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Average subentropy and coherence of random mixed quantum states

Lin Zhang, Uttam Singh, Arun Kumar Pati

Published Oct 29, 2015• 3 citations
OpenAlex

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.

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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.
We investigate the generic aspects of quantum coherence guided by the concentration of measure phenomenon. We find the average relative entropy of coherence of pure quantum states sampled randomly from the uniform Haar measure and show that it is typical, i.e., the probability that the relative entropy of coherence of a randomly chosen pure state is not equal to the average relative entropy of coherence (within an arbitrarily small error) is exponentially small in the dimension of the Hilbert space. We find the dimension of a random subspace of the total Hilbert space such that all pure states that reside on it almost always have at least a fixed nonzero amount of the relative entropy of coherence that is arbitrarily close to the typical value of coherence. Further, we show, with high probability, every state (pure or mixed) in this subspace also has the coherence of formation at least equal to the same fixed nonzero amount of the typical value of coherence. Thus, the states from these random subspaces can be useful in the relevant coherence consuming tasks like catalysis in the coherence resource theory. Moreover, we calculate the expected trace distance between the diagonal part of a random pure quantum state and the maximally mixed state and find that it does not approach to zero in the asymptotic limit. This establishes that randomly chosen pure states are not typically maximally coherent (within an arbitrarily small error). Additionally, we find the lower bound on the relative entropy of coherence for the set of pure states whose diagonal parts are at a fixed most probable distance from the maximally mixed state.
Abstract The Wishart ensemble is a useful and important random matrix model used in diverse fields. By realizing induced random mixed quantum states as a Wishart ensemble with fixed unit trace, using matrix integral technique we give a fast track to the average coherence for random mixed quantum states induced via partial-tracing of the Haar-distributed bipartite pure states. As a direct consequence of this result, we get a compact formula for the average subentropy of random mixed states. These compact formulae extend our previous work.
It is well known that random bipartite pure states are typically maximally entangled within an arbitrarily small error. Showing that the marginals of random bipartite pure states are typically extremely close to the maximally mixed state, is a way to prove the above. However, a more direct way to prove the above is to estimate the distance of random bipartite pure states from the set of maximally entangled states. Here, we find the average distance between a random bipartite pure state and the set of maximally entanglement states as quantified by three different quantifiers of the distance and investigate the typical properties of the same. We then consider random pure states of a single quantum system and give an account of the typicality of the average l1l_1l1​ norm of coherence for these states scaled by the maximum value of the l1l_1l1​ norm of coherence. We also calculate the variance of the l1l_1l1​ norm of coherence of random pure states to elaborate more on the typical nature of the scaled average l1l_1l1​ norm of coherence. Moreover, We compute the distance of a random pure state from the set of maximally coherent states and obtain the average distance.
We analyze mean fidelity between random density matrices of size NNN, generated with respect to various probability measures in the space of mixed quantum states: the Hilbert-Schmidt measure, the Bures (statistical) measure, the measure induced by the partial trace, and the natural measure on the space of pure states. In certain cases explicit probability distributions for the fidelity are derived. The results obtained may be used to gauge the quality of quantum-information-processing schemes.
Properties of random mixed states of dimension NNN distributed uniformly with respect to the Hilbert-Schmidt measure are investigated. We show that for large NNN, due to the concentration of measure, the trace distance between two random states tends to a fixed number $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}=1/4+1/\ensuremath{\pi}$, which yields the Helstrom bound on their distinguishability. To arrive at this result, we apply free random calculus and derive the symmetrized Marchenko-Pastur distribution, which is shown to describe numerical data for the model of coupled quantum kicked tops. Asymptotic value for the root fidelity between two random states, F=34\sqrt{F}=\frac{3}{4}F​=43​, can serve as a universal reference value for further theoretical and experimental studies. Analogous results for quantum relative entropy and Chernoff quantity provide other bounds on the distinguishablity of both states in a multiple measurement setup due to the quantum Sanov theorem. We study also mean entropy of coherence of random pure and mixed states and entanglement of a generic mixed state of a bipartite system.
Abstract We study the average skew information-based coherence for both random pure and mixed states. The explicit formulae of the average skew information-based coherence are derived and shown to be the functions of the dimension N of the state space. We demonstrate that as N approaches to infinity, the average coherence is 1 for random pure states, and a positive constant less than 1/2 for random mixed states. We also explore the typicality of average skew information-based coherence of random quantum states. Furthermore, we identify a coherent subspace such that the amount of the skew information-based coherence for each pure state in this subspace can be bounded from below almost always by a fixed number that is arbitrarily close to the typical value of coherence.