Quantum Mysteries Disentangled
Ron Garret
28 November 2001
Revised (slightly) August 2008
The most incomprehensible thing about the Universe is that it is
comprehensible.
Albert Einstein
We now know that the moon is demonstrably not there when
nobody looks.
N. David Mermin
… nobody understands quantum mechanics.
Richard P. Feynman
ABSTRACT
This paper attempts to dispel some of the “esential mystery” of quantum
mechanics (QM) by describing some recent (as of 201) results in quantum
information theory at a level acesible to the layman. The discusion is
motivated by first showing how informal acounts of QM’s mysteries (specifically,
entanglement and quantum erasers) lead to a contradiction of relativity. The
apparent contradiction is resolved with an elementary mathematical analysis.
Finally, I engage in wild philosophical speculation in order to allay fears that a
better understanding of QM runs the risk of taking all of the fun out of it.
1. The Magic Show
Quantum Mechanics (QM) is an enduring source of entertainingly intractable
philosophical puzles. After nearly a hundred years of pondering, the reality of
QM seems more and more like a magic trick that stubornly resists all atempts at
common-sense explanation.
At some level there is real magic in QM that will endure all atempts to
deconstruct it. But, like all good tricks, Q relies to some extent on sleight of
hand and misdirection. The pater used most often when talking about QM tels a
story about a mathematical theory that predicts with astounding acuracy the
outcomes of measurements made on particles. The magic arises because the
structure of the theory describes a world where (aparently) physical entities
literally do not have physical properties until those properties are measured.
The sleight-of-hand is that the term "measurement" is never defined. Of course,
this is not news. The fact that easureent is such a crucial part of the theory
but is nowhere reflected in the mathematics has long been the cause of varying
degres of uneasines. Einstein, of course, was the most uneasy of all, demanding
through the EPR "paradox" to see what was in the magician''s other hand [1]. John
Bel, in a stroke of uncomon genius, figured out how to open the magician''s
hand to show that it was, in fact, empty [2]. The hidden variables had truly
disappeared. The QM magician was vindicated, the mysteries endure, and the
philosophical arguments over such things as whether cats qualify as conscious
observers endure along with them. But because philosophical dilemmas do not
chalenge the scientific standing of the theory, and because maters as they stand
are the source of so much good clean fun, the world has been largely content to
obey the admonition to pay no attention to the man behind the curtain.
Unfortunately, it turns out that the story of QM has a fatal flaw. Not QM itself,
mind you, but the story, the pater that goes along with the theory. In particular,
the idea of measurement as described in the QM story leads directly to a physical
imposibility, specificaly faster-than-light comunication. To see this we have to
begin by reviewing the QM story.
2. A Gallery of Mysteries
2.1 The two-slit experiment
The grandmother of all quantum mysteries is the two-slit experiment. A beam of
monochroatic light, for exaple from a laser, shines upon a screen. Betwen
the scren and the light source is a barier in which two narrow slits have been
cut out. What apears on the scren is an interference patern, showing multiple
fringes of destructive and constructive interference, demonstrating the wave-like
nature of light.
If we examine these fringes closely we find, of course, that the light is made of
particles, photons. The particle-like nature of light becomes particularly evident
when the intensity of the beam is very low, in which case it is posible to observe
individual photons striking the scren. The interference pattern manifests itself
only in the cumulative effect of many photons striking the screen over time.
Quantum mystery manifests itself when we ask the question: which slit did a
photon pass through on its way to the scren. We find that any physical change
to the experimental setup that would allow us, even in principle, to determine the
photon''s route ends up changing the photon''s behavior: instead of an
interference pattern we now observe just two bright spots of light, one
corresponding to each slit.
This turns out, apparently, to be a fundamental feature of quantum mechanics.
We can repeat the experiment with different kinds of particles (e.g. electrons
instead of photons) and diferent methods of generating two paths for the
particles to folow (e.g. a Stern-Gerlach apparatus instead of two slits, or a Mach-
Zender interferometer) and the result is always the same: if there is no way to
determine which path the particle took, there is interference. If there is a way to
determine the path, the interference disapears. It doesn''t mater if the decision
to measure the particle''s path is made after the particle has already pased the
slits. It doesn''t matter if the way of determining the particle''s path involves direct
interaction with the particle or not. For example, John Gribbin writes:
... we only ned to lok at one of the two [slits] to change the
pattern appropriate to particles on the scren. Somehow, the
electrons going through the second [slit] ''know'' that we are looking
at the [first slit] and also behave like particles as a result.
One significant feature of the patter that acompanies the quantum magic show is
talking about particles mysteriously "knowing" what is going on somewhere else
in the world. We will see this trick again (with even more dramatic results) when
we take a closer look at the EPR paradox.
2.2 Quantum erasers
We don''t actualy have to make a measurement (whatever that means) in order to
make particles stop behaving like waves and start behaving like particles. We
only have to introduce some change that makes it posible in principle to
determine which slit a particular photon pased through on the way to the screen
and we will destroy the interference patern exactly as if we had actually
measured the particle''s position.
For example, if we use light that is polarized in a particular direction and put a
polarization rotator at one of the two slits then the interference patern will go
away as if we had actualy measured the position of the photon. This is because
the polarization rotator makes it posible in principle to determine which slit the
photon has gone through by measuring the photon''s polarization.
But this subtle "proto-measurement" is diferent from a "real" measurement
because it is reversible. The "information" about which slit the photon went
through can be "erased" by introducing a polarizing filter in front of the scren
oriented at 45 degres to the original polarization axis. The photons that pass
through this filter will all be polarized in the same direction, so it is no longer
possible to tel from which slit they came. Lo and behold the interference is
(mysteriously, of course) restored!
2.3 EPR pairs
If the two-slit experiment is the grandmother of all quantum mysteries then surely
the EPR paradox is the grandfather. It is posible to produce so-caled "entangled
pairs" of photons that have the property that measurements performed on both
photons are always perfectly corelated (or anti-corelated). Here''s how the
situation was described in a 1992 Scientific American article:
Spoky correlations betwen separate photons were demonstrated
in an experiment at the Royal Signals and Radar Establishment in
England. In this simplified depiction, a down-converter sends pairs
of photons in opposite directions. Each photon pases through a
separate two-slit aparatus and is directed by mirrors to a detector.
Because the detectors canot distinguish which slit a photon passes
through each photon goes both ways generating an interference
patern.. Yet each photon''s momentum is also corelated with its
partner''s. A measurement showing a photon going through the
uper left slit would instantaneously force its distant partner to go
through the lower slit on the right.
Mysterious indeed! And the clincher is that it isn''t magic, it''s physics. This is
really the way the world is.
Except that it isn''t. Wel, sort of. It isn''t a lie exactly. Quantum mechanics really
is the way the world is, but it''s not as mysterious as it''s been made out to be. To
see this we first have to see how the story as presented so far is internally
inconsistent.
3. Smoke and mirrors
Let''s review the essential elements of the story so far.
1. A two-slit experiment produces interference.
2. Any modification to the two-slit experiment that allows us to determine even
in principle which slit a particle went through (a which-way measurement)
destroys the interference.
3. Some modifications that might alow the position of the particle to be
deterined and thus destroy the interference can be "undone" or "erased" and
restore the interference pattern.
4. An EPR experiment consists of a pair of two-slit experiments. The outcome of a
measurement made on one side is always perfectly (anti)corelated to the
outcome of the same measurement on the other side.
To this we add a simple version of the Heisenberg uncertainty principle:
5. It is not possible to simultaneously know the position and velocity of a particle.
Now here is why all these things canot possibly be true. Consider one side of an
EPR experiment. It is a two-slit experiment, so there is interference (story element
1). Imagine that we perform a which-way measurement on that side of the
experiment, thus destroying the interference on that side (story element 2). What
hapens to the interference on the other side? Does it disapear or does it
remain?
It turns out that either posibility leads to a contradiction. If the interference
remains then we have a situation where we know which way the particle went
(because of story element 4 and the fact that we know which way its EPR partner
went) but we have interference nonetheless, which contradicts story element 2.
On the other hand, if the interference disappears then we can use this
phenomenon to do faster-than-light signaling. Recal the quote from Scientific
American:
A measurement showing a photon going through the uper left slit
would instantaneously force its distant partner to go through the
lower slit on the right. [Emphasis added.]
Because the efect is instantaneous and the two sides of the experiment can be
separated by an arbitrary distance the result would be a faster-than-light
comunications channel. Note that this is more than just spooky-action-at-a-
distance (which realy does ocur). In this case performing a volitional action
(choosing to take a measurement or not) on one side of the aparatus causes an
instantaneous observable change (presence or absence of interference) on the
other side. We could use this phenomenon to transmit classical information faster
than light, which would violate relativity.
There is another posibility: there might not have ben any interference to begin
with. It might be that having an EPR partner "counts" as a modification to the
experiment that alows us to determine the path of the photon in principle. But
this can''t be right either for two reasons. First, we know that in a standard two-
slit experiment it is not possible to determine which slit the photon passed
through (because we see interference). If it is not posible to determine the path
of the photon in one two-slit experiment then by symmetry it canot be posible
to determine the path of a photon in a second, identical two-slit experiment.
Actualy we are on somewhat shaky ground with this argument. It could be the
case that it is not posible to determine the path of the photon on either side
individualy, but it might stil be posible by some mathematical magic to
reconstruct the paths of the photons by combining information from both sides of
the experiment. I will return to this posibility later. But for now let us simply
supose that there is no interference. Fine. We can stil produce faster-than-light
comunication by creating interference instead of destroying it. How? By simply
measuring the velocity of one of the particles! By the Heisenberg uncertainty
principle if we know the velocity we canot know the position, even in principle.
A velocity measurement is a "quantum eraser" that eliminates whatever subtle
proto-measurement there might have been in the EPR pair and restores the
interference that (we are presuming) was destroyed by the entanglement. Now we
are asured that we canot know the position of either particle even in principle,
so we must have created interference where before there was none. Again we
have a way to transmit information faster than light.
This is a very strong argument for the possibility of superluminal communication.
The argument is in fact corect! But Einstein is safe because one of our premises
is false; the story we have ben told about quantum mechanics is wrong. This is
not to say that quantum mechanics is wrong, just the comonly told story about
it.
4. The man behind the curtain
Here be equations with funny Greek symbols. Don’t panic.
4.1 Entanglement
When faced with an apparent paradox in quantum mechanics it is usualy best to
go back to the mathematics and see what the theory actualy says would happen.
Let us begin with the siple two-slit experiment. The mathematical description of
the state of a photon in this experiment is:
(Ψ
U
+ Ψ
L
)/√2
where Ψ
U
represents the state of the photon in the uper slit and Ψ
L
represents
the state of the photon in the lower slit. The probability density is the squared
modulus of this quantity:
[|Ψ
U
|
2
+ |Ψ
L
|
2
+ (Ψ
U
Ψ
L
+ Ψ
L
Ψ
U
)]/2
The term (Ψ
U
Ψ
L
+ Ψ
L
Ψ
U
) is the mathematical manifestation of interference.
If you didn''t folow that it doesn''t really matter. The details of the mathematics
are not important. Only the overall structure of the equations matters, except for
one small detail: the √2 term, which is there to make the overal probability come
out to be 1. This will turn out to be important shortly.
Now let us add a detector at the slits to determine which way the photon went. To
describe this situation mathematically we have to add a description of the state of
the detector:
(Ψ
U
|D
U
> + Ψ
L
|D
L
>)/√2
where |D
U
> is the state of the detector when it has detected a photon at the upper
slit and |D
L
> is the state of the detector when it has detected a photon at the lower
slit. Now the probability density is:
[|Ψ
U
|^2 + |Ψ
L
|^2 + (Ψ
U
Ψ
L
U
|D
L
> + Ψ
L
Ψ
U
L
|D
U
>)]/2
This also has an interference term as before, but with the adition of
U
|D
L
> and
L
|D
U
> terms. These terms represent the amplitude of the detector to
spontaneously change from one of its two states to the other. If the detector is
working properly then these amplitudes are zero and the interference term
vanishes. This is the mathematical manifestation of the informal statement that if
information is available about the path of the photon then the interference
disappears.
Now consider the description of an EPR-entangled pair of photons:
(|↑↓> + |↓↑>)/√2
At first glance this looks very much like the single-photon case, except that where
before we had Ψ
U
and Ψ
L
we now have |↑↓> and |↓↑>, representing respectively
photon 1 being in the uper slit and photon 2 being in the lower slit and vice
versa. But this distinction is crucial because it turns out that there is some
notational sleight-of-hand going on here. First, |↑↓> is shorthand for |↑>|↓>.
Second, the arrow symbols have no semantic significance; they are just compact
mnemonic identifiers. We could just as well have written |UL> and |LU> (which of
course is shorthand for |U>|L> and |L>|U>) as |↑↓> and |↓↑>. Finaly, Ψ
U
is just
another way of writing |U>. So if we employ alternative notation we get the
following description of two entangled photons:
(Ψ
U
|U> + Ψ
L
|L>)/√2
which is precisely the same as the description of the single photon with a position
detector. Mathematicaly, measurement and entanglement look identical, so we
have the first half of an answer to our superluminal counications puzzle: there
is no interference to begin with because the entanglement destroys the
interference in exactly the same way (acording to the mathematics) that
measurement does.
But this still leaves open the possibility that we can aply a quantum eraser to
"undo" the measurement-like efects that entanglement has, restore interference,
and salvage superluminal communication and our Nobel Prize.
4.2 Quantum erasers
Let us take a closer look at how a quantum eraser is suposed to work. We start
with a classic two-slit experiment, but we use light that is initially polarized in one
direction, say vertical. At one of the slits we place a polarization rotator so that
any photon that pases through that slit becomes polarized in the horizontal
direction. The net efect of this change acording to the classic quantum
mechanical story is that it is now posible to determine in principle which slit the
photon passed through and the interference goes away. And indeed it does.
Now we place a polarizing filter oriented at 45 degres in front of the screen. The
photons that pas through this filter all have their polarizations oriented in the
same direction, the which-way information is lost, and interference is restored. It
certainly sems as if the measurement-like efects of the polarization rotator
(destruction of interference) have been undone by the filter.
Once again, let''s lok at the math. Let us supose that the state of the photon is
initialy polarized in the vertical (V) direction, and the polarization rotator is on
the uper slit. Then the state of the photon after passing through the slits and
the polarization rotator is:
(|UH> + |LV>)/√2
that is, the photon has either gone through the uper slit and is now horizontally
polarized, or it has gone through the lower slit and is now verticaly polarized.
This formula has the same form as the entangled/measured and therefore non-
interfering photons above. The photon is entangled with itself — one state
(position) has become entangled with a different orthogonal state (polarization).
Now we "erase" this entanglement by placing a 45 degre polarization filter in
front of the screen. After passing through this filter the state of the photon is:
(|U> + |L>)(|H> + |V>)/2√2
The (|H> + |V>) term denotes the fact that this photon is polarized at 45 degrees,
i.e. it is in a superposition of the horizontaly polarized |H> state and the
verticaly polarized |V> state. The (|U> + |L>) term denotes the fact that the
position of the photon is in a superposition of uper-slit and lower-slit states. The
position of the photon is now unentangled/unmeasured (so is the polarization) so
there is interference.
Now, notice the 2√2 term in the denominator. It''s there to make the total
probability come out to be 1. But if you actualy do the math the total probability
for this state is not 1, it''s 1/2! What happened? It appears that either we''ve made
a mistake or half of our photons have gone missing.
In fact half of the photons have gone missing. Where did they go? They were
filtered out by the polarization filter. This is no great surprise; filtering is what
filters do. But it turns out that the photons filtered out by the polarization filter
have a diferent wave function than the ones that the filter alowed to pass,
namely:
(|U> + |L>)(|H> - |V>)/2√2
The minus sign where before there was a plus sign indicates that these photons
are polarized on an axis that is rotated 90 degrees from the ones that pass
through the filter, which is no surprise. We could rotate the filter 90 degres and
the situation would be reversed; the photons that before were reflected would
now pass through, and the photons that were pasing through would now be
reflected. Nothing else would change. We would stil have interference. No
surprises so far.
Here''s the kicker: the interference patern that we get if we rotate the polarizing
filter by 90 degrees is a diferent patern (thanks to the minus sign) from the one
that we had originally. Moreover, if we take the two paterns and ad them
together they exactly cancel each other out. The peaks in one patern fall onto
the troughs of the other, and the net result loks exactly like the non-interference
patern that we had without the filter. So the filter isn''t really creating
interference, it''s just filtering out interference that was already there.
Even the simple two-slit experiment works this way. When we place the slits in
front of the laser we are actualy filtering out certain photons (the ones at the
slits) and blocking all the other photons. We can also produce anti-interference
by installing an anti-filter (two sticks where the slits would be).
So what happens if we apply our quantum eraser to a pair of EPR photons?
Exactly the same thing. If we "erase" the information on one photon we actually
can detect interference that we couldn''t before. But we don''t actualy create that
interference; it was there all along. We just filter it out. And it turns out that in
order to actualy perform the filtering operation we need to transmit information
from one side of the apparatus to the other, which is what closes the superluminal
loophole.
Here''s how it works. We send a pair of EPR photons through a pair of two-slit
apparati each of which has a polarization rotator on one of the slits. On one side
of the aparatus (side A) we install a polarization filter which filters out
interference on that side and makes it visible. We can filter out interference on
the other side (side B) of the aparatus as folows: on side A we kep a record of
which photons pased through the filter and which were reflected. On side B we
keep a record of where each photon landed on the scren. We then take these
two records and combine them: for each photon that was pased through the
filter on side A, we take the coresponding photon on side B and note where it
landed on the scren. The end result is a (visible) interference patern. It was
there all along, but the only way we can filter it out so we can se it is to combine
information from both sides of the experiment. And that is the last nail in the
coffin of superluminal communication via entangled photons.
5. Illusions and Reality
The bottom line of this little thought experiment is that measurement and
entanglement are realy the same thing. This is in stark contrast to the maner in
which these topics are usualy presented. Measurement is asued to be
something that everyone understands from everyday experience. In fact, it is
assued to be so thoroughly understod through comon sense that the fact that
it is in fact entirely outside of the theory is conveniently swept under the rug with
only a slight occasional hint of scientific embarrassment. Entanglement, by
contrast, is presented as the depest of mysteries, the canonical example of the
intractability of QM by comon sense, the very antithesis of something as simple
and easily understood as measurement.
It turns out that by thinking of measurement and entanglement as related
phenomena we can shed quite a bit of light (so to speak) on the nature of physical
reality. In fact, it can help us comprehend that which Einstein found most
incomprehensible: the comprehensibility of the Universe.
The Universe is comprehensible because large parts of it are consistent. This
consistency alows us to understand our experiences in terms of stories whose
explanatory power endures from one moment to the next. (When these stories
are told using mathematics we call the scientific theories.) Some of these
stories, like the idea of a material object, are hardwired into the huan brain.
Other stories, like the idea of a chemical or electricity, are not innate. One of the
triumphs of the human species is that we are able to comunicate these stories,
so that a new story once constructed can be propagated without having to be
encoded into our DNA.
Consistency defines reality. We distinguish betwen the perceptions that we have
while sleeping from those we have while awake precisely because our wakeful
perceptions are more amenable to consistent storytelling. We call our wakeful
perceptions "reality" and our sleepful ones "dreams" for precisely this reason.
It is so deply ingrained in our psyche to believe that the universe is consistent
because reality is in some sense real that the sugestion that reality is simply a
mental construct that our brains concoct to explain consistency in perception
sounds preposterous on its face. For one thing, our brains are real. If they
weren''t, they wouldn''t be around to do any concocting. I wil defer this isue for
now; for the moment let us simply acept that consistency and reality are
intimately conected without making any comitments to which way the
causality runs. The point is that the Universe is coprehensible because it is
consistent. This is important because comprehensibility cannot be described
mathematically, but consistency can.
Note: what folows is based almost entirely on Nicolas Cerf and Christoph Adami’s
development of quantum information theory of measurement [3].
5.1 The mathematics of consistency
Consistency can be quantified using an information-theoretic construct caled the
Shannon entropy. The Shannon entropy for a single system is defined as:
H(A) = -Σp(a) log p(a)
where p(a) is the probability that the system A is in state a. When the system has
an equal probability of being in one of N states this quantity works out to be
simply log(N). When N is 1 (the system is definitely in a single state) the entropy
is zero.
We can extend this definition straighforwardly to more than one system:
H(AB) = -Σp(ab) log p(ab)
where p(ab) is the probability that system A is in state a and system B is in state
b. We can also define the conditional entropy:
H(A|B) = -Σp(a|b) log p(a|b)
where p(a|b) is the probability that system A is in state a under the assumption
that system B is in state b. The conditional entropy is therefore a measure of the
corelation or the consistency of systems A and B. If A and B are perfectly
consistent (i.e. perfect knowledge of the state of B provides perfect knowledge of
the state of A) then the conditional entropy is zero. If there is no consistency at
all between systems A and B (i.e. knowing the state of B tells you nothing about
the state of A) then the conditional entropy H(A|B) is equal to H(A).
A different way of expresing the same thing is with the mutual entropy or
information entropy, which is a measure of the amount of information about the
state of system A contained in the state of system B. The information entropy is:
I(A:B) = I(B:A) = H(A) – H(A|B)
= H(A) + H(B) – H(AB)
= H(AB) – H(A|B) – H(B|A)
To ilustrate these quantities consider two extreme examples: a completely
uncorelated pair of systems (like two coins being flipped) and a perfectly
corelated pair of systems (like a single coin and a perfect sensor measuring
whether the coin has landed heads or tails). The various entropies can be
succinctly illustrated with Venn diagrams like those shown in figure 1.
H(A|B) I(A:B) H(B|A)
1 10
Coin A Coin B
Total entropy=2
0 01
Coin Sensor
Total entropy=1
(a) (b)
Figure 1: Entropy diagram for two classical binary systems that are
(a) perfectly uncorrelated and (b) perfectly correlated.
In the case of the two uncorelated coins (figure 1a) the conditional entropies
H(A|B) and H(B|A) are both 1 and the mutual entropy I(A:B) is 0. The individual
entropies (the sums of the numbers in the circles) H(A) and H(B) are both 1, and
the total entropy of the two-coin system (the sum of all the numbers in the
diagram) is 2. The conditional entropy of zero tells us that knowing the state of
one coin tells us nothing about the state of the other. The total entropy of 2 tells
us that there are "two bits of randomness" in the system.
In the case of the coin-sensor system (figure 1b) the information entropy is 1 and
the conditional entropies are both 0, that is, knowing the state of the sensor gives
you perfect knowledge of the state of the coin (and vice versa). The individual
entropies H(A), H(B) and the total entropy H(AB) are all 1. There is one bit of
randomness in the system, and it is "shared" between the coin and the sensor.
Since probabilities are always real numbers between 0 and 1 we can prove that
the conditional entropy H(A|B) is always greater than or equal to 0. This is not
surprising, since a conditional entropy of zero means that two systems are
perfectly consistent, and you can''t get any more consistent than that. Or so it
would seem.
5.2 Quantum information theory
It is posible to extend classical information theory to quantum mechanics. In
clasical information theory the measure of a system being in a particular state is
a probability, which is a real nuber between 0 and 1. In quantum mechanics
the measure of a system being in a particular state is an amplitude, which is a
complex number whose norm is between 0 and 1. It turns out that we can define
all the same information-theoretic quantities for quantum systems as we can for
classical systems. The quantum entropy (also caled the Von Neuman entropy) S
is defined:
S(A) = -Tr
A
(ρ
A
log ρ
A
)
The clasical probability p(a) has ben replaced with something caled a density
matrix ρ
A
which is a mathematical description of the quantum state of system A.
The sumation operator has ben replaced with the trace operator Tr, which is
the mathematical description of what hapens when you make a measurement on
a quantum system. It "colapses the wave function" described by ρ
A
and yields a
classical probability, that is, a real number between 0 and 1.
With a bit of mathematical wizardry
1
we can construct quantum analogs S(AB)
and S(A|B) to the classical joint and conditional entropies H(AB) and H(A|B).
When we turn the crank on the math we get a truly remarkable result by virtue of
the fact that we are now dealing with density matrices (that is, complex numbers)
rather than probabilities (real numbers): the conditional entropy of a quantum
system can be less than zero. In fact, it can be as low as –1. Remember that a
classical conditional entropy of zero implied that two systems were perfectly
corelated. A negative conditional entropy implies that two systems are better
than perfectly correlated; they are somehow supercorelated. It should come as
no great surprise to learn that negative joint entropies (supercorelations) arise
when (and only when) the density matrix describes an entangled quantum state
like an EPR pair.
The entropy diagram for an EPR pair is shown in figure 2. The conditional
entropies are both –1, the joint entropy is 2, and the total entropy of the system is
0. In other words, the particles are supercorelated (whatever that means) and
there is no randomness in the system.
Figure 2: Entropy diagram for a system of two entangled particles.
5.3 Measurement
Now let us take a closer look at what really hapens during a measurement.
Consider measuring the position of a photon by shining light on an ordinary
piece of white paper. When photons arrive they are absorbed by atoms in the
paper (actually they are absorbed by electrons, but atoms make the most
convenient carying cases for electrons). Because of the particular surroundings
that the paper atoms find themselves in they eventualy re-emit most of the
photons they absorb in some random direction. (This is the behavior that makes
a piece of white paper look like white paper.) In the process, the quantum state of
the atom in the paper becomes entangled with the quantum state of the photon.
1
The mathematical wizardry is the definition of the joint density matrix ρ
A|B
like
so:
ρ
A|B
= [ ρ
AB
1/n
(1
A
? ρ
B
)
-1/n
]
n
Actualy, calling this quantity the joint density matrix is a bit of a misnomer
because it doesn''t obey all the formal properties of a density matrix. Pay no
attention to the man behind the curtain.
The photon then continues on its merry way and eventualy lands on another
atom. This time the atom is part of a light sensitive cell in the retina of one of our
eyes. This atom absorbs the photon like the paper did, but instead of re-emitting
the photon, the atom enters an excited energy state which sets off a series of
electrochemical reactions that eventually results in a nerve impulse.
But we''re getting ahead of ourselves. Let''s go back to that second atom, the one in
our retina. As a result of absorbing the photon this atom''s quantum state also
becomes entangled with the that of the photon. We now have three mutually
entangled particles: the original photon, the atom in the paper, and the atom in
the retina.
The entropy diagram for a system of three mutualy entangled particles is shown
in figure 3a. Like the case of two mutually entangled particles the total entropy is
zero (no randomnes in the system) and the conditional entropy of each of the
particles is –1. But notice what happens if we lok at only two of the three
particles (figure Xb). The contribution to the entropy from the third particle
disapears, and what is left over looks exactly like a system of two classically
corelated (not quantum entangled) particles. The apparent entropy of these two
particles is 1 (that is, there apears to be randomnes in the system if we ignore
one of the particles), but the actual total entropy of the system of thre particles
is still zero.
Figure 3: Entropy diagrams for a system of thre mutualy entangled
particles (a) and the sae diagram with one particle ignored or
"traced over" (b). The total entropy in both cases is 0, but the
apparent total entropy in the second case is 1.
It turns out that this result generalizes to any number of mutualy entangled
particles. If we ignore any one particle, the entropy diagra of the remaining
particles looks like a system of N-1 particles in a clasicaly corelated state with a
non-zero entropy. In other words, quantum information theory provides a
mathematical description of the physical proces of measurement in terms of
quantum theory itself. It acounts for the aparent contradiction between
quantum theory, which says that entropy is conserved in unitary transformations,
and the apparent increase in entropy that arises from the randomnes in
quantum measurements.
5.4 Reversibility
Quantum mechanics predicts that all physical proceses are reversible. Since
quantu information theory is a purely quantum theory then under QIT
measurements are in principle reversible as well. This is apparently at ods with
the observed ireversibility of measurements (not to mention the second law of
thermodynamics).
The key is the phrase "in principle." In fact, measurements are reversible in
principle (and so quantum erasers are posible in principle). But let''s lok at
what it would take to actually reverse a measurement.
Under QIT, a measurement is just the propagation of a mutualy entangled state
to a large nuber of particles. To reverse this proces we would have to
"disentagngle" these quantum states. In principle this is posible. In practice it is
not. To see why, let''s go back to a single EPR pair of photons. These photons
were created when an atom absorbed a single photon and emitted a pair of
photons each with half the energy of the original. To reverse this proces we
would have to arange for both members of the EPR pair to be absorbed by a
single atom and then have that atom emit a single photon with twice the energy
of the two input photons.
The key requirement is that to disentangle an entangled state the particles that
participate in that state have to be physicaly brought together. If this were not
the case, if there were any way to disentangle a quantum state while the
component particles were far apart we could imediately use this to produce
faster than light comunication using the ethod described in section 3.
Intentionally arranging for this to hapen for even a single pair of particles would
be a significant engineering chalenge. The chance that it will hapen by
acident for even a single pair of particles is vanishingly smal. And the chance
that it wil hapen for every member of a system of 10
23
mutualy entangled
particles (which is what it would take to reverse a measureent) is (essentially)
zero.
6. Philosophical implications of QIT
Quantum information theory offers some atractive features as a story to tell
about quantum mechanics. It describes quantum measurement in terms of
quantum mechanics itself. It describes how classical corelations arise from
quantu entanglement. It provides an acount of the (apparent) increase in
entropy in the measurement proces that is consistent with entropy-conserving
unitary transformations. Most importantly, QIT completely explains the
"mystery" of spooky action at a distance by describing easurement in terms of
entanglement. The quantum-information-theoretical description of a pair of
measurements made on an EPR pair is exactly the same as a pair of measurements
ade on a single particle. “Spoky action at a distance” ought to be no more
(and no less) mysterious than the “spooky action acros time” which makes the
universe consistent with itself from one moment to the next.
Nonetheles, this story extracts a certain tol on our intuition because it insists
that we abandon our usual notions of physical reality. The mathematics of
quantum information theory tell us unambiguously that particles are not real. To
quote Cerf and Adami:
... the particle-like behavior of quantum systems is an illusion
[emphasis in original] created by the incoplete observation of a
quantum (entangled) system with a macroscopic number of degrees
of freedom.
and
... randomnes is not an essential cornerstone of quantum
measurement but rather an illusion created by it.
So Mermin was on the right track, but he didn’t get it quite right: not only is the
mon is not realy there when nobody loks, but it isn''t realy there even when
you do look! "Physical reality" is not "real", but information-theoretical reality is.
We are not physical entities, but informational ones. We are made of, to quote
Mermin, "corelations without corelata." We are not made of atos, we are made
of (quantum) bits. At the risk of stretching a metaphor beyond its breaking point,
what we usualy call reality is “really” a very high quality simulation running on a
quantum computer.
This is a very counterintuitive view of the world, but the mathematics of Quantum
Mechanics tel us unambiguously that it is corect, just as the mathematics of
relativity tell us that there is no absolute time and space. Entanglement, far from
being an obscure curiosity of QM, is in fact at its very heart. Entanglement is the
reason that measurement is possible, and thus the reason that the Universe is
comprehensible.
Enlightening as this new insight may be, it does leave us with the vexing question:
if what we perceive as reality is only an illusion, what is the "substrate" for this
ilusion? To quote Joe Provenzano: If reality is an ilusion, who (or what) is being
illused? If reality is a magic trick, who is the audience?
The best I can ofer as an answer to that question is a Zen koan from Douglas
Hofstadter:
Two monks were arguing about a flag. One said, "The flag is
moving." The other said, "The wind is moving." The sixth
patriarch, Zeno, hapened to be pasing by. He told them, "Not the
wind, not the flag. Mind is moving."
References:
[1]A. Einstein, B. Podolsky, and N. Rosen, "Can Quantum-Mechanical
Description of Physical Reality Be Considered Complete?" Physical review 47, 777
(1935).
[2]Bell, J. S., 1964, “On the Einstein Podolsky Rosen Paradox,” Physics, 1 (3),
195.
[3]C. H. Adami and N. J. Cerf, “Information Theory of Quantum Entanglement
and Measurement,” Physica D 120 (1998) 62-81.
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