[Paper Review] Average subentropy and coherence of random mixed quantum states

Lin Zhang, Uttam Singh+1 more|arXiv (Cornell University)|Oct 29, 2015

Quantum Information and Cryptography3 citations

TL;DR

This paper derives analytical expressions for the average subentropy and coherence of random mixed quantum states using Selberg's integrals across various probability measures. It shows that average subentropy approaches the maximum possible value (achieved by the maximally mixed state) as dimension increases, and demonstrates typicality of relative entropy of coherence for random mixed states via numerical analysis.

ABSTRACT

The generic aspects of the entanglement for random pure states are known to be established via the powerful phenomena of concentration of measure. Here, we find analytical expressions for the average subentropy over the set of random mixed states generated via various probability measures on it. The main ingredients in these results are Selberg's integrals. 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. As an application, we find the average coherence of random mixed quantum states sampled from various probability measures, analytically. In the special case of the random mixed states sampled from the induced measure via the partial tracing of the random bipartite pure states, we establish the typicality of the relative entropy of coherence for random mixed states, numerically. In particular, we show that almost all random mixed states have relative entropy of coherence equal to the average relative entropy of coherence.

Motivation & Objective

• To derive analytical expressions for average subentropy over random mixed quantum states under various probability measures.
• To investigate the behavior of average coherence in random mixed states generated via different sampling methods.
• To establish the typicality of relative entropy of coherence for random mixed states, particularly under the induced measure from partial tracing of pure states.
• To explore the asymptotic behavior of subentropy as system dimension increases.

Proposed method

• Utilization of Selberg's integrals to analytically compute average subentropy over random mixed states.
• Application of probability measures on the set of mixed states, including the induced measure from partial tracing of random pure states.
• Numerical analysis to verify the typicality of relative entropy of coherence in high-dimensional random mixed states.
• Use of concentration of measure phenomena to understand generic properties of quantum states.
• Derivation of closed-form expressions for average coherence based on the same probability measures used for subentropy.
• Leveraging symmetry and random matrix theory to simplify high-dimensional integrals over the space of mixed states.

Experimental results

Research questions

• RQ1 How does the average subentropy of random mixed quantum states behave as the system dimension increases?
• RQ2 What is the analytical expression for the average coherence of random mixed states under different probability measures?
• RQ3 To what extent do typical random mixed states exhibit coherence values close to the average relative entropy of coherence?
• RQ4 Does the relative entropy of coherence concentrate around its mean for random mixed states generated via partial tracing of pure states?
• RQ5 Can the maximum subentropy value be asymptotically approached by the average subentropy of random mixed states?

Key findings

• The average subentropy of random mixed states approaches the maximum possible subentropy value, which is attained by the maximally mixed state, as the system dimension increases.
• Analytical expressions for average coherence are derived for random mixed states sampled from various probability measures, including the induced measure.
• For random mixed states generated via partial tracing of random pure states, the relative entropy of coherence is numerically shown to be typical, meaning it concentrates around its average value.
• The typicality of coherence implies that almost all such random mixed states have a relative entropy of coherence equal to the average value.
• The use of Selberg's integrals enables exact analytical computation of average subentropy and coherence, revealing universal asymptotic behavior.