Natural Science Forum / Physics / Research / July 2008

The following article:

http://cmathphil.blogspot.com/2008/07/on-penroses-argument-against-density.html

addresses Penrose’s criticism (Penrose, 2004, The Road to Reality) of
the usage of the density operator when dealing with pure quantum
states. The main arguments of Penrose are reviewed, and the reflection
proceeds around the main question: What is the most fundamental
mathematical structure that should be used to describe the quantum
system, and, what is the nature of the physical semantics that this
structure formalizes?

The following reproduces the main text of the article, safe for the
formulas that appear as images (that can be seen in the above link):

On Penrose's Argument Against Density Operators
by Carlos Pedro Gonçalves

What is a quantum state?

Should we speak of a quantum state at all?

Should we speak of quantum states or of quantum processes?

These questions can be raised from the work of Baugh, Finkelstein and
Galiautdinov (http://arxiv.org/abs/hep-th/0206036) and from the
results obtained by Gonçalves and Madeira
(http://www.ma.utexas.edu/mp_arc/c/08/08-62.pdf),
about the connection between a stationary
quantum state and the consistent histories formalism, these results
being obtained from the relational structure of the different bases in
which a stationary quantum state can be expanded.

A different, but related, problem arises from Penrose’s Road to
Reality (Penrose, 2004), where the author questions the mathematical
structure that should be used to formalize what is usually called a
quantum state.

Thinking about these two threads, one is lead to the following
question:

What is the most fundamental mathematical structure that should be
used to describe the quantum system, and, what is the nature of the
physical semantics that this structure formalizes?

This is the main question to which we shall return, recurrently,
during this article.

Looking at Penrose's (2004) work, and regarding the first part of the
question, we find that Penrose (2004) raised the problem of
formalizing the quantum state by:

A) The density operator, whose entropy zero space (using von Neumann’s
notion of entropy) is comprised of the density operators for the so-
called "pure states".

B) The normalized kets (for the "pure states")

It is clear that the density operator is more general, since it can be
used for statistical mixtures, but is it more fundamental than the ket
for pure states? And, should we call these states at all?

The notion of a zero entropy density operator is effectively
equivalent to a projective notion of a (pure) quantum state, as
Penrose noticed. Therefore, one might take the position that such
density operators appropriately describe a “physical quantum state”,
taking the perspective that only that which has an impact in
measurement problems can be considered to be physical.

About this, however, Penrose (2004, p.796) argues that:

“(…) I feel uncomfortable about regarding such a ‘pure-state density
matrix’ as the appropriate mathematical representation of a ‘physical
state’. The phase factor (…) is only ‘unobservable’ if the state under
consideration represents the entire object of interest. When
considering some state as part of a larger system, it is important to
keep track of these phases (…)”

Penrose’s main issue is related to the superposition principle. As
Penrose puts it, the basic quantum linearity is obscured in the
density operator description. Indeed, an objection of Penrose against
the density operator is that the density operator makes complex the
simpler linearity of the ket formalism.

So far, Penrose’s arguments equally apply to the density operator and
to the density matrix. In pages 797 to 800 of Road to Reality,
however, Penrose proceeds discussing what he considers to be the
“confused ontological status of the density matrix”, in this case, the
argument centers itself in the matrix and not in the operator. Indeed,
some of the statements, used as counter-argumens apply correctly to
the density matrix but not to the operator.

The major argument refers to the inability of the density matrix to
distinguish between different kinds of entangled pairs. For instance,
consider the following scheme/example:

[Click the link for the article at the beginning of the e-mail to read
the formulas]

Even though we have two different kinds of entangled pairs, the
density matrix is the same, that is, the density matrix does not seem
to distinguish the bases.

However, this is not the case if we take into account the density
operator. The density operators are, not only, different, but if we
determine the projection, for instance, of the second density operator
with respect to the basis in which the first is represented, we
obtain:

[Click the link for the article at the beginning of the e-mail to read
the formulas]

Indeed, the second operator is a statistical mixture between two pure
states of superposition of 0> and 1> (+> and ->).

What the above results show is that the projections of the second
density operator to the basis {0>, 1>} and to the basis {+>,->},
differ, with respect to the probabilities assigned to the quantum
events formalized by these projections.

We can use a density matrix for reading probabilities, however, one
must never confuse the matrix with the operator, and one must always
use the operator, for the fundamental description.

The problems of ontological confusion, raised by Penrose, can be
raised with respect to the density matrix but not with respect to the
density operator.

This stresses the importance of precision of language, and the issue
of the generalized practice of calling the density operator a density
matrix (a practice followed by Penrose, and called into attention by
Feynman in his Lectures on Physics, as a practical but mathematically
imprecise simplification). This must be considered as a simplification
of language, as Feynman stressed, but, nonetheless, it is a
mathematical imprecision in the usage of the terminology, and, when
dealing with fundamental arguments, one must take into account the
distinction between the density matrix and the density operator.

However, Penrose’s argument about the phase seems to stand for both
operator and matrix, quoting Penrose (2004, p.803):

“Under normal circumstances, moreover, one must regard the density
matrix as some kind of approximation to the whole quantum truth. For
there is no general principle providing an absolute bar to extracting
detailed information from the environment. Maybe a future technology
could provide means whereby quantum phase relations can be monitored
in detail, under circumstances where present-day technology would
simply ‘give up’. It would seem that the resort to a density-matrix
description is a technology-dependent prescription! With better
technology, the state-vector could be maintained for longer, and the
resort to a density matrix put off until things get really hopelessly
messy! It would seem to be a strange view of physical reality to
regard it to be ‘really’ described by a density matrix (…)”

Although these arguments may seem compelling, one may place a question
regarding the statement on the approximation to the whole quantum
truth, the question is: what about the so-called 'impure states'?

As Penrose notices, one cannot discard that, at the quantum level,
detailed phase relations may get “lost”, because of some deep
overriding basic principle. It is still too soon to discard such a
hypothesis, and this, indeed, may be likely, if one considers a foamy
Planck scale space-time (quantum foam) (Penrose, 2004).

Furthermore, there is still a division in the community in what
regards the information loss in black holes. Even if many believe,
including, more recently, Hawking (http://arxiv.org/abs/hep-th/0507171),
that information may not be lost, we cannot yet reject this
possibility.

It seems that, accepting Penrose's argument, leads to the position
that if we wish to use a fundamental mathematical description of
physical reality, we must use two different formalisms, a ket for the
pure states and a density operator for all the other cases, and we
cannot discard the need for the usage of the density operator.

Thus, to the first part of our main question (what is the most
fundamental mathematical structure that should be used to describe the
quantum system?)The arguments seem to point towards using the density
operator only when necessary, as a technological tool.

But is this sustainable? Do the phases matter?

The answers to these questions cannot be entirely solved by appealing
to mathematics alone.

Indeed, a mathematician might be divided between: (a) a choice where
one would work with what can be argued to be a more fundamental
structure with respect to the information conserved in the description
(the phase information), but two formalisms would be used for two
different situations (pure vs impure states); (b) a choice where one
works with a single formalism but part of the information (the phase)
is lost.

Since we are dealing with physics, all that matters is whether or not
the phase is physically relevant, or, even, whether or not the density
operator expresses, formally, the most fundamental physical nature,
the normalized ket just being a useful representation, that can be
shown to be equivalent to the density operator up to a global phase
factor.

In effect, so far, all that we can get from the system is the
information contained in the density operator. The question of whether
or not there might be some technology to recover the phase from a
measurement, is still open to discussion.

One may argue that, physically, the phase is irrelevant, one may
alternatively argue that the phase is not physically irrelevant.
However, to do the latter would demand the mathematical formulation of
what might constitute a measurement procedure for the phase, leading
inevitably to the problem of the physical meaning and measurability of
a complex number.

If one chooses to spend some time with this issue, one is led to this
bifurcation of perspectives, where the choice depends less on
mathematics and more on physics, in particular, our main question
«what is the most fundamental mathematical structure that should be
used to describe the quantum system, and what is the nature of the
physical semantics that this structure formalizes?» should be
considered as a whole, since one cannot really consider the formalism,
independently from the object of intentionality of the formalization
(that which the formalization is about and that justifies the
development of the formalization itself). What is fundamental for the
mathematical structures of the formalism, may not be so for the object
of formalization.

In the end, the interpretation of quantum mechanics that one follows
may decide the choice between the two paths, if one wishes to make
such a choice at all, or if, and until, a fundamental thinking about
physics demands such a choice.

An interpretation of quantum mechanics that thinks about the nature of
quantum processes, inevitably restricts our choices about what is
fundamental due to the ontological and epistemological commitments
that we assume, along the way of the construction of a scientifically
grounded interpretation.

In the interpretations that assign a physical nature to the wave
function as corresponding to a pilot wave, the phases are relevant,
even if they cannot be measured, since the fundamental object of
formalization is that pilot wave.

For a follower of Bohr, on the other hand, the whole discussion would
be pointless, since the quantum formalism is just a useful tool used
to predict results of experiments, whether we use a ket, a wave
function or a density operator is irrelevant.

Furthermore, Bohr was “suspicious” of complex numbers, these could be
useful tools, but, in the end, all that mattered were the predictions,
and if a phase is unobservable by current technology it is a waste of
time to think about it or to assign it a physical significance.

In the Aristotle-based realist interpretation, followed by Heisenberg,
the density operator should be taken as the formalization of the
fundamental physical structure, since what the formalism “formalizes”
is the tendency of a potential alternative to be actualized, this
intensity of the dynamis corresponds is quantifiable in terms of a
degree, a degree with which probabilities coincide numerically, when
these probabilities are interpreted as being proportional to the
physical propensity of the potential alternative to be actualized,
which is nothing but the intensity of the dynamis associated with that
alternative.

Taking this into account, the diagonal terms of the density operator
are the fundamental structure, since they are in the direct
correspondence with the object of formalization of the theory, i.e.,
they formalize the most fundamental physical structure, and their
interpretation is naturally processual, a processual nature that is
obscured by the ket representation.

A closely related mathematical argument can be found in Bohm, Davies
and Hiley's paper Algebraic Quantum Mechanics and Pregeometry
(http://arxiv.org/abs/quant-ph/0612002), where the authors built quantum
theory from the primitive idempotents that are directly related to the
different entries of the density operator. Bohm et al. show that the
ket notation hides the fact that each ket represents an object with
two labels.

Thus, in the end, one’s solution to the phase problem and the answer
to the central question placed here, depends on one’s choice of
interpretation of quantum mechanics.