Strongly symmetric spectral convex bodies are Jordan algebra state spaces

Abstract

We show that the finite-dimensional convex compact sets having the properties of spectrality and strong symmetry are precisely the normalized state spaces of finite-dimensional simple Euclidean Jordan algebras and the simplices. Various assumptions are known characterizing complex quantum state spaces among the Jordan state spaces, which combine with this theorem to give simple characterizations of finite-dimensional quantum state space.Spectrality and strong symmetry arose in the study of "general probabilistic theories" (GPTs), in which convex compact sets are considered as state spaces of abstractly conceivable physical systems, though not necessarily ones corresponding to actual physics. We discuss some implications of our result-which is purely convex geometric in nature-for such theories. A major concern in the study of such theories, and also in the theory of operator algebras, has been the characterization of the state spaces of finite and infinite-dimensional Jordan algebras, and the characterization of standard quantum theory over complex Hilbert spaces, or the state spaces of von Neumann or C * -algebras, within the class of Jordan-algebraic state

General probabilistic theories: An introduction

“…• Diagonalization and existence of spectral decompositions were investigated [79][80][81][82]. It seems that some form of spectral decompositions is crucial for singling out quantum and quantum-like theories among other GPTs.…”

Non-standard entanglement structure of local unitary self-dual models as a saturated situation of repeatability in general probabilistic theories

“…As consequence of Corollary 7, the standard structure of entanglement can not be determined by saturated repeatability and local unitary symmetry. In other words, our results implies that global symmetry like [11] is more essential for determination of entanglement structure that self-duality with local unitary symmetry.…”

“…[1,3,4,5]. This problem is recently studied in the modern operational approach of foundation of quantum theory, called General Probabilistic Theories (GPTs) [1,2,3,4,5,12,7,8,9,10,13,11].…”

“…One of the important aims of studies of GPTs is finding reasonable operational postulates to choose the standard composite system among the above various models. One successful result [11] uniquely determines the standard structure of entanglement under a kind of global symmetry. However, the operational interpretation of the global symmetry is not clarified well so that the justification of the global symmetry of structure of entanglement is kept incomprehensive.…”

Entanglement and superposition are equivalent concepts in any physical theory

“…The crucial aspect of our techniques is that they by-pass the Hilbert space structure that underlies quantum mechanics, but that is not included in other possible non-classical theories. This gives a counter-example to possible axiomatizations of quantum theory [41]: for example, it is known that existence of purifications [21,42], certain symmetries [43,44] or self-duality and spectrality [45] are enough to single-out quantum theory among other non-classical theories. Our results show that existence of superpositions, entanglement, and availability of BB84 protocol do not restrict the set of possible theories at all.…”