Limited Holism and Real-Vector-Space Quantum Theory

Abstract

Quantum theory has the property of "local tomography": the state of any composite system can be reconstructed from the statistics of measurements on the individual components. In this respect the holism of quantum theory is limited. We consider in this paper a class of theories more holistic than quantum theory in that they are constrained only by "bilocal tomography": the state of any composite system is determined by the statistics of measurements on pairs of components. Under a few auxiliary assumptions, we derive certain general features of such theories. In particular, we show how the number of state parameters can depend on the number of perfectly distinguishable states. We also show that realvector-space quantum theory, while not locally tomographic, is bilocally tomographic.

The Feynman problem and fermionic entanglement: Fermionic theory versus qubit theory

“…23 First, we define an effect to be n-local if it can be written as a conic combination of tensor products of effects that are at most n-partite. Definition 2.3.…”

“…Well-known examples of opts are: (i) quantum theory (qt) (recently axiomatized within the operational framework [18][19][20] ), (ii) the classical information theory, 20 (iii) the box-world, 21 and (iv) the real quantum theory (rqt). 22,23 In Section 2 we review the operational framework and present the recent results of Ref. 24, where the superselection rule has been formalized in the general context of opts.…”

“…24 allows for many other theories. Among them we will discuss briefly the case of rqt-which also lacks local tomography 23 and monogamy of entanglement 26 -and the theory with number superselectionwhich only admits superposition of states having the same particle occupation number.…”

Sporadic SICs and the Normed Division Algebras

“…If the system in question is a set of three qubits, then we'd be talking about the lattice of subspaces of C 8 . To make this look exactly like the setting of Dedekind's lattice, we would have to do quantum mechanics over real vector space ("rebits" instead of qubits), but that's not so bad as far as pure math is concerned [51][52][53].…”

Information Theoretic Characterization of Physical Theories with Projective State Space

“…The works previously cited together with many others [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] contributed to define and consolidate a framework to model physical theories compatibly with the abandonment of determinism and with the assumption that physical laws are intrinsically random. This framework is currently known as probabilistic theories framework and constitutes a starting point to investigate the foundations of quantum theory and possible alternative or deeper theories.…”