# The importance of being spectral

This is a Perspective on "On the properties of spectral effect algebras" by Anna Jenčová and Martin Plávala, published in Quantum 3, 148 (2019).

By Alessandro Bisio (Dipartimento di Fisica, Pavia University, Italy).

The spectral theorem, a major result in functional analysis, states that any normal (A linear operator is normal if it is closed, densely defined, and it commutes with its adjoint.) operator on a Hilbert space $\mathcal{H}$ can be diagonalized, i.e. it is unitarily equivalent to a multiplication operator . The importance of the spectral theorem in Quantum Theory can hardly be underestimated. Observables are self-adjoint (and thus normal) operators and their eigenvalues, which are real numbers, represent the possible outcomes of a measurement. Mixed states, which are trace-class positive (and hence self-adjoint) operators, can also be diagonalized. Restricting to the finite dimensional case, any mixed state $\rho$ can then be written as a finite convex combination of orthogonal pure states, i.e. $\rho = \sum_{i=1}^d p_i |\psi_i\rangle\langle\psi_i|$ where $d$ is the dimension of the Hilbert space. This is a remarkable feature of Quantum Theory (Trivially, this is also a feature of classical theory.) and it allows us to interpret every density matrix as an ensemble of perfectly discriminable pure states, or, equivalently, as the average state of a classical information source. Moreover, the fact that every quantum state can be diagonalised is the reason why majorization theory  can be applied in Quantum Information [3,4]. Majorization is also a main ingredient of the recent frameworks for quantum thermodynamics (see e.g. the reviews [5,6]).

Spectrality plays an ubiquitous role in quantum theory but in a generic theory (Here we are referring to the framework of generalized and operational probabilistic theories [7,8,9]), as in the PR-boxes theory  (also called “boxworld” theory ) it may fail to hold. Therefore, it has an important place in the works dealing with the recostruction of the mathematical framework of quantum theory from operational axioms. On one hand, diagonalization can be derived from other axioms [12,13]. On the other hand, the fact that any state can be decomposed as a convex mixture of perfectly discriminable states can be one of the axioms which the Hilbert space formalism is derived from .

Other approaches (We are now referring to the works which predominantly belong to the Quantum Logic literature [15,16,17]) focus on the structure of the set of effects, rather than on the space of physical states. An effect  is a linear functional which maps states to probabilities and describes a yes/no mesurement. In quantum theory effects are represented by positive operators bounded by the identity. Therefore, the spectral theorem applies and we can decompose any effect into one-dimensional projectors. The notion of effect algebras has been introduced in Ref.  as an abstraction of the set of quantum effects. It consists of a set ${E}$ endowed with an abelian and associative binary operation “$+$” and elements $0$ and $1$ such that $i)$ $\forall a \in {E}$ $\exists! \, a’$ such that $a+a’=1$ and $ii)$ $a +1 \in {E}$ implies $a = 0$. Effect algebras are a tool which allow to analyze the fundamental features of the elementary binary physical quantities. In particular, they can be regarded as an algebraic model for a logic (where its elements play the role of propositions) which can encompass both the structure of classical Boolean Logic and the orthomodular lattice structure of Quantum Logic. Necessary and sufficient conditions for an effect algebra to be affinely isomorphic to the effect set of Quantum Theory have been derived in Ref. . The basic ingredients of this derivation are the assumption that the effect algebra is both convex, i.e. it posses a convex structure, and spectral. A convex effect algebra ${E}$ is spectral if its elements can be diagonalized. More precisely, we require that for any element $f \in {E}$, there exist a finite collection $\{ a_i \}$ of extremal one-dimensional effects such that $\sum_i a_i =1$ and $f = \sum_i \lambda_i a_i$ for some real coefficients $\lambda_i \in [0,1]$. Such a collection $\{ a_i \}$ of effects is called a context and in finite dimensional Quantum Theory they correspond to resolutions of the identity in terms of rank-one projectors.

How strong is the assumption that a convex effect algebra is spectral? A way to address this question is to examine the diversity of spectral convex effect algebras that are allowed. For example, it is not difficult to realise  that a spectral convex effect algebra which has exactly one context is isomorphic to the effect algebra of Classical Probability Theory. On the opposite, since a Hilbert space has continuously many orthonormal basis, Quantum Theory posseses infinitely many contexts. The results of A. Jencova and M. Plavala  suggest that spectrality is a rather strong assumption. They prove that a convex spectral effect algebra is either classical or it has infinitely many contexts. There exist no convex spectral effect algebras with a finite, greater than one number of contexts. Moreover, a convex spectral effect algebra is proved to recover the orthomodular lattice structure of Quantum Logic if the state space is sharply determining (See e.g. Definition $13$ of Ref.  for the notion of sharply determining state space).

Whichever the framework one chooses, the fact that states and effects can be diagonalized strongly affects the mathematical structure of a theory. This paper of A. Jencova and M. Plavala is a significant step forward in our
understanding of the role that spectrality plays in physical theories.

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