Transcript Document
Real-World Quantum Measurements:
Fun With Photons and Atoms
Aephraim M. Steinberg
Centre for Q. Info. & Q. Control
Institute for Optical Sciences
Dept. of Physics, U. of Toronto
CAP 2006, Brock University
DRAMATIS PERSONÆ
Toronto quantum optics & cold atoms group:
Postdocs: Morgan Mitchell ( ICFO)
Matt Partlow
Optics: Rob Adamson
Lynden(Krister) Shalm
Xingxing Xing
An-Ning Zhang
Kevin Resch(Zeilinger
Masoud Mohseni (Lidar)
Jeff Lundeen (Walmsley)
)
Atoms: Jalani Fox (...Hinds)
Stefan Myrskog (Thywissen)
Ana Jofre(Helmerson) Mirco Siercke
Samansa Maneshi
Chris Ellenor
Rockson Chang
Chao Zhuang
Current ug’s: Shannon Wang, Ray Gao, Sabrina Liao, Max Touzel, Ardavan Darabi
Some helpful theorists:
QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture.
Daniel Lidar, János Bergou, Pete Turner, John Sipe, Paul Brumer, Howard Wiseman, Michael Spanner,...
Quantum Computer Scientists
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!)
OUTLINE
The grand unified theory of physics talks:
“Never underestimate the pleasure people
get 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.”
OUTLINE
Measurement: 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}
1
Distinguishing the indistinguishable...
How to distinguish non-orthogonal
states optimally
vs.
H-polarized photon
45o-polarized photon
The view from the laboratory:
Use generalized
(POVM) quantum
measurements.
A measurement
of a two-state
system can only
[see,yield
e.g., Y.
Sun,
J. Bergou,
and M. Hillery, Phys.
two
possible
results.
Rev.
A 66, 032315isn't
(2002).]
If the
measurement
guaranteed to succeed, there
are 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.
The geometric picture
1
2
Q
Q
90o
1
Two non-orthogonal vectors
The same vectors rotated so their
projections onto x-y are orthogonal
(The z-axis is “inconclusive”)
A test case
Consider these three non-orthogonal states:
Projective measurements can distinguish these states
with 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.)
But a unitary transformation in a 4D space produces:
…and these states can be distinguished with certainty
up to 55% of the time
Experimental schematic
(ancilla)
A 14-path interferometer for
arbitrary 2-qubit unitaries...
Success!
"Definitely 3"
"Definitely 2"
"Definitely 1"
"I don't know"
The correct state was identified 55% of the time-Much better than the 33% maximum for standard measurements.
M. Mohseni, A.M. Steinberg, and J. Bergou, Phys. Rev. Lett. 93, 200403 (2004)
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?)
Conditional measurements
(Aharonov, Albert, and Vaidman)
AAV, PRL 60, 1351 ('88)
Prepare a particle in |i> …try to "measure" some observable A…
postselect the particle to be in |f>
i i
Measurement
of A
f 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.
Reconciliation: measure A "weakly."
Poor resolution, but little disturbance.
Aw
f Ai
f i
…. can be quite odd …
Predicting the past...
A+B
B+C
What are the odds that the particle
was in a given box (e.g., box B)?
It had to be in B, with 100% certainty.
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
X
Y
B+C=
X+B-Y
A redefinition of the redefinition...
So: the very same logic leads us to conclude the
particle was definitely in box X.
X + B'
= X+B+Y
X
Y
X + C' =
X+B-Y
The Rub
A (von Neumann) Quantum
Measurement of A
Initial State of Pointer
Final Pointer Readout
Hint=gApx
System-pointer
coupling
x
x
Well-resolved states
System and pointer become entangled
Decoherence / "collapse"
Large back-action
A Weak Measurement of A
Initial State of Pointer
Final Pointer Readout
Hint=gApx
x
System-pointer
coupling
x
Poor resolution on each shot.
Negligible back-action (system & pointer separable)
Mean pointer shift is given by <A>wk.
Need not lie within spectrum of A, or even be real...
The 3-box problem: weak msmts
Prepare 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?
Aw
f Ai
f i
PA = < |A><A| >wk = (1/3) / (1/3) = 1
PB = < |B><B| >wk = (1/3) / (1/3) = 1
PC = < |C><C|>wk = (-1/3) / (1/3) = -1.
Questions:
were these postselected particles really all in A and all in B?
can this negative "weak probability" be observed?
[Aharonov & Vaidman, J. Phys. A 24, 2315 ('91)]
A Gedankenexperiment...
ee-
e-
e-
A negative weak value for Prob(C)
Intensity (arbitrary units)
Perform weak msmt
on rail C.
Post-select either A,
B, C, or A+B–C.
Compare "pointer
states" (vertical
profiles).
1.4
1.2
A+B–C
(neg. shift!)
Rail C
(pos. shift)
1
0.8
0.6
Rails A and B (no shift)
0.4
220
200
180
160
140
Pixel Number
K.J. Resch, J.S. Lundeen, and A.M. Steinberg, Phys. Lett. A 324, 125 (2004).
120
100
2a
Seeing without looking
QuickTime™ and a
TIFF (Uncompressed) decompressor
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QuickT i me™ and a
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" 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)
Problem:
D
C
Consider a collection of bombs so sensitive that
a collision with any single particle (photon, electron, etc.)
Bomb absent:
is guarranteed to trigger it.
Only detector C fires
BS2 that certain of
Suppose
the bombs are defective,
but differ in their behaviour in no way other than that
Bomb present:
they will not blow up when triggered.
"boom!"
1/2 bombs (or
Is there any way to identify
the working
C up? 1/4
some of them)
without blowing them
BS1
D
1/4
The bomb must be there... yet
my photon never interacted with it.
Hardy's Paradox
(for Elitzur-Vaidman “interaction-free measurements”)
C+
D+
D-
CD+
Outcome
–> e- was
Prob
in
Din
D+–>
ande+
C-was
1/16
BS2+
BS2I+
I-
O-
O+
W
BS1+
e+
BS1e-
D- and C+ 1/16
D+D- –> ?
C+ and C- 9/16
But
D+ …
and
if they
D- 1/16
were
both
in, they 4/16
Explosion
should
have annihilated!
But what can we say about where the particles
were or weren't, once D+ & D– fire?
Probabilities e- in
e- out
e+ in
0
1
1
e+ out
1
-1
0
1
0
In fact, this is precisely what Aharonov et al.’s weak measurement
formalism predicts for any sufficiently gentle attempt to “observe”
these probabilities...
Weak Measurements in Hardy’s Paradox
Ideal Weak Values
N(I-)
N(O-)
N(I+)
N(O+)
0
1
1
1
-1
0
1
0
Experimental Weak Values (“Probabilities”?)
N(I-)
N(O-)
N(I+)
0.243±0.068
0.663±0.083
0.882±0.015
N(O+)
0.721±0.074
-0.758±0.083
0.087±0.021
0.925±0.024
-0.039±0.023
3
Quantum tomography: what & why?
1.
2.
3.
4.
5.
Characterize unknown quantum states & processes
Compare experimentally designed states & processes to design goals
Extract quantities such as fidelity / purity / tangle
Have enough information to extract any quantities defined in the future!
• or, for instance, show that no Bell-inequality could be violated
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
Quantum Information
What's so great about it?
What makes a computer quantum?
(One partial answer...)
If a quantum "bit" is described by two numbers:
|> = c0|0> + c1|1>,
then n quantum bits are described by 2 n coeff's:
|> = 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.
We need to sensitive
understandtothe
nature of quantum information itself.
It is also exponentially
decoherence.
How tocarriers
characterize
and compare
quantum states?
Photons are ideal
of quantum
information-they
to most manipulated,
fully describe their
in and
a given system?
can be easilyHow
produced,
andevolution
detected,
to manipulate
them?
don't interactHow
significantly
with
the environment. They
are already used to transmit quantum-cryptographic
The danger ofthrough
errors & fibres
decoherence
with
system size.
information
undergrows
Lakeexponentially
Geneva, and
soon
across the Danube
The only the
hopeair
forup
QIto
is quantum
error correction.
through
satellites.
We must learn how to measure what the system is doing, and then correct it.
Density matrices and superoperators
()
( )
One photon: H or V.
State: two coefficients
CH
CV
Density matrix: 2x2=4 coefficients
CHH
CVH
CHV
CVV
Measure
intensity of horizontal
intensity of vertical
intensity of 45 o
intensity 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.
Some density matrices...
Much work on reconstruction of optical density matrices in the Kwiat
group; theory advances due to Hradil & others, James & others, etc...;
now a routine tool for characterizing new states, for testing gates or
purification protocols, for testing hypothetical Bell Inequalities, etc...
Spin state of Cs atoms (F=4),
Polarisation state of 3 photons
in two bases
(GHZ state)
Klose, Smith, Jessen, PRL 86 (21) 4721 (01)
Resch, Walther, Zeilinger, PRL 94 (7) 070402 (05)
Wigner function of atoms’ vibrational
quantum state in optical lattice
QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.
J.F. Kanem, S. Maneshi, S.H. Myrskog, and A.M. Steinberg, J. Opt. B. 7, S705 (2005)
Superoperator provides information
needed to correct & diagnose operation
Measured superoperator,
in Bell-state basis:
(expt)
Leading Kraus operator allows
us to determine unitary error.
Superoperator after transformation
to correct polarisation rotations:
(predicted)
Residuals allow us to estimate
degree of decoherence and
other errors.
M.W. Mitchell, C.W. Ellenor, S. Schneider, and A.M. Steinberg, Phys. Rev. Lett. 91 , 120402 (2003)
QPT of QFT
Weinstein et al., J. Chem. Phys. 121, 6117 (2004)
To the trained eye, this is a Fourier transform...
The status of quantum cryptography
4a
Measurement as a tool:
Post-selective operations for the construction of
novel (and possibly useful) entangled states...
Highly number-entangled states
("low-noon" experiment).
States such as |n,0> + |0,n> ("noon" states) have been proposed for
high-resolution interferometry – related to "spin-squeezed" states.
Important factorisation:
+
=
A "noon" state
A 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!
Theory: H. Lee et al., Phys. Rev. A 65, 030101 (2002); J. Fiurásek, Phys. Rev. A 65, 053818 (2002)
Trick #1
Okay, we don't even have single-photon sources*.
But we can produce pairs of photons in down-conversion, and
very weak coherent states from a laser, such that if we detect
three photons, we can be pretty sure we got only one from the
laser and only two from the down-conversion...
SPDC
|0> + e |2> + O(e2)
laser
* But
|0> + |1> + O(2)
e |3> + O(3) + O(e2)
+ terms with <3 photons
we’re working on it (collab. with Rich Mirin’s quantum-dot group at NIST)
Postselective nonlinearity
How to combine three non-orthogonal photons into one spatial mode?
"mode-mashing"
Yes, it's that easy! If you see three photons
out one port, then they all went out that port.
The basic optical scheme
+ e i3
Dark ports
PBS
DC
photons
HWP
to
analyzer
PP
Phas e
s hifte r
QWP
Ti:s a
It works!
Singles:
Coincidences:
Triple
coincidences:
Triples (bg
subtracted):
M.W. Mitchell, J.S. Lundeen, and A.M. Steinberg, Nature 429, 161 (2004)
4b
4b
Complete characterisation
when you have incomplete information
Fundamentally Indistinguishable
vs.
Experimentally Indistinguishable
But what if when we combine our photons,
there is some residual distinguishing information:
some (fs) time difference, some small spectral
difference, some chirp, ...?
This will clearly degrade the state – but how do
we characterize this if all we can measure is
polarisation?
Quantum State Tomography
Indistinguishable
Photon Hilbert Space
2H ,0V , 1H ,1V , 0 H , 2V
HH
, HV VH , VV
?
Distinguishable Photon
Hilbert Space
H1H2 , V1H2 , H1V2 , V1V2
Yu. I. Bogdanov, et al
Phys. 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, given
that we don’t know how to tell the particles apart ?
The Partial Density Matrix
The answer: there are only 10 linearly independent parameters which
are invariant under permutations of the particles. One example:
HH , HH
HV VH, HH
HH , HV VH HV VH, HV VH
HV VH,VV
HH ,VV
Inaccessible
VV , HH
VV , HV VH
VV ,VV
information
HV -VH, HV -VH
Inaccessible
information
The sections of the density matrix labelled “inaccessible” correspond to
information about the ordering of photons with respect to inaccessible
degrees of freedom.
(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
Experimental Results
No Distinguishing Info
Distinguishing Info
When distinguishing
information is introduced the
HV-VH component increases
without affecting the state in
the symmetric space
HH + VV
Mixture of
45–45 and –4545
The moral of the story
1. 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
Predicted Wigner-Poincaré
function for a variety of
“triphoton states” we are
starting to produce:
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