Generalized probabilistic theory
Generalized probabilistic theories (GPTs) are a general framework to describe the operational features of arbitrary physical theories. A GPT must specify what kind of physical systems one can find in the lab, as well as rules to compute the outcome statistics of any experiment involving labeled preparations, transformations and measurements. The framework of GPTs has been used to define hypothetical non-quantum physical theories which nonetheless possess quantum theory's most remarkable features, such as entanglement or teleportation. Notably, a small set of physically motivated axioms is enough to single out the GPT representation of quantum theory.[1] Any such set of axioms is dubbed a reconstruction.
Definition
A GPT is specified by a number of mathematical structures, first of which is a set of types, each of which represents a class of physical systems. Each of these types comes with a vector space (the state space) and a convex set within that is the set of allowed states. Physical states (of that system type) correspond to elements of , while measurements are identified with some set of linear maps from to probability distributions (the distribution of the outcomes of the measurement). Physical operations are a subset of all linear maps which transform states into states, even when they just act on parts thereof (the notion of "part: is subtle: it is specified by explaining how different system types compose and how the global parameters of the composite system are affected by local operations). Any such a set of structures defines a GPT.
For instance, quantum mechanics is a GPT where system types are described by a natural number (the Hilbert space dimension); states of systems of Hilbert space dimension live in a state space of dimension (the set of Hermitian matrices of size ) and belong to the set of normalized positive semidefinite matrices; measurements are identified with Positive Operator valued Measures (POVMs); and the physical operations are completely positive maps. Systems compose via the tensor product and thus all properties of the composite system are accessible via local operations (namely, there are no global parameters). Classical physics, on the other hand, is a GPT where states correspond to probability distributions and both measurements and physical operations are stochastic maps.
Counterexamples to quantum theory
The framework of GPTs has provided examples of consistent physical theories which cannot be embedded in quantum theory and indeed exhibit very non-quantum features. One of the first ones was Box-world, the theory with maximal non-local correlations.[2] Other examples are theories with third-order interference[3] and the family of GPTs known as generalized bits.[4] Not only have these theories incompatible measurements, but their dynamics are also limited by a no-cloning theorem.[5] These two features, which were previously thought to be the mark of quantum theory, are in fact present in any non-classical GPT.[2]
See also
References
- ^ Hardy, L. "Quantum Theory From Five Reasonable Axioms". arXiv:quant-ph/0101012.
- ^ a b Barrett, J. (2007). "Information processing in generalized probabilistic theories". Phys. Rev. A. 75: 032304. doi:10.1103/PhysRevA.75.032304.
- ^ Dakić, B.; Paterek, T.; Brukner, C. (2014). "Density cubes and higher-order interference theories". New J. Phys. 16 (2): 023028. doi:10.1088/1367-2630/16/2/023028.
- ^ Pawłowski, M.; Winter, A. (2012). ""Hyperbits": The information quasiparticles". Phys. Rev. A. 85: 022331.
- ^ Barnum, H.; Barrett, J.; Leifer, M.; Wilce, A. (2012). Abramsky, S.; Mislove, M. (eds.). Teleportation in General Probabilistic Theories. AMS Proceedings of Symposia in Applied Mathematics. American Mathematical Society.