Unitary evolution disobeyed by null measurement?
Neil B.
true measurement occurs. The proposed example is where one orthogonal
state of a co-evolving pair (like, |x> and |y> polarized light together
in a system) is altered by "null measurement", i.e. a possible
interaction failed to occur (AKA Renninger negative result.) A NM isn't
a "true" measurement since we don't get a detection or collapse into a
final state. An example of a NM is the effect of barriers that change
interference, as explained below.
First, "unitary evolution", as defined at
http://www.iscid.org/encyclopedia/Unitary_Evolution:
> A superposition of states always evolves in the same way as each
> of the two states would evolve individually.
> A superpositional state - with w and z
> as complex numbers - can be notated as:
> w x (alternative A) + z x (alternative B).
Maybe you heard of NM effects through the phrase "quantum seeing in the
dark." NMs can alter statistical output (such as interference patterns)
despite the lack of a detection event or conventional interaction
(absorption, reflection, etc.) Here's a good explanation of the issue
and the device used:
http://en.wikipedia.org/wiki/Mach-Zehnder_interferometer
Skip what's familiar. Simple example: send photons, one at a time,
through a tuned Mach-Zehnder interferometer. We get hits (detection
events) only in the A-channel, because first beamsplitter split the wave
function of the photon in two parts. Those portions interfere with each
other when they recombine at the second Bs, making the equivalent of
bright (output A-channel) and dark (output B-channel) fringes. But now,
put a black barrier in the lower leg LL of the MZ, some distance away
from the first Bs. The barrier causes detector "hits" in the previously
dark B-channel. Of course, these detections occur when the barrier
wasn't
hit! If the barrier absorbs the photon, the photon is localized
("collapsed") - it's gone and can't be anywhere else. The barrier
didn't
reflect the wave either, since it's dark. The explanation (so far as
any is possible in the weird world of QM!) is that the barrier forced
the disappearance of the photon's WF in LL, and "reallocated" it all
into the upper leg UL. (It must all be there, since missing the barrier
now makes probability of detection 100% in that leg.) (Note: a mirror
in the way would also disrupt the interference pattern, but wouldn't
reallocate the amplitude of the WF. The mirror would just redirect the
other portion away from the second Bs. This distinction matters to the
argument below.)
OK, what's the bearing on unitary evolution? The novelty is
polarization. We input diagonal polarized light; a superposition of the
|x> and |y> states. Write as, w|x> + z|y>. In this case, w = z = (sqrt
2)/2 ~ 0.707. After being split, intensities (ensemble equivalent, note
I = amplitude squared) are 0.5 for each beam. We label amplitudes as
"one" for LL and 'two" for upper, and now new w1 = w2 = z1 = z2 = 0.5.
The chances for finding as x or y in the LL, or for x or y in the UL,
are 25% each. Interference works the same as before. But replace the
"dark barrier" with a polarizing filter P that absorbs LP light along
orientation y. To the |x> state, the filter is like clear glass. To
the |y> state, P is a dark barrier. P has a 25% chance of absorbing the
photon (1/2 x 1/2), but consider what happens if it doesn't. The |x>
portion, if acting alone, should be unaffected by what happens to |y>.
That's the very essence of "unitary evolution." The |y> portion, if
acting alone, would be reallocated to double intensity around the top
leg, and thus sqrt(2) times more amplitude. So UE predicts that after
the lower wave encounters P, the new values are:
w1� = w2� = 0.5
z1� = 0, z2� = 0.707
The total remaining probability (0.5^2 + 0.5^2 + 0.707^2) still adds to
one as it must, since the photon remains unabsorbed. This follows the
explanation of UE as given above: "A superposition of states always
evolves in the same way as each of the two states would evolve
individually." Of course, the "absolute" amplitudes aren't the same as
if for separate cases; they can't be. It's the relative behavior that
is supposed to stay the same. So, we multiply the amplitudes for each
independent evolution by a factor (such as 0.707 in this case) to get
the values for the combination.
However, notice something odd here: the relative amplitudes of |x> and
|y> have changed. Final polarization of a beam depends on the way |x>
and |y> go together, so in this case the polarization would have to
"suddenly" change from 45� to arctan (0.707/0.5) = 54.7�. This isn't
just an abstraction or "interpretation" issue, it predicts real
consequences we can measure. We could measure the new polarization
angle with filters, and the statistics would *suddenly* change. Not
only that, we could start getting hits in a previously impassable filter
(135�, the perp. diagonal to the incoming.) This has problems such as
allowing FTL communication: if a "machine-gun" train of photons is used,
pulling the barrier in and out must immediately affect the polarization
of the split wave at any distance. It also violates the classical wave
equivalent, which says that the filter in LL has no effect on what we
find in the upper leg. The polarization in the UL should stay at 45�.
For that to happen, we need to "reallocate" not just the |y> state, but
the |x> state as well. IOW, an effect on one state "contaminated" the
other one and forced it to change for reasons of consistency - all this,
before a genuine detection happened. We can visualize as follows: in
the absence of absorption at P, the remaining chances go from all 25%
each, to 1/3 each: for finding x in LL, x in UL, or y in UL (because
with no absorption, the remaining chance of detection still adds to
one.) The amplitudes are the sqrts. of the chances. So, instead of the
way predicted by UE, the actual amplitude changes must be:
w1� = w2� = 0.577
z1� = 0, z2� = 0.577
Compare to the definition of UE above. Now, we find that the condition
of "(alternative A)" and its relation to (alternative B) was modified by
the evolution of (alternative B). Yes, w1� has the same proportion to
w2� as before. Yet we couldn't continue to simply use the separate
state evolutions times a constant multiplier, and z2� - z1� is different
too. Note we can also use cases where P doesn't absorbs all of a linear
state, etc. Hence, this result violates UE in the strict sense: the
failure to be absorbed at P wasn't really a "measurement" - it left the
photon still without a "final resting place" or
thermodynamically-irreversible detector "reading" etc. IOW, it isn't
even as much as a genuine null "reading" from a real detector - P is
just a filter, not attached to any macroscopic observation process. We
would have to so broadening the range of what brings UE to an end, it
would perhaps fatally weaken the postulate. Almost any potential
interaction could count as a null "measurement" which forced a
UE-breaking reallocation of multiple states. The UE postulate needs
revision.
Can any readers do this experiment? Does it fit into your research,
instrumental or theoretical?
Has something similar been done already? Please let me know.
Just use the NG, Google, or contact me at:
neil_delver[at][high temperature]mail dot com.
Thanks.
Edward Green
I haven't followed you in detail, but all quantum weirdness ... well,
at least all possibly excepting Bell like results ... can be resolved
into the idea that quantum systems act _exactly_ like a wave until
there is a detection, whereupon the wave is implicitly decomposed onto
basis states, giving us the probability of measurement. I suspect
your "null measurement" falls pretty squarely under this rubric. To
put it another way, a polarizing filter is a measurement, even when a
photon passes through unmolested.
Neil B.
"Edward Green" <spamsp...@netzero.com> wrote in message
news:b9cdf7ba-2b35-42cf...@g23g2000vbr.googlegroups.com...
Your point has some merit "as far as it goes" but isn't really the core
issue. Unitary evolution says that if the states continue to evolve,
their respective evolutions continue *independently.* So regardless of
what may have happened to them meanwhile, pseudo-measurement or not,
what happens to the wave state |y> isn't supposed to meddle in the
evolution of state |x>.
Also, the |y> state doesn't pass through the P filter anyway, only the
|x> state does - there is no literal interaction (internal effect even
in principle) between |y> and the filter. And if passing through a
filter or etc. was a "measurement" because of preferential chance of
interaction, then it's a "measurement" when a photon reflects off a
colored surface etc. I mean, the distinction between a measurment and
mere "interaction" would become nearly pointless. (Interactions are not
measurements per se, unless an "irreversible" result is obtained. REM
that a photon in an interferometer is bouncing off mirrors, being split
by thin films, etc. There must be a special interaction which
irreversibly takes the wave out of circulation, in effect. Without a
"true measurement" the WF is still available to be split again into more
portions at a Bs, etc. We could e.g. split the photon waves in my
proposal yet more, have more intereference arrangments later, etc.
But regardless of the semantics of what to include as a "measurement",
unitarity means that any remaining WFs of the states should be
independent. I gave two formulas for predicted amplitudes and
polarization states: the first, as theorized by UE (I'm rather sure on
that); and the second, as I predict per avoiding FTL communication and
in complementarity with classical optics. We can test for who's right.
Anyone care to make a bet?
Neil B.
"Neil B." <neil_...@caloricmail.com> wrote in message
news:ktidnXxVLa-3OP7X...@posted.widowmaker...
>
> "Edward Green" <spamsp...@netzero.com> wrote in message
> news:b9cdf7ba-2b35-42cf...@g23g2000vbr.googlegroups.com...
> On Jul 19, 2:41 pm, "Neil B." <neil_del...@caloricmail.com> wrote:
>
>
> But regardless of the semantics of what to include as a "measurement",
> unitarity means that any remaining WFs of the states should be
> independent. I gave two formulas for predicted amplitudes and
> polarization states: the first, as theorized by UE (I'm rather sure on
> that); and the second, as I predict per avoiding FTL communication and
> in complementarity with classical optics. We can test for who's right.
> Anyone care to make a bet?
>
WFs should be independent of each other. Unitarity merely means, the
total probability adds to one. In this case, it does anyway regardless
of whether unitary *evolution* is upheld or not. The bottom line: if UE
is true, the expectation value for the polarization angle in the upper
leg is different than if we demand consistency with classical optics
etc., and this can be tested.
Neil B.
>I argue we can disrupt unitary evolution in certain cases, even before
>a
> true measurement occurs. The proposed example is where one orthogonal
> state of a co-evolving pair (like, |x> and |y> polarized light
> together
> in a system) is altered by "null measurement", i.e. a possible
> interaction failed to occur (AKA Renninger negative result.) A NM
> isn't
> a "true" measurement since we don't get a detection or collapse into a
> final state. An example of a NM is the effect of barriers that change
> interference, as explained below.
>
> First, "unitary evolution", as defined at
> http://www.iscid.org/encyclopedia/Unitary_Evolution:
>
>> A superposition of states always evolves in the same way as each
>> of the two states would evolve individually.
>> A superpositional state - with w and z
>> as complex numbers - can be notated as:
>> w x (alternative A) + z x (alternative B).
>
> Can any readers do this experiment? Does it fit into your research,
> instrumental or theoretical?
> Has something similar been done already? Please let me know.
> Just use the NG, Google, or contact me at:
> neil_delver[at][high temperature]mail dot com.
>
> Thanks.
Actually, I think there's a simpler and possibly better example, which I
thought of years ago but just didn't appreciate or promote well enough.
It might count as a "measurement" so I'm not sure if it violates UE,
that's for discussion. If UE is true, the evolution of the combined
states should be a linear superposition of the individual cases. That
should presumably include any interaction which does not "collapse" the
particle into a final state, but in this case it's debatable whether we
did that.
Consider first the case of a diagonal (45�) LP photon approaching a LP
filter which fully passes that orientation. So, it will go through at
nearly 100% chance. We can write the incoming amplitude as "one" and the
exit amplitude as "one." Then consider components that would superpose
to make that prior state: the separate |x> and |y> basis states. So,
imagine an |x> photon passing the diagonal filter. It has a 50% chance
of passing (amplitude A = cos theta = cos 45� , chance = relative exit
intensity I = cos^2 theta.) The same values apply for a |x> photon. If
either were by itself, we could write the amplitudes as "one" being
attenuated to ~0.707.
A diagonal photon can be considered a superposition:
|45�> = sqrt(2)/2 ( |x> + |y> )
So if we renormalize, and if UE is true, we should find that in the
cases where the photon was not absorbed, the before and after go like
this:
0.707 |x> ~~~> 0.5 |x>
0.707 |y> ~~~> 0.5 |y>
Clearly, the before/after proportions don't match the individual cases.
This brings up Mr. Green's objection even more vividly, but I still am
not convinced that saves us from unitary evolution failing here. There
is still an "evolving wave function" and not a "hit" on a detector.
OTOH, some say that the projection postulate says that once we measure a
particle it can continue in the new state fixed by the "measuring
device". OK, debate that, but I wanted to put up a simpler example. The
original one I OPed is better because there isn't ever an actual
measurement, nothing is altered or collapsed or "projected" into any
different state.
Neil B.
>
> "Neil B." <neil_...@caloricmail.com> wrote in message
> news:Otmdna6pnpTK9P7X...@posted.widowmaker...
>> I argue we can disrupt unitary evolution in certain cases, even
>> before a true measurement occurs. The proposed example is where
>> one orthogonal state of a co-evolving pair (like, |x> and |y>
>> polarized light together in a system) is altered by "null
>> measurement", i.e. a possible interaction failed to occur
>> a "true" measurement since we don't get a detection or collapse
>> into a final state. An example of a NM is the effect of barriers
>> that change interference, as explained below.
>>
>> First, "unitary evolution", as defined at
>> http://www.iscid.org/encyclopedia/Unitary_Evolution:
>>
>>> A superposition of states always evolves in the same way as each
>>> of the two states would evolve individually.
>>> A superpositional state - with w and z
>>> as complex numbers - can be notated as:
>>> w x (alternative A) + z x (alternative B).
>>
> <snip>
>> Can any readers do this experiment? Does it fit into your research,
>> instrumental or theoretical?
>> Has something similar been done already? Please let me know.
>> Just use the NG, Google, or contact me at:
>> neil_delver[at][high temperature]mail dot com.
>>
>> Thanks.
>
> Actually, I think there's a simpler and possibly better example, which
> I thought of years ago but just didn't appreciate or promote well
> enough. It might count as a "measurement" so I'm not sure if it
> violates UE, that's for discussion. If UE is true, the evolution of
> the combined states should be a linear superposition of the individual
ironically, if you vector-add the |x> + |y> evolutions, you do get the
correct vectors for the diagonal |/> state (because the |x> + |y>
themselves add to sqrt(2) times their original amplitude, but the
outputs were both in the same direction and make for double the output.
But it is still a problem that the "amplitudes" of the outcome "absorbed
by the filter" do not add up: there are 50% each amplitudes of
absorption for |x> and |y>, but no amplitude of absorption for the |/>
state even though it is a linear superposition of the former LP bases.
Neil B.
"Edward Green" <spamsp...@netzero.com> wrote in message
news:b9cdf7ba-2b35-42cf...@g23g2000vbr.googlegroups.com...
I have a better example of the paradox, in which there isn't even that
kind of pseudo-measurement. Instead of a polarizing filter in one leg,
make the first Bs a polarizing beam splitter. That really isn't a
"measuring device" at all, because it does not "project" the state into
the measured eigenstate or base. IOW, an x-LP filter will convert
(collapse or project) incoming e.g. 20� photon into the x-state. You
lose the ability to find out that photon started as a 20� photon, it is
unrecoverable. But a PBS simply separates the base states into different
directions. We can recombine and get back the original LP angle. So we
use the PBS, and then we make a second split of the lower leg (holding
|y> to face a black barrier. If the barrier isn't hit, an independent
|y> beam would have to increase its amplitude, leading to similar
problems as in the original example at top.
I'm still worried though, about the fact that there really is an
amplitude for "absorption" and we're ignoring it by saying, "in the
cases it doesn't occur ...." But even then, that subset should still be
governed by linear superposition - but it isn't.
I'm thinking, maybe to try circular pol entering a QWP. Say RH enters a
QWP with fast |y> axis. Hence it comes out as 45� linear (QWP in front
of LP filter is how most "CP filters" are built. Note, they don't do the
same job when reversed!) Note that we have lost angular momentum from
the outside world, and it must (on average) build up in the QWP as
transits accumulate (but no absorptions: wave plates are not filters.
That is also why they aren't "measuring" the photon - they don't destroy
the original information by "projection" like filters do.) But each
individual basis |x> or |y> lack angular momentum by themselves. I'm not
sure if that violates UE, but the generator of the original RH light
would be changing its AM, whereas individual generation of LP would not.
Hence the mechanical vectors of the constituent "evolutions" (from
long-term addition a la Beth experiment) would not be consistent. All
this stuff is rather tricky.