Quantum Measurements:
some technical background
(AKA: the boring lecture)
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“Measurement postulate”
“Projection postulate”
The two aspects of measurement
Density matrices, environments, et cetera
von Neumann measurements
(entanglement and decoherence)
Slides, and some other useful links, to be posted:
http://www.physics.utoronto.ca/~steinberg/QMP.html
14 Oct 2003
The measurement postulate
...measurement outcomes...
“Collapse of the wavefunction”
Future measurements of A will of course agree with this ai
Measurement
State preparation
What are the effects of
measurement?
Suppose we have two pawns, one black and one white, and I put
one in each hand – we can write this state as something like
=
+
Obviously, if I open my left hand and measure the colour of
its pawn, I find either black or white, not both – from that point
on, I describe the left pawn as one colour eigenstate,
or
.
Is the other pawn still in a state of uncertain colour?
No – obviously, its state has also been “affected” by this measurement.
More refined version:
the “projection postulate”
When ai is found, the state
Of course, this is not normalized, so the final state is actually
Finding the left pawn to be black leaves the system in a
state where the right pawn is known to be white (unsurprisingly).
Two effects of measurement
1. One thing happens, as opposed to all other possibilities
2. Interference between the different possibilities becomes impossible.
100%
50%
det. 2
A
50%
What’s the state of the particles before the final beam-splitter?
det. 1
If no bomb was present, half the particles are in path A and
half are in path B.
B
If the bomb “measures” which path each is in... then still,
half are in path A and half are in path B.
The measurement didn’t change the probabilistic description
of the state... but without the bomb, interference caused all
particles to interfere at the final beam splitter and go left;
with the bomb, there is no such interference.
Measurement destroyed phase information, but left the
probabilities unchanged. |A + ei|B  |A “OR” |B .
How does the bomb cause the
other detector to fire?
The probability is given by the absolute square of this inner product,
which is 1/4 + 1/4 = 1/2 (because the orthogonality of "peace" and
"BOOM" cause the cross-terms to vanish).
Sneaky fact...
No one knows why one thing happens instead of many
simultaneous things... in fact, no one knows whether
this is true (cf. “relative-state”, “many-worlds”, “many-minds”
interpretations).
No “collapse” process has ever been observed – i.e., no case
where we would make the wrong predictions if we didn’t
assume collapse. Yet to make sense of probabilities, one
typically assumes that by the time you measure something,
it’s one thing or another. (But how do you know that when
I measured it, I wasn’t still in a probabilistic state? “Wigner’s friend.”)
We can try
(a) to understand what measurements do to coherence
and/or (b) to search for a real “collapse” process, supplementary
to quantum mechanics as we know it.
We need a formalism for this...
Note that in that interferometer,
|A  |det. 1 and |det. 2;
|B  |det. 1 and |det. 2;
but |A + |B  |det. 2 only
(because of interference).
The state "|A OR |B" might be |A (and get to det. 1 half the time)... or it
might be |B (and get to det. 1 half the time). It's not |A + |B, |A – |B, etc.
Any QM wave function you write down which is half A and
half B will exhibit some interference; no wave function can
describe the state after such a measurement.
Technical example: there is no spin-1/2 state with <SZ> = <SX> = 0.
"Pure states"
individual QM wave functions
"Mixed states"
probabilistic mixtures of QM states.
(e.g., results of measurements)
"Density Matrices"
Intro to density matrices...
Interpretation of matrix elements
Diagonal elements = probabilities
Off-diagonal elements = "coherences"
(provide info. about relative phase)
Connection to observables
And what about mixed states?
• The essential property of a statistical mixture is
that all expectation values are just the weighted
averages of those for the individual pure states.
• Our expression for expectation values is linear
in the density matrix – i.e., we can keep using
that expression with mixed states, if we define
the mixed-state density matrix itself as a
weighted average.
Density matrices for mixed states
Note: probabilities still 50/50, but no coherence.
What happens if you don't look at
part of your system?
When you calculate expectation values, you trace over the system.
If your operators depend only on a subsystem, then it makes no
difference whether you trace over other systems before or after:
Decoherence arises from throwing
away information
Taking this 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 
...
coherence
lost
There is still coherence between  and , but if the
environment is not part of your interferometer, you may
as well consider it to have "collapsed" to  or  .
This means there is no effective coherence if you look
only at the system.
Decoherence: the party line
When a particle interacts with a measurement device, the two
subsystems become entangled (no separable description).
Coherence is still present, but only in the entire system; if there is
enough information in the measurement device to tell which path
your subsystem followed, then it is impossible to observe interference
without looking at both parts of the system.
The effective density matrix of your system (traced over states of
the measuring apparatus) is that of a mixed state.
Coherence is never truly lost, as unitary evolution preserves the purity
of states. In principle, this measurement interaction is reversible.
In practice, once the system interacts with the "environment", i.e., anything
with too many degrees of freedom for us to handle, we cannot reverse it.
Just as in classical statistical mechanics, it is the approximation of an open
system which leads to effective irreversibility, and loss of information (increase
of entropy).
Loss of Information = Loss of Coherence
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 (entangled)
Hint=gApx
A
System-pointer
coupling
x
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).
Summary
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.