Against the Tyranny of ‘Pure States’ in Quantum Theory

Abstract

In this paper we provide arguments against the dominant role played by the notion of pure sate within the orthodox account of quantum theory. Firstly, we will argue that the origin of this notion is intrinsically related to the widespread empirical-positivist understanding of physics according to which 'theories describe actual observations of subjects (or agents)'. Secondly, we will show how within the notion of pure state there is a scrambling of two mutually incompatible definitions. On the one hand, a contextual definition which attempts to provide an intuitive physical grasp in terms of the certain prediction of a measurement outcome; and on the other hand, a noncontextual purely abstract mathematical definition which has no clear physical content. We will then turn our attention to the way in which pure states and mixtures have been considered by two categorical approaches to QM, namely, the topos approach originally presented by Chris Isham and Jeremy Butterfield [27, 28, 29] and the more recent logos categorical approach presented by the authors of this article [11, 12]. While the first approach presents serious difficulties in order to produce an orthodox understanding of pure states and mixtures, the latter presents a new scheme in which the distinction between pure states and mixtures becomes completely irrelevant.

Beyond Purity and Mixtures in Categorical Quantum Mechanics

“…The abstract vector Ψ makes reference to the state |ψ , but also to any rotation a i |ϕ i (see for discussion [4]). As remarked in [13], this latter definition is not consistent with the previous one. While the first operational definition -which has the purpose to secure the existence of an observable which will be certain if measured-is explicitly contextual (i.e.…”

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In a recent paper [12], we discussed the serious inconsistency present within the operational and mathematical definition(s) of the notion of pure state. Continuing this analysis, in this work we attempt to address the role of 'purity' and 'mixtures' within two different categorical approaches to QM, namely, the topos approach originally presented by Chris Isham and Jeremy Butterfield [26,27,28] and the more recent logos categorical approach presented by the authors of this article [10,11,13]. While the first approach exposes the difficulties to produce a consistent understanding of pure states and mixtures, the latter approach presents a new scheme in which their reference is erased right from the start in favor of an intensive understanding of projection operators and quantum superpositions.

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“…Later on, this same idea 3 became explicitly introduced within the main postulates of the theory through the definition of the notion of pure state. As we have discussed in detail in [13], the notion of purity is operationally defined: 'If a quantum system is prepared in such way that one can devise a maximal test yielding with certainty (probability = 1) a particular outcome, then it is said that the quantum system is in a pure state.' In turn, the notion of maximal test allows to interpret a quantum observable as being an actual property -i.e., a property that will yield the answer yes when being measured [37].…”

“…However, as remarked in [13], there is also a mathematical definition according to which, the pure state of a quantum system is described by a unit vector, |ψ , in a Hilbert space. This definition has no physical counterpart and makes reference to a purely abstract mathematical feature of vectors, namely, that when considered in terms of density operators their norm is 1, that is, ρ is a pure state if Tr(ρ 2 ) = 1, or equivalently 4 when ρ = ρ 2 .…”

The Power of Inconsistency in Anti-Realist Realisms about Quantum Mechanics (Or: Lessons on How to Capture and Defeat Smoky Dragons)

“…This condition is fulfilled in classical mechanics via the Galielan transformations and in relativity theory via the Lorentz transformations. On the contrary, in QM the orthodox interpretation of probability in terms of binary measurement outcomes together with the non-invariant definition of quantum state has destroyed the operational invariance present within Heisenberg's original matrix formulation (see for a detailed analysis and discussion [21,22]). This mathematical condition can be translated in conceptual terms as a phenomenologicalobjective condition which imposes the need to discuss about the same object of experience independently of the particular viewpoint taken by an agent.…”