Quantum measurements
and quantum erasers
(AKA: no more dull than the last lecture?)
• von Neumann measurements
– (entanglement and decoherence)
• The Quantum Eraser
– Equivalence of collapse and correlation pictures
– EPR correlations
• An application of the collapse picture
– EPR correlations for nonlocal dispersion cancellation
(AKA: something to leave over for next week again...)
Slides, and some other useful links, are still being posted at:
http://www.physics.utoronto.ca/~steinberg/QMP.html
21 Oct 2003
Recap: decoherence arises from
throwing away information
Taking the trace over the environment retains only terms diagonal
in the environment variables – i.e., no cross-terms (coherences) remain
if they refer to different states of the environment.
(If there is any way – even in principle – to tell which of two
paths was followed, then no interference may occur.)


 
s when env is 
s when env is 
So, how does a system become
"entangled" with a measuring device?
• First, recall: Bohr – we must treat measurement classically
Wigner – why must we?
• von Neumann:there are two processes in QM: Unitary and Reduction.
He shows how all the effects of measurement we've described so far
may be explained without any reduction, or macroscopic devices.
• [Of course, this gets us a diagonal density matrix – classical
probabilities without coherence – but still can't tell us how
those probabilities turn into one occurrence or another.]
To measure some observable A, let a "meter" interact
with it, so the bigger A is, the more the pointer on the
meter moves.
P is the generator of translations, so this just means we
allow the system and meter to interact according to
Hint  A P.
An aside (more intuitive?)
Suppose instead of looking at the position of our pointer,
we used its velocity to take a reading.
In other words, let the particle exert a force on the pointer,
and have the force be proportional to A; then the pointer's
final velocity will be proportional to A too.
F=gA
U(x) = g A X
Hint = g A X
This works with any pair of conjugate variables.
In the standard case, Hint = g A Px , we can see
The pointer position evolves at a rate
proportional to <A>.
A von Neumann measurement
Initial State of System
A
Initial State of Pointer
x
Final state of both
Hint=gApx
A
System-pointer
coupling
x
A von Neumann measurement
Initial State of System
A
Initial State of Pointer
x
Final state of both (entangled)
Hint=gApx
A
System-pointer
coupling
x
Entangled (nonseparable) states
Consider the state resulting from this interaction with a pointer P:
If the different states Pi are orthogonal, no such product could
yield terms like 1P1 and 2P2 without yielding 1P2 etc. The
canonical example is the EPR spin state | - |.
IN OTHER WORDS: if you ask a question just about the system
on its own, there exists no quantum state vector which can fully
describe it.
Effectively, we have a mixed state, and need the density matrix
obtained by tracing over the pointer.
A von Neumann measurement
Initial State of System
Effective state of system
(if pointer ignored)
+
+
+
A
Initial State of Pointer
x
Hint=gApx
“OR”
System-pointer
coupling
“OR” “OR”
A
Unless the pointer is somehow included
in the interferometer, interference will
never again be observed between these
different peaks; we may as well suppose
a collapse has really occurred, and one peak
or another has been selected at random.
Back-Action
In other words, the measurement does not simply cause the
pointer position to evolve, while leaving the system alone.
The interaction entangles the two, and as we have seen, this
entanglement is the source of decoherence.
It is often also described as "back-action" of the measuring
device on the measured system. Unless Px, the momentum
of the pointer, is perfectly well-defined, then the interaction
Hamiltonian Hint = g A Px looks like an uncertain (noisy)
potential for the particle.
A high-resolution measurement needs a well-defined pointer position X.
This implies (by Heisenberg) that Px is not well-defined.
The more accurate the measurement, the greater the back-action.
Measuring A perturbs the variable conjugate to A "randomly"
(unless, that is, you pay attention to entanglement).
(For future thought: note that my entanglement argument
needed to assume that the pointer states were orthogonal.)
Summary so far...
We have no idea whether or not "collapse" really occurs.
Any time two systems interact and we discard information about
one of them, this can be thought of as a measurement, whether
or not either is macroscopic, & whether or not there is collapse.
The von Neumann interaction shows how the two systems become
entangled, and how this may look like random noise from the point
of view of the subsystem.
The "reduced density matrix" of an entangled subsystem appears
mixed, because the discarded parts of the system carry away
information. This is the origin of decoherence of the measured
subsystem.
Quantum Eraser
(Scully, Englert, Walther)
Suppose we perform a which-path measurement using a
microscopic pointer, z.B., a single photon deposited into
a cavity. Is this really irreversible, as Bohr would have all
measurements? Is it sufficient to destroy interference? Can
the information be “erased,” restoring interference?
Some mathematics...
A superposition state:
Probabilities:
Interference terms
Now consider a larger Hilbert space, including a Measuring Apparatus:
New probabilities:
NO INTERFERENCE!
But what if we select (project) out, not A, and not B, but an equal superposition?
INTERFERENCE RETURNS!
A microscopic measurement
i2
M1
SOURCE
det. 1
s1
BS
s2
i1
det. 2
M2
The "i" photons provide which-path information, and destroy the interference.
Can this information be "erased"?
If it's no longer possible to tell whether the photon came
from s1 or s2, then interference is restored!
But it is still possible...
(i1+i2)
+ (i1- i2)
= i1
i1+i2
i2
M1
SOURCE
s1
BS
s2
i1- i2
(i1+i2)
- (i1- i2)
= i2
i1
M2
In fact, this should have been obvious.
If combining the i photons at a beam-splitter could restore fringes
on the right, nothing would prevent me from combining them a year
after you looked at your detectors. Could I change whether or not you
had seen fringes ?!
UNITARY EVOLUTION CANNOT DESTROY INFORMATION!
ORTHOGONAL STATES REMAIN ORTHOGONAL FOR ALL TIME.
Obviously, nothing you do to the idlers can affect the signals.
Sorry, that was another lie.
Nothing unitary I do to the idlers affects the signals.
Measurement is not unitary – in other words, if I only keep some events and
throw out others, perhaps I can restore your interference.
Trigger on "i1+i2" events – no longer any way
to tell whether they were i1 or i2, no matter what!
i2
M1
SOURCE
det. 1
s1
BS
s2
i1
det. 2
M2
"i1+i2"
"i1- i2"
Together
Don't overlook the symmetry...
Detectors 1 and 2 are equally likely to fire, regardless of the phase setting.
When the "i1-i2" detector fires, this may tell me that detector 1 will fire
instead of detector 2.
Of course, have the time, the "i1+i2" detector fires, telling me that detector 2
will fire instead of detector 1.
...or is it that half the time, detector 1 fires, collapsing the "i" photon
into "i1-i2"...
...and that half the time, detector 2 fires, collapsing the "i"
into "i1+i2"...?
Which is the system and which is the measuring apparatus?
Making it look more complicated...
Ou, Wang, Zou, & Mandel,
Phys Rev A 41, 566 (1990).
Plus ça change...
What if you combine the idlers so
they've got nowhere else to go?
Zou, Wang, Mandel,
PRL 67, 318 (1991).
A polarisation-based quantum
eraser...
First, a familiar picture: the Hong-Ou-Mandel interferometer
t2 +r2 = 1/2 - 1/2 = 0;
no coincidence counts.
The polarisation quantum eraser
Half-wave plate
H
H
V
H
Polarizers (why 2?)
tt
rr
H
V
V
H
distinguishable;
no interference.
Interference going away...
And coming back again!
How complicated you have to make
it sound if you want to get it published
"Calculations are for those who don't trust their intuition."
Simple collapse picture
M1
SOURCE
HWP
signal
V
BS
idler
Suppose I detect a photon at
 here. This collapses my photon
into H cos  + V sin .
This means an amplitude of cos 
that the other photon was V, and
of sin  that it was H.
Being careful with reflection phase
shifts, this collapses the other output
port into V cos  - H sin , which of
course is just ( + p/2).
H
M2
Here I'm left with a photon 900 away from whatever
I detected. Now I just have linear optics to think about.
Of course I get sinusoidal variation as I rotate this polarizer.
"...and experiment is for those who don't trust their calculations."
Polarisation-dependence of rate
at centre of H-O-M dip...
But did I need to invoke collapse?
(and if so, which photon did the work?)
M1
SOURCE
HWP
1
signal
BS
idler
2
250
M2
Vs Hi
(V2 + i V1) (H1 + i H2)
= 1H 2V - 1V 2H + i [1H 1V + 2H 2V]
In coincidence, only see |HV> - |VH> .... that famous EPR-entangled state.
Of course we see nonlocal correlations between the polarisations.
These joint-detection probabilities can be calculated directly, without collapse;
add the amplitudes from HV and VH: P(1,2) = |cos(1)sin(2)-sin(1)cos(2)|2
= sin2(1 - 2).
This is the Bell-Inequality experiment done by Shih&Alley and Ou&Mandel.
Hong-Ou-Mandel Interference
as a Bell-state filter (Viennese delicacy)
r
r
+
t
t
r2+t2 = 0; total destructive interf. (if photons indistinguishable).
If the photons begin in a symmetric state, no coincidences.
{Exchange effect; cf. behaviour of fermions in analogous setup!}
The only antisymmetric state is the singlet state |HV> – |VH>, in
which each photon is unpolarized but the two are orthogonal.
Nothing else gets transmitted.
This interferometer is a "Bell-state filter," used
for quantum teleportation and other applications.
Some references
Quantum measurement theory;
the quantum eraser;
some early QE experiments.
Bell-inequality tests;
dispersion cancellation;
newer QEs (atom interferometry;
delayed choice).
Y.H. Kim et al., Phys. Rev. Lett. 84, 1 (2000)
T. Pfau et al., Phys. Rev. Lett. 73, 1223 (1994)