Stephen Parrott
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 Posted: Mon Nov 17, 2008 2:14 pm Post subject: Strange results of "weak" quantum measurement |
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There has been considerable interest in recent years in
the so-called theory of "weak" quantum measurements,
whose aim seems to be to measure the average value of
a quantum observable while negligibly disturbing the measured system.
The theory seems to have started in a paper provocatively titled
"How the Result of a Measurement of a Component of the Spin
of a Spin-1/2 Particle Can Turn Out to be 100",
by Aharonov, Albert, and Vaidman [Phys. Rev. Lett. 14 (1988),
1351-1354].
This paper was subsequently severely criticized by A.J. Leggett and
by A. Peres in Comments to PRL.
However, the theory initiated by Aharonov, et al.,
seems to have been somewhat accepted in the intervening years.
For example, a more recent paper reaches similar (but not
identical) conclusions in a more general context:
K.J. Resch and A.M. Steinberg, "Extracting Joint Weak Values with Local,
Single-Particle Measurements", PRL 92 (2004), 130402.
Recently several authors have claimed
to have actually measured weak values:
J. S. Lundeen and A. M. Steinberg, "Experimental joint weak
measurement on a photon pair as a probe of Hardy>s paradox,
arXiv:0812.4229vi
K. Yokota, et al., "Direct observation of Hardy>s paradox
by joint weak measurement with an entangled photon pair",
arXiv: 0811.1625.
These seem experimentally very clever, though I have doubts
that they are actually measuring what they believe.
[For example, Hardy>s paradox involves two particles
which can jointly have one of four locations in
an interferometer. The authors claim to resolve
the paradox by weakly measuring (-1) particle pairs
in one of the locations!
Normally, one would assume that the number of particles
in a given location would have to be a non-negative integer!
You can have 0, 1, 2, ... people in a room,
but what would it mean to have -1 person in the room?
Maybe the next person to enter the room is annihilated,
resulting in -1 + 1 = 0 people in the room? I suppose
it could be, but the authors have not observed that for
particles.]
Depending on the author, the theoretically "derived" weak value
of a quantum observable *X* when the initial state was *i*
and the final state *f* is either
<i|X|f>/<i|f>
(according to the above paper of Aharonov, et al.)
or the real part of that (according to Resch and Steinberg).
(It seems obvious that if either is correct,
it has to be Resch and Steinberg,
since average values of quantum observables have to be real!)
Note that the above expression is meaningless
(or infinite, according to taste) when *i* and *f* are orthogonal.
Can anyone tell me what is the standard interpretation of this?
That you get an arbitrarily large result when you weakly measure
the average value of X with *i* and *f* nearly orthogonal
and <i|X|f> nonzero??
Stephen Parrott |
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a student
Guest
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| Posted: Tue Nov 18, 2008 4:23 pm Post subject: Re: Strange results of "weak" quantum measurement |
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On Nov 18, 1:14 am, Stephen Parrott <postn...@email.toast.net> wrote:
[quote]There has been considerable interest in recent years in
the so-called theory of "weak" quantum measurements,
whose aim seems to be to measure the average value of
a quantum observable while negligibly disturbing the measured system.
The theory seems to have started in a paper provocatively titled
"How the Result of a Measurement of a Component of the Spin
of a Spin-1/2 Particle Can Turn Out to be 100",
by Aharonov, Albert, and Vaidman [Phys. Rev. Lett. 14 (1988),
1351-1354].
This paper was subsequently severely criticized by A.J. Leggett and
by A. Peres in Comments to PRL.
However, the theory initiated by Aharonov, et al.,
seems to have been somewhat accepted in the intervening years.
For example, a more recent paper reaches similar (but not
identical) conclusions in a more general context:
K.J. Resch and A.M. Steinberg, "Extracting Joint Weak Values with Local,
Single-Particle Measurements", PRL 92 (2004), 130402.
Recently several authors have claimed
to have actually measured weak values:
J. S. Lundeen and A. M. Steinberg, "Experimental joint weak
measurement on a photon pair as a probe of Hardy>s paradox,
arXiv:0812.4229vi
K. Yokota, et al., "Direct observation of Hardy>s paradox
by joint weak measurement with an entangled photon pair",
arXiv: 0811.1625.
These seem experimentally very clever, though I have doubts
that they are actually measuring what they believe.
[For example, Hardy>s paradox involves two particles
which can jointly have one of four locations in
an interferometer. The authors claim to resolve
the paradox by weakly measuring (-1) particle pairs
in one of the locations!
Normally, one would assume that the number of particles
in a given location would have to be a non-negative integer!
You can have 0, 1, 2, ... people in a room,
but what would it mean to have -1 person in the room?
Maybe the next person to enter the room is annihilated,
resulting in -1 + 1 = 0 people in the room? I suppose
it could be, but the authors have not observed that for
particles.]
[/quote]
Nice discussion, and some good points!
[quote] Depending on the author, the theoretically "derived" weak value
of a quantum observable *X* when the initial state was *i*
and the final state *f* is either
<i|X|f>/<i|f
(according to the above paper of Aharonov, et al.)
or the real part of that (according to Resch and Steinberg).
(It seems obvious that if either is correct,
it has to be Resch and Steinberg,
since average values of quantum observables have to be real!)
[/quote]
Yes, you are right - if the system is weakly coupled to an "A-meter"
with a broad initial pointer distribution, and later measured to be
in state |f> (eg, via a strong measurement of some observable F
for which |f> is an eigenstate), then the average of the readouts of
the A-meter is the REAL part of the above quantity. It is not
clear to me why Aharonov et al consider the whole complex
number.
It is interesting, however, that the imaginary part may also be
given an interpretation, as I note further below.
[quote] Note that the above expression is meaningless
(or infinite, according to taste) when *i* and *f* are orthogonal.
Can anyone tell me what is the standard interpretation of this?
That you get an arbitrarily large result when you weakly measure
the average value of X with *i* and *f* nearly orthogonal
and <i|X|f> nonzero??
[/quote]
I think that the standard interpretation might be that this occurs
with probability zero. Physically, one can imagine that a very
weak coupling to an A-meter is simply unable to change the
initial state |i> to an orthogonal state |f>.
Your point is a good one even so, if one tries
to interpret the mathematical weak value as somehow "real" -
as Aharonov et al certainly try to. In fact, I think you have
come up with a real "weakness" of their interpretation! :)
However, there is also a different interpretation of weak
values, unrelated to weak couplings, where there is no problem
for orthogonal initial and final states - see the eprint
http://arxiv.org/abs/quant-ph/0308137 by Johansen (also
published in Phys. Lett. A 322 (2004) 298). In this approach
one asks: if I know the system is described by state |i>, and
I make a measurement of F to obtain result f (corresponding
to eigenstate |f>), then what is the best estimate I can make
of A, on the basis of this information?
It turns out that
1. The best estimate, for a natural measure of error in the
estimate, is given by the real part of the weak value, and
2. The square of the error in the estimate is given by the
imaginary part of the weak value.
3. The probability that |i> and |f> are orthogonal is zero (they
would both have to be eigenstates of F, but then the
measurement of F would have to return the value associated
with eigenstate |i>, i.e., they would be the same state). |
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