Dear Colleagues,
I am delighted to have an opportunity to “listen” to the
discussion of quantum information. Thank you Andrei and
Jonathan for the informative and concise introduction, thank
you Andrei for your skillful moderation, and many thanks to
others for interesting contributions. Since the discussion
drifted towards the issues of the foundations of quantum
theory, the subject of my old (first) research love, I
cannot resist temptation to add my two pennies. I hope that
my comments may become helpful in answering some questions.
For instance, the degree in which quantum mechanics deviates
from classical mechanics, or the question about the
difference between quantum and classical probability.
HOW THE PHYSICISTS INTERPRET QUANTUM THEORY?
However, I would like to start from one more general
comment. Some of the earlier contributors where referring to
the way quantum theory is interpreted by physicists, as if
there was such one interpretation. I am afraid, there is not
such thing as uniform interpretation of quantum theory and
never has been any uniformity in this regard. What united
(temporarily) physicists was common agreement about the
formalism. But even in this case there has been some
diversity of the views.
Physicists have been always divided in their views on the
subject of interpretations. Usually the division is
presented as if it was only a solitary giant Einstein who
refused to play the dice with God against Copenhagen
Interpretation promoted by Bohr, Heisenberg, et al.
Actually, the person who exhausted Heisenberg and Bohr in
the discussions the most was Schroedinger, who never
accepted their views.
Later, in the second half of the XXth century when QM has
become the usual stuff of undergraduate physics program, the
primary division of the physicists has developed not along
the borders of different interpretations of QM, but between
the vast majority who simply did not have any view and did
not care much about it (at least not after graduating from
college,) and those few deviants who still wanted to discuss
it in their professional activities. When I was a physics
student, majority of my professors where openly hostile to
the discussions of QM interpretation. “If you continue to
waste your time on QM interpretation, you will never do
anything important in physics!” was the usual dictum. Well,
it has become true in my case. But I do not regret it. If I
listened to them, I would have missed a lot of fun which I
experienced in trying to understand quantum mechanics. After
all, it is not likely that any “schroederons” would have
been collided in the accelerators today, if I gave up
asking “stupid” questions about foundations of QM. Later
when I was already a faculty member, I could hear frequently
in informal conversations carried in between conference
talks the criticisms like: “Did you listen to Brown. Too
bad, the guy is finished. He has nothing interesting to say.
No results, just philosophy.”
The negative attitude to the matters of interpretation can
be found even in the famous Feynman’s Lectures, where he
wrote humorously, but approvingly (I have never forgiven him
this comment) that the interpretation of QM is no more a
matter of interest to physicists, as they do not talk about
it at lunches and dinners anymore.
DIFFERENCES IN INTERPRETATION VERSUS DIFFERENCES IN
FORMALISMS
Those who still considered foundations of QM as a valid
subject of research have been divided not so much by
differences in interpretation, but in the choice of
formalism, which of course influenced the way of
interpretation. In our discussion the attention of
contributors has been drawn to the interpretation of the
wave functions. Before the two bibles of QM have been
published (Andrei used this very good metaphoric expression
to emphasize the importance of von Neumann’s book
Foundations… published in 1932, but there was another
influential book published two years earlier, Dirac’s
Principles…; both deserve to be considered the holy
scriptures) quantum theory has been discussed in terms
either of wave functions (Schroedinger’s picture) or
matrices (Heisenberg’s picture). The pictures were roughly
equivalent (however not from the strict mathematical point
of view), but since wave functions could be easily
associated with the conventional knowledge through Born’s
interpretation of the “square” of a wave function as
probability amplitude, most discussions were based on
Schroedinger’s approach. What von Neumann really did in
his “bible” was that he got rid of the wave functions from
the picture. The reason was that there is no one to one
correspondence between the quantum states and wave
functions. Multiply the wave function by any complex number
of modulus one and the state is the same, but wave function
different. Also, you can use functions of different
variables to describe the same state. It has become clear
that the wave function is a theoretical construct, not an
object with direct physical interpretation. In von Neumann’s
formalism the central role is played by a Hilbert space, and
wave functions were just elements of the convenient
representation of such a space. It is true, generations of
experimental physicists, chemists, etc. were using wave
functions and Schroedinger’s equations without ever thinking
about Hilbert spaces, but it is a matter of practical
applications in which the choice of concepts and their
understanding are secondary issues. Those who were looking
for understanding QM knew that you cannot interpret one
concept without interpretation of all formalism. And the
concept of a wave function does not furnish a full formalism
of quantum theory. Von Neumann provided such a consistent
formalism in terms which did not use the concept of a wave
function as fundamental. Actually, he even assumed that
quantum state is not described by a vector in the Hilbert
space, but by a one dimensional subspace or equivalently by
a projector on such a subspace. So, a wave function
disappeared from the view. It was long way from de Broglie
who believed that the wave functions are actual physical
waves.
After von Neumann published his Foundations several other
different formalisms have been introduced, such as algebraic
formalism based on C* algebras (started 1934 by Jordan,von
Neumann, and Wigner, developed later in different variations
by Segal, Haag and Koestler, Gelfand and Naimark), convex
state space approach (initiated by Stone, and von Neumann
and Morgenstern, and developed by Mielnik, Ludwig, Davies
and Lewis), quantum logic approach (initiated 1936 by
Birkhoff and von Neumann, developed by Jauch and Piron and
many others). It is amazing that each time the name of von
Neumann appears among initiators of the very different
approaches.
Why were physicists looking for new formalisms?
There were two main motivations. First, there was no fully
general and consistent relativistic QM formalism. Quantum
electrodynamics has been developed giving amazingly accurate
results, but by the price of some mathematically suspicious
steps and without a perspective of one universal
relativistic quantum theoretical formalism which could serve
all physics (including gravitation). Even now there is no
such such formalism available. This I think is a main reason
for saying that physics is in a crisis. If you compare the
success of putting together all physical interactions except
gravitation with the failure of developing a uniform
mathematical model of the reality in which forces may be
distinct, but methods of inquiry uniform, which one is more
important? But "crisis" means "judgment" in Greek, so maybe
soon we will find the eternal bliss.
The second motivation was to get rid of the elements of the
theory which are clearly redundant. I have mentioned that
the wave functions has been removed from the picture (but of
course not from the tool kit of those who were applying
physics to solving particular problems) because they
involved many elements which did not have any physical or
philosophical interpretation. The formalism presented in von
Neumann’s Foundations still had a lot of redundancy. Of
course, any new formalism had to have special case which
could be interpreted in Hilbert space language, but the
formalism itself should be free of all what was not
necessary.
WHAT IS COMMON IN DIFFERENT APPROACHES TO QM?
So, what was necessary? By the time of Foundations it was
clear that any formalism has to describe the two notions, of
the state(s) of a system and of the observable(s). It is
interesting that before QM was born the state of the system
has been almost completely neglected. It was considered
obvious that the state of the system is directly given by
the values of observables. In classical mechanics, when we
have enough functions on the state space (phase space), the
state can be identified uniquely as the unique element of
the intersection of inverse images of the values of
functions representing observables. In QM the separation of
the roles of the concept of a state and of observables has
become fundamental. In classical mechanics (CM) every
observable has some unique and clearly determined value in
each state of the system (the state was identified as a
point in the phase space, and the observable was a function
on this space).
Now, in quantum mechanics (QM) it turned out that in some
states the system may have specific value of an observable,
in some states not. It seems strange and un-intuitional.
However strange it is, no physicist would object this formal
fact (as long as either of the formalisms of QM are
accepted.) Now, the question is whether it is because of
inadequacy of the description of the state (knowing some
additional hidden variables describing the state it would be
possible to identify the unique values of all observables,)
or the description is the best possible and the system in
some states simply does not posses the properties which
determine uniquely the values of observables. Here are
differences in views.
Of course, no formalism can answer this question itself. The
only way to resolve the dillema (if at all possible) is to
check which formalism fits the best what we know from
experimental testing.
The formalisms which I mentioned above choose as the most
fundamental either the concept of an observable, the concept
of a state, or neither. The algebraic approach (which I
called C* algebra approach, but which actually involves in
its different variations different algebras) starts from an
algebra of observables, and proceeds to secondary concept of
the state. The convexity approach starts from the concept of
a convexity space of states and proceeds to observables.
The "biblical" approach involved both concepts of the state
and of the observable with equal status (states as
projectors, observables as selfadjoint operators). There is
finally the fourth possibility to use neither state, nor
observable as fundamental concept.
INTRO TO QUANTUM LOGIC
My favorite, the quantum logic approach starts from the
concept of the logic of events (in early times called often
yes-no propositions or yes-no experiments, but such names
may be confusing). There are many different ways the logic
of events and events themselves are interpreted. It is a
topic for a separate and long story. The simplest will be to
use the concept of an event as primitive concept and to use
for the explanation an analogy to the concept of an event in
probability. Here too, the question is not as trivial as it
may seem. What is an event that you get even score when you
roll a die. The naïve answer is based on the formal
representation of an event by the set {2,4,6}. But it does
not make sense to say that the set {2,4,6} occurred or
happened when the outcome was for instance 4. So an event is
not just a subset of the outcome set, but the fact that the
outcome belongs to the set representing event. Something
like that is in the case of quantum logic.
In the following I will risk several oversimplifications,
for which I have to apologize to those who know quantum
logics. I believe that clarity and limited space should go
before precision and accurateness this time.
The analogy to probability will be the best starting point.
In probability theory, we usually start from the set of
outcomes and events are defined as subsets of this set. In
preparation for more general concept of a quantum (or
empirical) logic, we will have to use a little bit more
abstract but equivalent formulation in which the power set
(the set of all subsets) of the outcome space is replaced by
an (atomic) Boolean algebra. All proability theory can be
formulated this way (Mazurkiewicz, Los, and many others).
Then we can introduce a probability measure on this Boolean
algebra. In the infinite case, we have to consider
probability measure only on measurable subsets, so the
Boolean algebra would correspond to the structure of
measurable subsets.
Now, in elementary classical probability we are introducing
the concept of random variables as (measurable) functions
from the outcome space to real numbers. Since we do not work
with subsets of the outcome space, but with Boolean algebra
of events, we will introduce first the concept of an
observable as a function from Borel sets of the set of real
numbers B(R) to our logic of events L (in this case having
the structure of Boolean algebra) satisfying some natural
conditions of structure preservation. Thus an observable is
in this case a measure on B(R) with values in Boolean
algebra L.
It turns out that in this classical probability case, when
we represent L by subsets of some set (of outcomes), to each
observable corresponds some random variable (i.e. function
from the set of outcomes to real numbers).
This part is little bit confusing (why observables “go”
opposite direction to random variables?),but it is just a
technical issue which does not have much influence on the
general concepts. We can eliminate it from our
consideration, if remember that observables and random
variables are in some correspondence, but they are not
completely identical.
Thus we have as a central concept the logic of events and a
probabilistic measures on (atomic) Boolean algebras in the
classical case. In classical mechanics the logic of events
is basically the Boolean algebra of subsets of a phase
space. The state is a probability measure on the subsets of
a phase space.
Now, how can we get to quantum mechanics? Consider the logic
of events which has the structure more general than Boolean
algebra. It has a binary operation of so called “meet”
(generalizing the intersection of sets in the Boolean
algebra of subsets, or the connective “and” in sentential
logic), a binary operation of join (union of sets or the
connective “or”), and a unary operation of orthocomplement
(Complement set for subsets, or the negation in sentential
logic). This kind of more general structure is called an
orthomodular lattice, when some conditions are satsfied.
Boolean algebras can be considered as special cases of
orthomodular lattices which are distinguished by the fact
that they satisfy the law of distributivity. This law is an
obvious rule in sentential logic or in a Boolean algebra of
subsets: A &(B or C) = (A&B) or (A&C). In general
orthomodular lattices do not satisfy the rule of
distributivity, but much weaker condition of
orthomodularity. Its exact formulation is here not important
(it is not difficult, but would not help much in
understanding quantum logics).
The quantum logic of Hilbert space formulation is the
orthomodular lattice of subspaces of the Hilbert space (or
equivalently the orthomodular lattice of projectors). But in
this approach we do not need any Hilbert space. We do not
have any concepts which do not correspond to physical
reality.
In some sense QM in the quantum logic approach can be
considered just a quantum version of probability theory.
IS QUANTUM PROBABILITY DIFFERENT FROM CLASSICAL PROBABILITY?
Yes and no. Classical probability is on Boolean algebras,
quantum probability on more general orthomodular lattices.
Thus, not all properties of classical probability can be
extended to this more general case. Please do not let
deceive yourself by the fact that in each case we have
basically the same axioms of the measure, or because we
assign some numbers between 0 and 1 to events. The axioms of
Boolean algebra belong (implicitely) to axioms of
probability. These stronger axioms give many properties to
classical probability which cannot be reproduced in the more
general (weaker axioms for the structure of logic) quantum
probability on quantum logics. Thus, the probabilities are
the same as bounded measures on orthomodular lattices. But,
they are different because the measures are defined on
different structures. There are many fundamental theorems
from classical probability which are not valid in the
quantum case.
SUPERSELECTION RULES – PHYSICAL SYSTEMS ARE ONLY PARTIALLY
QUANTUM THEORETICAL
First important advantage of the quantum logic approach is
that we can recognize that there are two extreme cases of
these (empirical) quantum logics, one is purely classical
(Boolean algebra,) one is purely quantum (in the formal
language of this theory completely irreducible,) and then
big number of intermediate cases.
How to understand it?
Boolean algebra can be decomposed completely into the direct
product of two element Boolean algebras (into “bits”). In
the pure quantum case, the orthomodular lattice is non-
decomposable (irreducible) into direct product at all. It
corresponds to the fact that each two states can be
superposed (entangled). For the cases in between, partial
decomposition into direct product is possible, but only into
the components which are irreducible but different from two
element Boolean algebras (bits). Entaglement is possible
only within completely irreducible "sector" of quantum
logic. Superselection rules describe such partial
decomposition into irreducible sectors. They can be
introduced by an observable (called superobservable) which
is compatible with all other observables (using Hilbert
space language, which commute with all other observables).
On purely quantum system there are no superobservables. In
physics we have several superobservables, such as mass,
charge, etc.
It corresponds to the fact that you cannot entangle electron
and proton, or electron and photon. Thus the actual physical
systems have frequently logics which are somewhere between
classical and purely quantum ones.
CAN WE FIND ENTANGLEMENT IN LIVING ORGANIZMS?
Are this irreducible quantum logics unique to quantum
theory? Not. We have many simple examples of irreducibility
around. The logic of subspaces of any linear space is
irreducible. Or, we can describe geometry in terms of such a
logic. Then we can talk about “entangled points”. This
means that when we consider any two different points, they
form a line. Now this line has many other points and can be
described using these other points.
Observe that it cannot happen in the Boolean algebra of
subsets. If you take two one element sets, they will form a
two element set, but this set has only these two elements.
There is only one way you can “dis-entangle” such a set
into elements generating it. In geometry, you can dis-
entangle the line into many different pairs of generating
points.
BUT, please be careful, this entanglement is not exactly the
same as quantum entanglement as the structures are
essentially different. They are similar in the fact that
they share the same property of irreducibility.
Now, the question is: Is it possible that an organism
(human) could be the result of entanglement of its parts? I
doubt it. It is possible that the organism is decomposable
into parts which are irreducible. But physics gives quite
straight answer. There are definitely some superobservables
in the description of the organism as a physical object.
On the other hand, it is possible that some aspects of human
organism (but not the body) can form a structure which is
irreducible (entangled). There is nothing in quantum logic
formulation of QM which makes it exclusively physical, or
which restricts it to elementary particles. Quantum
character is simply a property of irreducibility of the
logical structure, and such logical structures can be found
in many non-physical contexts (e.g. geometry on the plane).
However, every nontrivial (dimension higher than two)
quantum logic (i.e. logic satisfying all axioms) must be
infinite. I am not sure how this infinity could fit the
description of an organism).
Whoops! I am sorry. It’s too long.
Regards,
Marcin
Marcin J. Schroeder, Ph.D.
Professor and Dean of Academic Affairs
Akita International University
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Received on Mon Jun 5 04:56:40 2006
