Average subentropy, coherence and entanglement of random mixed quantum states

TL;DR

This paper derives compact formulas for average subentropy and coherence of random mixed quantum states, revealing their typicality and limitations in higher dimensions, with implications for quantum resource usefulness and entanglement measures.

Contribution

It provides new analytical expressions for average subentropy and coherence of random mixed states and demonstrates their typicality and boundedness in high dimensions.

Findings

Average subentropy approaches maximum with increasing dimension.

Average coherence of mixed states is bounded and less than that of pure states in high dimensions.

Most states in a specific class have fixed entanglement measures, simplifying their computation.

Abstract

Compact expressions for the average subentropy and coherence are obtained for random mixed states that are generated via various probability measures. Surprisingly, our results show that the average subentropy of random mixed states approaches to the maximum value of the subentropy which is attained for the maximally mixed state as we increase the dimension. In the special case of the random mixed states sampled from the induced measure via partial tracing of random bipartite pure states, we establish the typicality of the relative entropy of coherence for random mixed states invoking the concentration of measure phenomenon. Our results also indicate that mixed quantum states are less useful compared to pure quantum states in higher dimension when we extract quantum coherence as a resource. This is because of the fact that average coherence of random mixed states is bounded uniformly, however, the average coherence of random pure states increases with the increasing dimension. As an important application, we establish the typicality of relative entropy of entanglement and distillable entanglement for a specific class of random bipartite mixed states. In particular, most of the random states in this specific class have relative entropy of entanglement and distillable entanglement equal to some fixed number (to within an arbitrary small error), thereby hugely reducing the complexity of computation of these entanglement measures for this specific class of mixed states.