. Steinberg Centre for Q. Info. & Q. Control
Institute for Optical SciencesDept. of Physics, U. of Toronto
Real-World Quantum Measurements:Fun With Photons and Atoms
CAP 2006, Brock University
![Page 2: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/2.jpg)
QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
DRAMATIS PERSONÆToronto quantum optics & cold atoms group: Postdocs: Morgan Mitchell ( ICFO)
Matt Partlow An-Ning Zhang Optics: Rob Adamson Kevin Resch(Zeilinger )
Lynden(Krister) Shalm Masoud Mohseni (Lidar)Xingxing Xing Jeff Lundeen (Walmsley)
Atoms: Jalani Fox (...Hinds) Stefan Myrskog (Thywissen)Ana Jofre(Helmerson) Mirco SierckeSamansa Maneshi Chris Ellenor Rockson Chang Chao Zhuang
Current ug’s: Shannon Wang, Ray Gao, Sabrina Liao, Max Touzel, Ardavan DarabiSome helpful theorists: Daniel Lidar, János Bergou, Pete Turner, John Sipe, Paul Brumer, Howard Wiseman, Michael Spanner,...
![Page 3: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/3.jpg)
The 3 quantum computer scientists:see nothing (must avoid "collapse"!)hear nothing (same story)say nothing (if any one admits this thing
is never going to work, that's the end of our funding!)
Quantum Computer Scientists
![Page 4: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/4.jpg)
OUTLINEThe grand unified theory of physics talks:
“Never underestimate the pleasure peopleget from being told something they already know.”
Beyond the standard model:“If you don’t have time to explain something
well, you might as well explain lots of things poorly.”
![Page 5: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/5.jpg)
OUTLINEMeasurement: this is not your father’s observable!
• {Forget about projection / von Neumann}
• “Generalized” quantum measurement• Weak measurement (postselected quantum systems)
• “Interaction-free” measurement, ...• Quantum state & process tomography• Measurement as a novel interaction (quantum logic)• Quantum-enhanced measurement• Tomography given incomplete information
• {and many more}
![Page 6: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/6.jpg)
Distinguishing the indistinguishable...
1
![Page 7: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/7.jpg)
How to distinguish non-orthogonal states optimally
Use generalized (POVM) quantum measurements. [see, e.g., Y. Sun, J. Bergou, and M. Hillery, Phys.
Rev. A 66, 032315 (2002).]
The view from the laboratory:A measurement of a two-state system can onlyyield 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 to use an expanded (3D or more)system. To distinguish 3 states, we need 4D or more.
H-polarized photon 45o-polarized photon
vs.
![Page 8: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/8.jpg)
The geometric picture
Two non-orthogonal vectorsThe same vectors rotated so theirprojections onto x-y are orthogonal
(The z-axis is “inconclusive”)
90o
1
2
1
![Page 9: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/9.jpg)
But a unitary transformation in a 4D space produces:
…and these states can be distinguished with certaintyup to 55% of the time
A test caseConsider 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.)
![Page 10: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/10.jpg)
Experimental schematic
(ancilla)
![Page 11: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/11.jpg)
A 14-path interferometer for arbitrary 2-qubit unitaries...
![Page 12: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/12.jpg)
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"
M. Mohseni, A.M. Steinberg, and J. Bergou, Phys. Rev. Lett. 93, 200403 (2004)
![Page 13: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/13.jpg)
Can we talk about what goes on behind closed doors?
(“Postselection” is the big new buzzword in QIP...but how should one describe post-selected states?)
![Page 14: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/14.jpg)
Conditional measurements(Aharonov, Albert, and Vaidman)
Prepare a particle in |i> …try to "measure" some observable A…postselect the particle to be in |f>
Does <A> depend more on i or f, or equally on both?Clever answer: both, as Schrödinger time-reversible.
Conventional answer: i, because of collapse.
ii ffMeasurement of A
Reconciliation: measure A "weakly."Poor resolution, but little disturbance.
…. can be quite odd …
ifiAf
A =w
AAV, PRL 60, 1351 ('88)
![Page 15: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/15.jpg)
Predicting the past...
A+B
What are the odds that the particlewas in a given box (e.g., box B)?
B+C
A+B
It had to be in B, with 100% certainty.
![Page 16: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/16.jpg)
Consider some redefinitions...
In QM, there's no difference between a box and any other state (e.g., a superposition of boxes).
What if A is really X + Y and C is really X - Y?
A + B= X+B+Y
B + C =X+B-Y
X Y
![Page 17: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/17.jpg)
A redefinition of the redefinition...
X + B'= X+B+Y
X + C' =X+B-Y
X Y
So: the very same logic leads us to conclude theparticle was definitely in box X.
![Page 18: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/18.jpg)
The Rub
![Page 19: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/19.jpg)
A (von Neumann) Quantum Measurement of A
Well-resolved statesSystem and pointer become entangled
Decoherence / "collapse"Large back-action
Initial State of Pointer
x x
Hint=gApx
System-pointercoupling
Final Pointer Readout
![Page 20: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/20.jpg)
A Weak Measurement of A
Poor resolution on each shot. Negligible back-action (system & pointer separable)
Mean pointer shift is given by <A>wk.
Hint=gApx
System-pointercouplingx
Initial State of Pointer
x
Final Pointer Readout
Need not lie within spectrum of A, or even be real...
![Page 21: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/21.jpg)
The 3-box problem: weak msmtsPrepare a particle in a symmetric superposition of
three boxes: A+B+C.Look to find it in this other superposition:
A+B-C.Ask: between preparation and detection, what was
the probability that it was in A? B? C?
Questions: were these postselected particles really all in A and all in B? can this negative "weak probability" be observed?
PA = < |A><A| >wk = (1/3) / (1/3) = 1PB = < |B><B| >wk = (1/3) / (1/3) = 1
PC = < |C><C|>wk = (-1/3) / (1/3) = 1.ifiAf
A =w
[Aharonov & Vaidman, J. Phys. A 24, 2315 ('91)]
![Page 22: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/22.jpg)
A Gedankenexperiment...
e-
e-
e-
e-
![Page 23: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/23.jpg)
0.4
0.6
0.8
1
1.2
1.4
100120140160180200220
Intensity (arbitrary units)
Pixel Number
Rails A and B (no shift)
Rail C(pos. shift)
A+B–C(neg. shift!)
A negative weak value for Prob(C)Perform weak msmt on rail C.
Post-select either A, B, C, or A+B–C.
Compare "pointer states" (vertical profiles).
K.J. Resch, J.S. Lundeen, and A.M. Steinberg, Phys. Lett. A 324, 125 (2004).
![Page 24: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/24.jpg)
Seeing without looking
2a
QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
![Page 25: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/25.jpg)
Problem:Consider a collection of bombs so sensitive that
a collision with any single particle (photon, electron, etc.)is guarranteed to trigger it.
Suppose that certain of the bombs are defective,but differ in their behaviour in no way other than thatthey will not blow up when triggered.
Is there any way to identify the working bombs (orsome of them) without blowing them up?
" Quantum seeing in the dark "(AKA: The Elitzur-Vaidman bomb experiment)
A. Elitzur, and L. Vaidman, Found. Phys. 23, 987 (1993)P.G. Kwiat, H. Weinfurter, and A. Zeilinger, Sci. Am. (Nov., 1996)
BS1
BS2
DC
Bomb absent:Only detector C fires
Bomb present:"boom!" 1/2 C 1/4 D 1/4
The bomb must be there... yetmy photon never interacted with it.
![Page 26: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/26.jpg)
BS1-
e-
BS2-
O-
C-D-
I-
BS1+
BS2+
I+
e+
O+
D+C+
W
Outcome ProbD+ and C- 1/16
D- and C+ 1/16
C+ and C- 9/16
D+ and D- 1/16
Explosion 4/16
Hardy's Paradox(for Elitzur-Vaidman “interaction-free measurements”)
D- –> e+ was in
D+D- –> ?
But … if they wereboth in, they shouldhave annihilated!
D+ –> e- was in
![Page 27: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/27.jpg)
Probabilities e- in e- oute+ in 1
e+ out 0
1 0
0 1
1 1
But what can we say about where the particles were or weren't, once D+ & D– fire?
In fact, this is precisely what Aharonov et al.’s weak measurementformalism predicts for any sufficiently gentle attempt to “observe”these probabilities...
![Page 28: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/28.jpg)
N(I-) N(O)
N(I+) 0 1 1N(O+) 1 1 0
1 0
0.039±0.0230.925±0.024
0.087±0.0210.758±0.0830.721±0.074N(O+)
0.882±0.0150.663±0.0830.243±0.068N(I+)
N(O)N(I-)
Experimental Weak Values (“Probabilities”?)
Ideal Weak Values
Weak Measurements in Hardy’s Paradox
![Page 29: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/29.jpg)
Quantum tomography: what & why?
3
1. Characterize unknown quantum states & processes2. Compare experimentally designed states & processes to design goals3. Extract quantities such as fidelity / purity / tangle4. Have enough information to extract any quantities defined in the future!
• or, for instance, show that no Bell-inequality could be violated5. Learn about imperfections / errors in order to figure out how to
• improve the design to reduce imperfections• optimize quantum-error correction protocols for the system
![Page 30: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/30.jpg)
Quantum InformationWhat's so great about it?
![Page 31: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/31.jpg)
What makes a computer quantum?
If a quantum "bit" is described by two numbers: |Ψ> = c0|0> + c1|1>,
then n quantum bits are described by 2n coeff's:|Ψ> = c00..0|00..0>+c00..1|00..1>+...c11..1|11..1>;
this is exponentially more information than the 2n coefficients it would take to describe n independent (e.g., classical) bits.
It is also exponentially sensitive to decoherence.
Photons are ideal carriers of quantum information theycan be easily produced, manipulated, and detected, anddon't interact significantly with the environment. Theyare already used to transmit quantumcryptographicinformation through fibres under Lake Geneva, and soonthrough the air up to satellites.
Unfortunately, they don't interact with each other very mucheither! How to make a logic gate?
across the Danube
(...Another talk, or more!)
We need to understand the nature of quantum information itself.
How to characterize and compare quantum states? How to most fully describe their evolution in a given system? How to manipulate them?
The danger of errors & decoherence grows exponentially with system size.The only hope for QI is quantum error correction.We must learn how to measure what the system is doing, and then correct it.
(One partial answer...)
![Page 32: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/32.jpg)
Density matrices and superoperatorsOne photon: H or V.State: two coefficients ()CHCV
( )CHH
CHV
CVH
CVV
Density matrix: 2x2=4 coefficientsMeasure
intensity of horizontalintensity of verticalintensity of 45ointensity of RH circular.
Propagator (superoperator): 4x4 = 16 coefficients.
Two photons: HH, HV, VH, VV, or any superpositions.State has four coefficients.Density matrix has 4x4 = 16 coefficients.Superoperator has 16x16 = 256 coefficients.
![Page 33: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/33.jpg)
Some density matrices...
Resch, Walther, Zeilinger, PRL 94 (7) 070402 (05)Klose, Smith, Jessen, PRL 86 (21) 4721 (01)
Polarisation state of 3 photons(GHZ state)
Spin state of Cs atoms (F=4),in two bases
Much work on reconstruction of optical density matrices in the Kwiatgroup; theory advances due to Hradil & others, James & others, etc...;now a routine tool for characterizing new states, for testing gates orpurification protocols, for testing hypothetical Bell Inequalities, etc...
![Page 34: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/34.jpg)
QuickTime™ and aPhoto - JPEG decompressor
are needed to see this picture.
Wigner function of atoms’ vibrational quantum state in optical lattice
J.F. Kanem, S. Maneshi, S.H. Myrskog, and A.M. Steinberg, J. Opt. B. 7, S705 (2005)
![Page 35: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/35.jpg)
Superoperator after transformationto correct polarisation rotations:
Residuals allow us to estimatedegree of decoherence andother errors.
Superoperator provides informationneeded to correct & diagnose operation
Measured superoperator,in Bell-state basis:
Leading Kraus operator allowsus to determine unitary error.
(expt) (predicted)
M.W. Mitchell, C.W. Ellenor, S. Schneider, and A.M. Steinberg, Phys. Rev. Lett. 91 , 120402 (2003)
![Page 36: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/36.jpg)
QPT of QFTWeinstein et al., J. Chem. Phys. 121, 6117 (2004)
To the trained eye, this is a Fourier transform...
![Page 37: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/37.jpg)
The status of quantum cryptography
![Page 38: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/38.jpg)
Measurement as a tool:Post-selective operations for the construction of
novel (and possibly useful) entangled states...
4a
![Page 39: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/39.jpg)
Theory: H. Lee et al., Phys. Rev. A 65, 030101 (2002); J. Fiurásek, Phys. Rev. A 65, 053818 (2002)
Highly number-entangled states("low-noon" experiment).
Important factorisation:
=+
A "noon" stateA really odd beast: one 0o photon,one 120o photon, and one 240o photon...but of course, you can't tell them apart,let alone combine them into one mode!
States such as |n,0> + |0,n> ("noon" states) have been proposed for high-resolution interferometry – related to "spin-squeezed" states.
![Page 40: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/40.jpg)
Trick #1
Okay, we don't even have single-photon sources*.
But we can produce pairs of photons in down-conversion, andvery weak coherent states from a laser, such that if we detectthree photons, we can be pretty sure we got only one from thelaser and only two from the down-conversion...
SPDC
laser
|0> + e |2> + O(e2)
|0> + a |1> + O(a2)
ea |3> + O(a3) + O(e2) + terms with <3 photons
* But we’re working on it (collab. with Rich Mirin’s quantum-dot group at NIST)
![Page 41: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/41.jpg)
How to combine three non-orthogonal photons into one spatial mode?
Postselective nonlinearityPostselective nonlinearity
Yes, it's that easy! If you see three photonsout one port, then they all went out that port.
"mode-mashing"
![Page 42: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/42.jpg)
-
+ei3φ
HWP
QWPPhaseshifter
PBS
DCphotons PP
Dark ports
Ti:sa
to analyzer
The basic optical scheme
![Page 43: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/43.jpg)
It works!
Singles:
Coincidences:
Triplecoincidences:
Triples (bgsubtracted):
M.W. Mitchell, J.S. Lundeen, and A.M. Steinberg, Nature 429, 161 (2004)
![Page 44: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/44.jpg)
Complete characterisationwhen you have incomplete information
4b 4b
![Page 45: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/45.jpg)
Fundamentally Indistinguishablevs.
Experimentally Indistinguishable
But what if when we combine our photons,there is some residual distinguishing information:some (fs) time difference, some small spectraldifference, some chirp, ...?
This will clearly degrade the state – but how dowe characterize this if all we can measure ispolarisation?
![Page 46: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/46.jpg)
Quantum State Tomography
{ }21212121 ,,, VVVHHVHH
Distinguishable Photon Hilbert Space
{ }, ,HH HV VH VV+
{ }2 ,0 , 1 ,1 , 0 ,2H V H V H V
Indistinguishable Photon Hilbert Space ?
Yu. I. Bogdanov, et alPhys. Rev. Lett. 93, 230503 (2004)
If we’re not sure whether or not the particles are distinguishable,do we work in 3-dimensional or 4-dimensional Hilbert space?
If the latter, can we make all the necessary measurements, giventhat we don’t know how to tell the particles apart ?
![Page 47: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/47.jpg)
The sections of the density matrix labelled “inaccessible” correspond to information about the ordering of photons with respect to inaccessible
degrees of freedom.
The Partial Density Matrix
Inaccessible
information
Inaccessible
information
€
rHH ,HH ρ HV +VH ,HH ρVV ,HH
ρ HH ,HV +VH ρ HV +VH ,HV +VH ρVV ,HV +VH
ρ HH ,VV ρ HV +VH ,VV ρVV ,VV
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
ρ HV −VH ,HV −VH( )
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
The answer: there are only 10 linearly independent parameters which are invariant under permutations of the particles. One example:
(For n photons, the # of parameters scales as n3, rather than 4n)
R.B.A. Adamson, L.K. Shalm, M.W. Mitchell, and A.M. Steinberg, quant-ph/0601134
![Page 48: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/48.jpg)
When distinguishing information is introduced the HV-VH component increases without affecting the state in the symmetric space
Experimental ResultsNo Distinguishing Info Distinguishing Info
HH + VVMixture of45–45 and –4545
![Page 49: Real-World Quantum Measurements: Fun With Photons and Atoms](https://reader035.fdocuments.us/reader035/viewer/2022081513/56815c94550346895dcaa20b/html5/thumbnails/49.jpg)
The moral of the story1. Post-selected systems often exhibit surprising behaviour which
can be probed using weak measurements.
2. Post-selection can also enable us to generate novel “interactions” (KLM proposal for quantum computing), and for instance to produce useful entangled states.
3. POVMs, or generalized quantum measurements, are in some ways more powerful than textbook-style projectors
4. Quantum process tomography may be useful for characterizing and "correcting" quantum systems (ensemble measurements).
5. A modified sort of tomography is possible on “practically indistinguishable” particles
QuickTime™ and aYUV420 codec decompressorare needed to see this picture.
Predicted Wigner-Poincaréfunction for a variety of“triphoton states” we arestarting to produce: