Abstract
We argue that the notion of pure sate within Standard Quantum Mechanics is presently applied within the specialized literature in relation to two mutually inconsistent definitions. While the first (operational purity) provides a basis-dependent definition which makes reference to the certain prediction of measurement outcomes, the latter (trace-invariant purity) provides a purely abstract invariant definition which lacks operational content. In this work we derive a theorem which exposes the serious inconsistencies existent within these two incompatible definitions of purity.
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Notes
Even though there are many interpretations of QM, some of which change the mathematical formalism of the theory (e.g., Bohm and GRW), there is a “Standard” version of QM which is taught in Universities all around the world.
The reference to ‘mixtures’—as contra-posed to ‘pure states’—is extremely problematic for it erases the fundamental distinction between quantum mixture and classical mixture; a well known distinction in the specialized literature which Bernard d’Espagnat termed proper and improper (see D’Espagnat 1976, chap. 6).
In more general terms, as discussed in de Ronde et al. (2018), it is exactly this formal aspect which allows us to talk in terms of an Actual State of Affairs (ASA) that evolves in time; i.e., a dynamical description in terms of the variation of (objective) definite valued observables (or ‘dynamical properties’) independent of the (subjective choice of the) perspective (or reference frame) from which they are being observed. Even in relativity theory, due to the Lorentz transformations, one can still consider ‘events’ as the building blocks of physical reality.
Today, the reference to states has become explicitly instrumental. As pointed out by Timpson (2010), for many contemporary researchers, the quantum state does not represent “how things are in an external, objective world, it merely represents what information one has. Mermin (2001), Peierls (1991), Wheeler (1990) and Zeilinger (1999) have all endorsed this kind of view. Hartle (1968, p. 709) provides an excellent summary: ‘The state is not an objective property of an individual system but is that information, obtained from a knowledge of how a system was prepared, which can be used for making predictions about future measurements’.”
A density matrix can be diagonalized, thus giving a set of eigenvalues \(0\le \lambda _1\le \ldots <\lambda _n\le 1\) with \(\sum _i \lambda _i = 1\). If \({\text{ Tr }}(\rho ^2)=1\), then \(\lambda _1=\ldots =\lambda _{n-1}=0\) and \(\lambda _n=1\). Hence, \({\text{ rk }}(\rho )=1\) and then \(\rho =|v\rangle \langle v|\) and \(\rho =\rho ^2\). Conversely, if \(\rho =\rho ^2\) it has eigenvalues 0 or 1, but from \(\sum _i \lambda _i = 1\) it follows \(\lambda _1=\ldots =\lambda _{n-1}=0\) and \(\lambda _n=1\). Hence, \({\text{ Tr }}(\rho ^2)=1\).
We distinguish here between the purely abstract vector \(\Psi\) and its specific representation in a basis \(|\psi \rangle\). As shown in detail in de Ronde and Massri (2020, Sect. 4) this distinction becomes explicitly visualizable through the use of graph theory (see figures 1 and 6 of the mentioned reference).
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Acknowledgements
We want to thank the insightful suggestions of three anonymous referees which allowed us to make the main point of the text substantially more clear. This work was partially supported by the following grants: FWO project G.0405.08 and FWO-research community W0.030.06. CONICET RES. 4541-12, the Project UNAJ 80020170100058UJ and PIO-CONICET-UNAJ 15520150100008CO “Quantum Superpositions in Quantum Information Processing”. The authors state that there is no conflict of interest.
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de Ronde, C., Massri, C. Against the Tyranny of ‘Pure States’ in Quantum Theory. Found Sci 27, 27–41 (2022). https://doi.org/10.1007/s10699-020-09720-x
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DOI: https://doi.org/10.1007/s10699-020-09720-x