Von Neumann and Luders postulates and quantum information theory

TL;DR

This paper investigates the foundational role of the projection postulate in quantum information theory, revealing that von Neumann's and L"uders' postulates are effectively equivalent in the context of quantum measurements involving entangled states.

Contribution

It demonstrates that von Neumann's measurement description implies L"uders' postulate, resolving a long-standing ambiguity in the foundational understanding of quantum measurement in quantum information.

Findings

Von Neumann's measurement description implies L"uders' postulate.

The equivalence resolves a 70-year-old ambiguity in quantum measurement theory.

Supports the continued use of L"uders postulate in quantum information applications.

Abstract

This note is devoted to some foundational aspects of quantum mechanics (QM) related to quantum information (QI) theory, especially quantum teleportation and ``one way quantum computing.'' We emphasize the role of the projection postulate (determining post-measurement states) in QI and the difference between its L\"uders and von Neumann versions. These projection postulates differ crucially in the case of observables with degenerate spectra. Such observables play the fundamental role in operations with entangled states: any measurement on one subsystem is represented by an observable with degenerate spectrum in the Hilbert space of a composite system. If von Neumann was right and L\"uders was wrong the canonical schemes of quantum teleportation and ``one way quantum computing'' would not work. Surprisingly, we found that, in fact, von Neumann's description of measurements via refinement implies (under natural assumptions) L\"uders projection postulate. It seems that this important observation was missed during last 70 years. This result closed the problem of the proper use of the projection postulate in quantum information theory. One can proceed with L\"uders postulate (as people in quantum information really do).