Multi-Observables and Multi-Instruments

This article introduces the concepts of multi-observables and multi-instruments in quantum mechanics. A multi-observable $A$ (multi-instrument $\mathcal{I}$) has an outcome space of the form $\Omega =\Omega _1\times\cdots\times\Omega _n$ and is denoted by $A_{x_1\cdots x_n}$ ($\mathcal{I}_{x_1\cdots x_n}$) where $(x_1,\ldots ,x_n)\in\Omega$. We also call $A$ ($\mathcal{I}$) an $n$-observable ($n$-instrument) and when $n=2$ we call $A$ ($\mathcal{I}$) a bi-observable (bi-instrument). We point out that bi-observables $A$ ($\mathcal{I}$) and bi-instruments have been considered in past literature, but the more general case appears to be new. In particular, two observables (instruments) have been defined to coexist or be compatible if they possess a joint bi-observable (bi-instrument). We extend this definition to $n$ observables and $n$ instruments by considering joint marginals of $n$-observables and joint reduced marginals of $n$-instruments. We show that a $n$-instrument measures a unique $n$-observable and if a finite umber of instruments coexist, then their measured observables coexist. We prove that there is a close relationship between a nontrivial $n$-observable and its parts. Moreover, a similar result holds for instruments. We next show that a natural definition for the tensor product of a finite number of instruments exist and possess reasonable properties. We then discuss sequential products of a finite number of observables and instruments. We present various examples such as Kraus, Holevo and L\"uders instruments.