Measurement Induced Randomness and State Merging

TL;DR

This paper introduces Measurement Induced Randomness (MIR), a quantum-specific measure of randomness arising from measurement, which differs from classical entropy and relates to quantum state merging costs.

Contribution

It defines MIR as a quantum phenomenon distinct from classical entropy, generalizes it to N-qubit systems, and links it to quantum discord and state merging cost changes.

Findings

MIR differs from classical conditional entropy in quantum systems.

MIR reduces to quantum discord for two-qubit systems.

MIR equals the change in quantum state merging cost due to measurement.

Abstract

In this work we introduce the randomness which is truly quantum mechanical in nature arising as an act of measurement. For a composite classical system we have the joint entropy to quantify the randomness present in the total system and that happens to be equal to the sum of the entropy of one subsystem and the conditional entropy of the other subsystem given we know the first system. The same analogy caries over to the quantum setting by replacing the Shannon entropy by the Von Neumann entropy. However, if we replace the conditional von Neumann entropy by the average conditional entropy due to measurement, we find that it is different from the joint entropy of the system. We call this difference Measurement Induced Randomness (MIR) and argue that this is unique of quantum mechanical systems and there is no classical counterpart to this. In other words the joint Von Neumann entropy gives only the total randomness that arises because of the heterogeneity of the mixture and we show that it is not the total randomness that can be generated in the composite system. We generalize this quantity for N-qubit systems and show that it reduces to quantum discord for two qubit systems. Further, we show that it is exactly equal to the change in the cost of quantum state merging that arises because of the measurement. We argue that for a quantum information processing tasks like state merging the change in the cost as a result of discarding prior information can also be viewed as a rise of randomness due to measurement.