Almost Quantum Theory

Abstract

Modal quantum theory (MQT) is a “toy model” of quantum theory in which amplitudes are elements of a general field. The theory predicts, not the probabilities of a measurement result, but only whether or not a result is possible. In this paper we review MQT and extend it to include mixed states, generalized measurements and open system dynamics. Even though MQT does not have density operators, superoperators or any concept of “positivity”, we can nevertheless establish a precise analogue to the usual representation theorem for CP maps. We also embed MQT in a larger class of modal theories. We show that the possibility assignments for separate measurements on a bipartite system in MQT are always weakly consistent with some probability assignment that respects the no-signaling principle.

Appendix

Here we fill in the details of the argument in Sect. 5.2. For convenience, we will suppose that modal quantum systems 1 and 2 are both described by the state space ${\mathcal {V}}$, and that the same two-outcome measurement is performed on each. The effect subspaces $\mathsf {E}$ and $\mathsf {F}$ in ${{\mathcal {V}}}^{*}$ are non-overlapping, so that $\mathsf {E} \cap \mathsf {F} = \left\langle 0 \right\rangle$. Finally, we assume that the joint possibility table for the state $\left| \Psi \right)$ is as follows:

(81)

(Each sub-table of Eq. 66 is of this form.)

We can find a basis for ${{\mathcal {V}}}^{*}$ of the form $\{ \left( e_{i} \right| , \left( f_{m} \right| \}$, where the $\{ \left( e_{i} \right| \}$ spans $\mathsf {E}$ and $\{ \left( f_{m} \right| \}$ spans $\mathsf {F}$. The dual basis $\{ \left| e_{i} \right) , \left| f_{m} \right) \}$ of ${\mathcal {V}}$ therefore has the property that $\left| e_{i} \right)$ is annihilated by every $\left( f_{m} \right|$ and $\left| f_{m} \right)$ is annihilated by every $\left( e_{i} \right|$. In fact, $\{ \left| e_{i} \right) \}$ spans the annihilator $\mathsf {F}^{\circ }$ and $\{ \left| f_{m} \right) \}$ spans $\mathsf {E}^{\circ }$. We can expand the composite state $\left| \Psi \right)$ in this way:

$$\begin{aligned} \left| \Psi \right) = \sum _{\textit{ij}} \alpha _{\textit{ij}} \left| e_{i} e_{j} \right) + \sum _{\textit{in}} \beta _{\textit{in}} \left| e_{i} f_{n} \right) + \sum _{\textit{mj}} \gamma _{\textit{mj}} \left| f_{m} e_{j} \right) + \sum _{\textit{mn}} \delta _{\textit{mn}} \left| f_{m} f_{n} \right) . \end{aligned}$$

(82)

From Eq. 81, we can see that the effect $\mathsf {E} \otimes \mathsf {F}$ is impossible, which implies that $\left( e_{i} f_{n} \left| \Psi \right. \right) = \beta _{\textit{in}} = 0$ for every i, n. In the same way, because $\mathsf {F} \otimes \mathsf {E}$ is impossible, $\gamma _{\textit{mj}} = 0$ for every m, j. Therefore,

$$\begin{aligned} \left| \Psi \right) = \left| \Psi _{\textit{ee}} \right) + \left| \Psi _{\textit{ff}} \right) , \end{aligned}$$

(83)

where $\left| \Psi _{\textit{ee}} \right) \in \mathsf {F}^{\circ } \otimes \mathsf {F}^{\circ }$ and $\left| \Psi _{\textit{ff}} \right) \in \mathsf {E}^{\circ } \otimes \mathsf {E}^{\circ }$.

Though we have supposed that the two systems are of the same type and that the same measurement is made on each, it is easy to adapt this argument to more general situations, provided the effect subspaces are non-overlapping.