Quantum instruments as a foundation for both states and observables

TL;DR

This paper proposes a unified operational framework for quantum theory based on quantum instruments, which directly correspond to laboratory devices, offering an experimentally grounded perspective that encompasses states, observables, and their time evolution.

Contribution

It demonstrates that quantum instruments can serve as a foundational element for quantum theory, unifying states and observables as derived concepts from measurement processes.

Findings

All measurable probabilities can be expressed using quantum instruments alone.

States and observables emerge as conditioned quantities from measurement sequences.

The approach naturally incorporates detector loss and generalizes time-symmetric quantum formalisms.

Abstract

We demonstrate that quantum instruments can provide a unified operational foundation for quantum theory. Since these instruments directly correspond to laboratory devices, this foundation provides an alternate, more experimentally grounded, perspective from which to understand the elements of the traditional approach. We first show that in principle all measurable probabilities and correlations can be expressed entirely in terms of quantum instruments without the need for conventional quantum states or observables. We then show how these states and observables reappear as derived quantities by conditioning joint detection probabilities on the first or last measurement in a sequence as a preparation or a post-selection. Both predictive and retrodictive versions of states and observables appear in this manner, as well as more exotic bidirectional and interdictive states and observables that cannot be easily expressed using the traditional approach. We also revisit the conceptual meaning of the Heisenberg and Schr\"{o}dinger pictures of time evolution as applied to the various derived quantities, illustrate how detector loss can be included naturally, and discuss how the instrumental approach fully generalizes the time-symmetric two-vector approach of Aharonov \emph{et al.} to any realistic laboratory situation.