Information processing in generalized probabilistic theories

I introduce a framework in which a variety of probabilistic theories can be defined, including classical and quantum theories, and many others. From two simple assumptions, a tensor product rule for combining separate systems can be derived. Certain features, usually thought of as specifically quantum, turn out to be generic in this framework, meaning that they are present in all except classical theories. These include the nonunique decomposition of a mixed state into pure states, a theorem involving disturbance of a system on measurement (suggesting that the possibility of secure key distribution is generic), and a no-cloning theorem. Two particular theories are then investigated in detail, for the sake of comparison with the classical and quantum cases. One of these includes states that can give rise to arbitrary nonsignaling correlations, including the superquantum correlations that have become known in the literature as nonlocal machines or Popescu-Rohrlich boxes. By investigating these correlations in the context of a theory with well-defined dynamics, I hope to make further progress with a question raised by Popescu and Rohrlich, which is why does quantum theory not allow these strongly nonlocal correlations? The existence of such correlations forces much of the dynamics in this theory to be, in a certain sense, classical, with consequences for teleportation, cryptography, and computation. I also investigate another theory in which all states are local. Finally, I raise the question of what further axiom(s) could be added to the framework in order to identify quantum theory uniquely, and hypothesize that quantum theory is optimal for computation.

Figure 1

The space of normalized states for a gbit in GNST corresponds to the square. If the measurements $X=1$ and $X=2$ are associated with spin measurements in the $z$ and $x$ directions, then the space of states for a quantum-mechanical qubit corresponds to the circle.

Figure 2

An allowed transformation.

Figure 3

Another allowed transformation.

Figure 4

A forbidden transformation.

Figure 5

Transformations of single systems in GNST and GLT can always be represented as the appending of classical circuits as shown here, or as convex combinations of transformations of this type. If a fiducial measurement $X$ is performed on the transformed system, this can be thought of as performing fiducial measurement $X^{\prime }$ on the original system, where $X^{\prime }=F1\left(X\right)$ for some function $F1$. When measurement $X^{\prime }$ is performed on the original system, outcome $a^{\prime }$ is obtained with some probability. The probability of obtaining outcome $a$ for the measurement $X$ on the transformed system is equal to the probability of obtaining an outcome $a^{\prime }$ such that $a=F2(X,a^{\prime })$, for some function $F2$.

Figure 6

Measurements on single systems in GNST or GLT can always be performed via a procedure of the type illustrated here, or via a convex combination of such procedures. First, fiducial measurement $X$ is performed and outcome $a^{\prime }$ is obtained. The outcome $a$ of the complete measurement is then $a=F1\left(a^{\prime }\right)$. This result applies to measurements with an arbitrary number of outcomes.

Figure 7

In GNST, transformations on bipartite systems comprised of two gbits can always be represented by the appending of classical circuits as shown here, or by a similar construction inverted with respect to the two subsystems, or by a convex combination of such. For the construction shown here, this means that if fiducial measurements $X,Y$ are performed on the transformed system, one may think of this as first performing a fiducial measurement $X^{\prime }$ on one half of the original system, where $X^{\prime }=F1(X,Y)$. This gives an outcome $a^{\prime }$. Then, perform a fiducial measurement $Y^{\prime }$ on the other subsystem, where $Y^{\prime }=F2(X,Y,a^{\prime })$. The final outcome pair $(a,b)$ is determined by a function $F3$ of $X$, $Y$, $a^{\prime }$ and $b^{\prime }$.

Figure 8

In GNST, measurements on bipartite systems of two gbits can always be carried out by a procedure like that illustrated here, by a similar procedure inverted with respect to the two subsystems, or by a convex combination of such. For the procedure shown here, this means that first, a fiducial measurement $X$ is performed on one subsystem, and outcome $a^{\prime }$ is obtained. Then fiducial measurement $Y^{\prime }$ is performed on the other subsystem, where $Y^{\prime }=F1\left(a^{\prime }\right)$, and outcome $b^{\prime }$ is obtained. The outcome $a$ of the complete measurement is given by $a=F2(a^{\prime },b^{\prime })$. This result applies to measurements with an arbitrary number of outcomes.