The Quantum Completeness Problem

Quantum mechanics (QM) is complete in a precise sense. It cannot be supplemented by more informative descriptions of physical systems given certain reasonable assumptions; this is what the no-hidden-variables proofs show. Theorems of Kochen-Specker-type (KS theorems) crucially employ an assumption of context-independence of the observables considered while those of Bell-type theorems use an assumption of locality. Since locality amounts to context-independence of local observables the former theorems can be considered as more general than the latter. What exactly do these theorems show in terms of physics? In which sense do they prove QM to be complete? The standard answer to these questions is indeed older than the theorems themselves. It was given by the inventors of QM, notably von Neumann (who himself devised a no-hiddenvariables argument) and Dirac. It says that the mathematical entity representing the maximal QM information we can have about a system S represents all the physical properties that S has. This is an informative physical interpretation of the theorems. Since it embodies the idea that the QM information about S is the complete representation of its physical state it may be called the completeness assumption. In this paper, I will try to show that the completeness assumption is not the correct interpretation of the KS theorems for an ultimately simple reason: it cannot be squared with QM itself. The argument proceeds as follows. Initially, I explicate four properties of QM – properties not mentioned but represented in standard axiomatisations – that will drive the argument, and moreover cast the completeness assumption into a precise form: COMP (basically a weakened version of the well-known eigenstate-eigenvalue link) (sec.2). Then I consider the central equation of the QM statistical algorithm: the trace formula. From the QM properties I conclude that the formula must be explicated in one of two ways. Neither option, however, can be harmonised with both QM and a general probability principle (sec.3). I discuss whether the argument is just an involved form of the measurement problem or else unduly neglects the notion of QM measurement – both with a negative result (sec.4). I consider a fundamental objection that indeed circumvents the argument and show that it violates one of the four QM properties (sec.5). Finally, I briefly consider what the KS theorems tell us about QM given that the argument against COMP is correct (sec.6).