RealWorld Quantum Measurements: Fun With Photons and Atoms

1
Real-World Quantum MeasurementsFun 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
2
DRAMATIS PERSONÆ Toronto 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 Siercke Samansa
Maneshi Chris Ellenor Rockson Chang Chao
Zhuang Current ugs Shannon Wang, Ray Gao,
Sabrina Liao, Max Touzel, Ardavan Darabi Some
helpful theorists Daniel Lidar, János Bergou,
Pete Turner, John Sipe, Paul Brumer, Howard
Wiseman, Michael Spanner,...
3
Quantum Computer Scientists
4
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 dont have
time to explain something well, you might as well
explain lots of things poorly.
5
OUTLINEMeasurement this is not your fathers
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

6
Distinguishing the indistinguishable...
7
How to distinguish non-orthogonal states optimally
vs.
The view from the laboratory A measurement of a
two-state system can only yield two possible
results. If the measurement isn't 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.
Use generalized (POVM) quantum measurements.
see, e.g., Y. Sun, J. Bergou, and M. Hillery,
Phys. Rev. A 66, 032315 (2002).
8
The geometric picture
1
2
Two non-orthogonal vectors
9
A test case
10
Experimental schematic
(ancilla)
11
A 14-path interferometer for arbitrary 2-qubit
unitaries...
12
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)
13
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?)
14
Conditional measurements(Aharonov, Albert, and
Vaidman)
AAV, PRL 60, 1351 ('88)
Prepare a particle in igt try to "measure" some
observable A postselect the particle to be in fgt
Does ltAgt depend more on i or f, or equally on
both? Clever answer both, as Schrödinger
time-reversible. Conventional answer i, because
of collapse.
15
Predicting the past...
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.
16
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?
17
A redefinition of the redefinition...
So the very same logic leads us to conclude
the particle was definitely in box X.
18
The Rub
19
A (von Neumann) Quantum Measurement of A
20
A Weak Measurement of A
Poor resolution on each shot.
Negligible back-action (system
pointer separable) Mean pointer shift is given
by ltAgtwk.
Need not lie within spectrum of A, or even be
real...
21
The 3-box problem weak msmts
Prepare a particle in a symmetric superposition
of three boxes ABC. Look to find it in this
other superposition AB-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?
Aharonov Vaidman, J. Phys. A 24, 2315 ('91)
22
A Gedankenexperiment...
23
A negative weak value for Prob(C)
Perform weak msmt on rail C. Post-select
either A, B, C, or ABC. Compare "pointer
states" (vertical profiles).
K.J. Resch, J.S. Lundeen, and A.M. Steinberg,
Phys. Lett. A 324, 125 (2004).
24
Data for PA, PB, and PC...
Rails A and B
Rail C
WEAK
STRONG
STRONG
25
Seeing without looking
26
" 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 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 that they will not blow up when
triggered. Is there any way to identify the
working bombs (or some of them) without blowing
them up?
Bomb absent Only detector C fires
Bomb present "boom!" 1/2 C 1/4
D 1/4
27
Hardy's Paradox(for Elitzur-Vaidman
interaction-free measurements)
D gt e- was in
D- gt e was in DD- gt ? But if they
were both in, they should have annihilated!
28
But what can we say about where the particles
were or weren't, once D D fire?
0
1
1
-1
In fact, this is precisely what Aharonov et al.s
weak measurement formalism predicts for any
sufficiently gentle attempt to observe these
probabilities...
29
Weak Measurements in Hardys Paradox
Ideal Weak Values
30
Quantum tomography what why?

• 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

31
Quantum Information
What's so great about it?
32
Quantum Information
What's so great about it?
33
What makes a computer quantum?
(One partial answer...)
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.
across the Danube
(...Another talk, or more!)
34
The Serious Problem For QI

• The danger of errors grows exponentially with the
  size of the quantum system.
• Without error-correction techniques, quantum
  computation would be a pipe dream.
• A major goal is to learn to completely
  characterize the evolution (and decoherence) of
  physical quantum systems in order to design and
  adapt error-control systems.
• The tools are "quantum state tomography" and
  "quantum process tomography" full
  characterisation of the density matrix or Wigner
  function, and of the "uperoperator" which
  describes its time-evolution.

35
Density matrices and superoperators
36
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 (F4), in two bases
Polarisation state of 3 photons (GHZ state)
Klose, Smith, Jessen, PRL 86 (21) 4721 (01)
Resch, Walther, Zeilinger, PRL 94 (7) 070402 (05)
37
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)
38
Superoperator provides informationneeded to
correct diagnose operation
(expt)
(predicted)
M.W. Mitchell, C.W. Ellenor, S. Schneider, and
A.M. Steinberg, Phys. Rev. Lett. 91 , 120402
(2003)
39
QPT of QFT
Weinstein et al., J. Chem. Phys. 121, 6117 (2004)
To the trained eye, this is a Fourier transform...
40
The status of quantum cryptography
41
Measurement as a tool Post-selective operations
for the construction of novel (and possibly
useful) entangled states...
42
Highly number-entangled states("low-noon"
experiment).
States such as n,0gt 0,ngt ("noon" states) have
been proposed for high-resolution interferometry
related to "spin-squeezed" states.
Important factorisation
43
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...
0gt e 2gt O(e2)
?? 3gt O(?3) O(?2) terms with lt3 photons
0gt ? 1gt O(?2)
But were working on it (collab. with Rich
Mirins quantum-dot group at NIST)
44
Postselective nonlinearity
How to combine three non-orthogonal photons into
one spatial mode?
45
Trick 3
But how do you get the two down-converted photons
to be at 120o to each other? More post-selected
(non-unitary) operations if a 45o photon gets
through a polarizer, it's no longer at 45o. If
it gets through a partial polarizer, it could be
anywhere...
46
The basic optical scheme
47
It works!
Singles
Coincidences
Triple coincidences
Triples (bg subtracted)
M.W. Mitchell, J.S. Lundeen, and A.M. Steinberg,
Nature 429, 161 (2004)
48
4b
Complete characterisation when you have
incomplete information
49
Fundamentally Indistinguishablevs.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?
50
Quantum State Tomography
If were 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 dont know how to tell the
particles apart ?
51
The Partial Density Matrix
The answer there are only 10 linearly
independent parameters which are invariant under
permutations of the particles. One example
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
52
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

Mixture of ?45??45? and ?45??45?
?H??H? ?V??V?
53
More Photons
If you have a collection of spins, what are the
permutation-blind observables that describe the
system?
They correspond to measurements of angular
momentum operators J and mj ... for N photons, J
runs to N/2
So the total number of operators accessible to
measurement is
54
The moral of the story

• Post-selected systems often exhibit surprising
  behaviour which can be probed using weak
  measurements.
• Post-selection can also enable us to generate
  novel interactions (KLM proposal for quantum
  computing), and for instance to produce useful
  entangled states.
• POVMs, or generalized quantum measurements, are
  in some ways more powerful than textbook-style
  projectors
• Quantum process tomography may be useful for
  characterizing and "correcting" quantum systems
  (ensemble measurements).
• 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