Measurement incompatibility and quantum steering via linear programming

TL;DR

This paper introduces a polynomial-time hierarchy of linear programs to efficiently determine measurement incompatibility and quantum steering, enabling analysis of large measurement sets and high-dimensional systems.

Contribution

It transforms the SDP approach into a hierarchy of linear programs with guaranteed convergence, applicable to arbitrary measurements and dimensions, improving scalability and accuracy.

Findings

Efficient bounds on incompatibility robustness for hundreds of measurements on qubits.

Non-trivial bounds for qutrit systems that are computationally intractable with standard SDP.

Comparable or superior performance to existing methods in certifying quantum steering.

Abstract

The problem of deciding whether a set of quantum measurements is jointly measurable is known to be equivalent to determining whether a quantum assemblage is unsteerable. This problem can be formulated as a semidefinite program (SDP). However, the number of variables and constraints in such a formulation grows exponentially with the number of measurements, rendering it intractable for large measurement sets. In this work, we circumvent this problem by transforming the SDP into a hierarchy of linear programs that compute upper and lower bounds on the incompatibility robustness with a complexity that grows polynomially in the number of measurements. The hierarchy is guaranteed to converge and it can be applied to arbitrary measurements -- including non-projective POVMs -- in arbitrary dimensions. While convergence becomes impractical in high dimensions, in the case of qubits our method reliably provides accurate upper and lower bounds for the incompatibility robustness of sets with several hundred measurements in a short time using a standard laptop. We also apply our methods to qutrits, obtaining non-trivial upper and lower bounds in scenarios that are otherwise intractable using the standard SDP approach. Finally, we show how our methods can be used to construct local hidden state models for states, or conversely, to certify that a given state exhibits steering; for two-qubit quantum states, our approach is comparable to, and in some cases outperforms, the current best methods.