Quantum Measurement Complexity

TL;DR

This paper introduces the concept of quantum measurement complexity, analyzing how many copies of a quantum state are needed for measurement, with implications for quantum cryptography and biological systems.

Contribution

It defines quantum measurement complexity and explores its theoretical limits, including relations for photon polarization measurements in quantum cryptography.

Findings

Quantum measurement complexity depends on the number of state copies examined.

Approximate measurements are possible within Heisenberg uncertainty limits.

Relations for measurement complexity are derived using information theory.

Abstract

This paper explores the problem of quantum measurement complexity. In computability theory, the complexity of a problem is determined by how long it takes an effective algorithm to solve it. This complexity may be compared to the difficulty for a hypothetical oracle machine, the output of which may be verified by a computable function but cannot be simulated on a physical machine. We define a quantum oracle machine for measurements as one that can determine the state by examining a single copy. The complexity of measurement for a realizable machine will then be respect to the number of copies of the state that needs to be examined. A quantum oracle cannot perform simultaneous exact measurement of conjugate variables, although approximate measurement may be performed as circumscribed by the Heisenberg uncertainty relations. When considering the measurement of a variable, there might be residual uncertainty if the number of copies of the variable is limited. Specifically, we examine the quantum measurement complexity of linear polarization of photons that is used in several quantum cryptography schemes and we present a relation using information theoretic arguments. The idea of quantum measurement complexity is likely to find uses in measurements in biological systems.