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 . Any such set of axioms is dubbed a reconstruction.

Definition

A GPT is specified by a number of mathematical structures. First, we need a set of types, each of which represents a class of physical systems. Each of these types comes with a vector space $V$ (the state space) and a convex set $S$ within (the set of allowed states). Physical states (of that system type) correspond to elements of $S$, while measurements are identified with some set $M$ of linear maps from $S$ 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 $D$ (the Hilbert space dimension); states of systems of Hilbert space dimension $D$ live in a state space of dimension $D^{2}$ (the set of Hermitian matrices of size $D$) and belong to the set $S$ 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 . Other examples are theories with third-order interference and the family of GPTs known as generalized bits . Not only have these theories incompatible measurements, but their dynamics are also limited by a no-cloning theorem . These two features, which were previously thought to be the mark of quantum theory, are in fact present in any non-classical GPT .

Reconstructions of quantum theory

L. Hardy introduced the concept of GPT in 2001, in an attempt to re-derive quantum theory from basic physical principles, which he did, up to a point (see below) . Hardy’s axioms were:

1. Probabilities. It is possible to prepare systems independently. Namely, experimental frequencies tend to definite numbers (the probabilities) as the number of experiments grows.
2. Simplicity. For each system type, the dimension of the state space is a function of the maximum number of states that can be distinguished with certainty. Such a function is the minimum value consistent with the axioms.
3. Subspaces. A system whose state is constrained to have support on only $M$ of a set of $N$ possible distinguishable states behaves like a system of where there are just $M$ distinguishable states.
4. Composite systems. The overall state of a composite system can be established by conducting local measurements in each subsystem. The number of distinguishable states of a composite system is the product of the number of distinguishable states in each subsystem.
5. Continuity. For any two pure states of a system [i.e., states which are not statistical mixtures of other physical states], there exists a continuous reversible transformation mapping one to the other.

Although his work is highly acclaimed, Hardy’s simplicity axiom was controversial. Not only it looks quite ad hoc, but it is also dependent from the other axioms. When removed, Hardy’s axioms give rise to an infinite hierarchy of hypothetical GPTs, with quantum mechanics at the bottom .

The work of Dakic and Brukner eliminated the axiom of simplicity and claimed a reconstruction of quantum theory based on three physical principles . This was followed by the more rigorous reconstruction of Masanes and Müller .

Axioms common to these three reconstructions are:

• The subspace axiom. Systems which can store the same amount of information are physically equivalent.
• Local tomography. To characterize the state of a composite system it is enough to conduct measurements at each part.
• Reversibility. For any two pure states, there exists a reversible physical transformation that maps one into the other.
• The no-restriction hypothesis. All linear homomorphisms mapping states to real numbers in [0,1] correspond to physical dichotomic measurements.

An alternative reconstruction, proposed by Chiribella et al. around the same time, is also based on the

• Purification axiom. For any state $S_{A}$ of a physical system $A$ there exists a bipartite physical system $A-B$ and an extremal state (or purification) $T_{AB}$ such that $S_{A}$ is the restriction of $T_{AB}$ to system $A$. In addition, any two such purifications $T_{AB},T_{AB}^{\prime }$ of $S_{A}$ can be mapped into one another via a reversible physical transformation on system $B$.

All such reconstructions just manage to recover finite dimensional quantum mechanics. In fact, infinite dimensional quantum theory violates the axioms of some such reconstructions. This was already noticed in Hardy’s original paper , where he argues that this feature

[...] may be further support for the case against giving continuous dimensional spaces a role in any fundamental theory of nature.

See also

• Quantum foundations