Intepretations of models for quantum mechanics

An important aspect of models for physics is their interpretation. This answers the question of how the elements of the model correspond to the 'reality' of the physical world. From a philosophical point of view, this is a problematic question by itself. There is no consensus about what this correspondence is. From an operational point of view, there is least debate about how to interpret the Hilbert space formalism.

The Hilbert space formalism and compact categories

Hilbert spaces describe the state space of a physical object. Unitary operators describe all 'physical operations'. By this, we mean all possible transitions from one state to another, that are not considered as measurements. Measurements are represented by collections of linear, self-adjoint operators that sum to the identity. Similarly, in a symmetric monoidal compact closed category, these entities are described by abstract concepts that generalise those in the Hilbert Space formalism. The objects represent the state of a physical system, morphisms correspond to physical operations and modules on classical structures represent measurements.

The mixed state formalism and CPM

In the mixed state formalism, quantum states are represented by morphisms between observables and possible outcomes of observables, given the specific quantum state.
The expectation of a projector, given a quantum state, corresponds to the probability of getting the image of the projector as the outcome of a measurement that contains this projector.
By the dynamics of quantum mechanics, these morphisms need to be linear, positive and self-adjoint; i.e. positive, self-adjoint matrices. Physical operations correspond therefore to operators on matrices which preserve positivity and self-adjointness. These operators are called completely positive maps.
Objects in CPM are morphisms in a symmetric monoidal compact closed category which precisely satisfy the abstract conditions for self-adjointness and positivity. similarly, morphisms are given by the abstract generalisation of completely positive maps. It follows that we can interpret CPM in the same way as we interpret the mixed state formalism.

Abstract C*-algebras: a level up

Rather than work with Hilbert spaces directly, many mathematical physicists, including von Neumann himself, work with algebras of operators on the Hilbert space. These correspond to the observables of the system, and states are then a derived notion. Quantum channels find their natural home in this formalism called algebraic quantum theory, namely as completely positive maps. This formalism is also more general than quantum mechanics proper, and also allows quantum field theory.

Thanks to the fact that compact categories are closed, we can model this inside the category. This is basically what the CPM construction does: it turns a category of finite-dimensional Hilbert spaces C^n and pure quantum channels between them into the category of matrix algebras M_n and arbitrary quantum channels between them.

Superselection: how can we interpret (abstract) C*-algebras?

However, not every operator on a Hilbert space is a bona fide observable, due to superselection. Therefore it is not enough to just work with matrix algebras or the CPM construction. Instead, we need direct sums of matrix algebras: one matrix algebra for each superselection sector, and a direct sum indexed over the superselection sectors. Another way to look at it is that instead of all matrices, we consider algebras of block matrices. This way we get all finite-dimensional C*-algebras.

This still works abstractly, and that is what the CP* construction does: it turns the category of finite-dimensional Hilbert spaces and pure quantum channels into the category of all C*-algebras and arbitrary quantum channels.

Classical structures

As a special case, commutative special dagger Frobenius algebras in the category of Hilbert spaces are algebras of matrices that are diagonal in some fixed basis. Therefore, we may think of them as maps that 'copy' and 'delete' physical states. These algebras are also known as classical structures. This name is derived from the fact that classical information/states can be copied and deleted, while this is not the case for quantum information/states, and classical statistical information/states.

Classical objects in CP*, the category of special dagger Frobenius algebras, can therefore be interpreted as those states that can be copied and deleted.

An important fact to note, is that the copy map in the non-commutative case is not a completely positive map. Therefore, according to the mixed state formalism, it is not a physical operation. This plays a key role in the categorical reformulation of the no-cloning theorem.


This state of affairs is rather unsatisfactory. We start with Hilbert spaces, which are unjustified physically; they just "happen to work", but have no derivation from operational first principles. This is abstracted into compact categories, which have more of an operational justification. But then we apply another ad hoc construction, to end up with the category of quantum channels that "just works", without much motivation. It would be much preferable to have an axiomatization of categories of the form CP*[C], ideally with operational justification for the axioms. Then we could just start from a physically justified setup in the first place, without having to construct "the right one" ad hoc.

Nonstandard models

Additionally, the CP* construction connects to very nice theory for nonstandard models, i.e. categories other than Hilbert spaces. For example, the objects in CP*[Rel] are precisely groupoids.

Open questions

This leads to the following list of open questions:

  • What is the operational categorical interpretation of superselection rules?
  • Can we find an axiomatization of categories of the form CP*[C]?
  • What is the physical significance of nonstandard models like CP*[Rel]?