Quantifying Measurement Incompatibility of Mutually Unbiased Bases

Area of Science:

• Quantum Information Science
• Quantum Measurement Theory
• Quantum Foundations

Background:

• Mutually Unbiased Bases (MUB) are fundamental in quantum information processing.
• MUB are typically considered maximally incompatible and complementary.
• Quantifying this incompatibility is crucial for understanding quantum limits.

Purpose of the Study:

• To precisely quantify the degree of incompatibility of MUB using noise robustness.
• To establish upper and lower bounds for the noise robustness of k MUB in dimension d.
• To investigate the operational inequivalence of MUB sets.

Main Methods:

• Utilizing the concept of noise robustness to quantify measurement incompatibility.
• Deriving analytical upper and lower bounds for MUB noise robustness.
• Analyzing specific cases, including complete sets of MUB for prime power dimensions.
• Proving the existence of operationally inequivalent MUB sets.

Main Results:

• Provided upper and lower bounds on the noise robustness for sets of k MUB in dimension d.
• Achieved tight bounds for complete sets of MUB (k=d+1) when d is a prime power.
• Derived a general upper bound on noise robustness for any set of quantum measurements.
• Demonstrated the existence of operationally inequivalent MUB sets and provided a lower bound on their number.

Conclusions:

• The noise robustness framework offers a precise measure of MUB incompatibility.
• The findings reveal operational distinctions between different MUB sets.
• Results have potential applications in understanding Einstein-Podolsky-Rosen (EPR) steering.