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://papers.ssrn.com/sol3/papers.cfm?abstract_id=1438013), 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:




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:

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.