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arXiv:1507.05123 · Physical Review A

Distinguishability of generic quantum states

Zbigniew Puchała, Łukasz Pawela, Karol Życzkowski

Published Jun 9, 2016• 48 citations
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Abstract

Properties of random mixed states of dimension $N$ distributed uniformly with respect to the Hilbert-Schmidt measure are investigated. We show that for large $N$, 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, $\sqrt{F}=\frac{3}{4}$, 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.

DOI
10.1103/physreva.93.062112

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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 D\ifmmode~\else˜\fi=1/4+1/\ensuremathπ\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}=1/4+1/\ensuremath{\pi}D\ifmmode~\else˜\fi​=1/4+1/\ensuremathπ, 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.
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 We study the properties of the random quantum states induced from the uniformly random pure states on a bipartite quantum system by taking the partial trace over the larger subsystem. Most of the previous studies have adopted a viewpoint of ‘concentration of measure’ and have focused on the behavior of the states close to the average. In contrast, we investigate the large deviation regime, where the states may be far from the average. We prove the following results: first, the probability that the induced random state is within a given set decreases no slower or faster than exponential in the dimension of the subsystem traced out. Second, the exponent is equal to the quantum relative entropy of the maximally mixed state and the given set, multiplied by the dimension of the remaining subsystem. Third, the total probability of a given set strongly concentrates around the element closest to the maximally mixed state, a property that we call conditional concentration. Along the same line, we also investigate an asymptotic behavior of coherence of random pure states in a single system with large dimensions.
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.
Abstract We investigate the spectral statistics of the difference of two density matrices, each of which is independently obtained by partially tracing a random bipartite pure quantum state. We first show how a closed-form expression for the exact joint eigenvalue probability density function for arbitrary dimensions can be obtained from the joint probability density function of the diagonal elements of the difference matrix, which is straightforward to compute. Subsequently, we use standard results from free probability theory to derive a relatively simple analytic expression for the asymptotic eigenvalue density (AED) of the difference matrix ensemble, and using Carlson’s theorem, we obtain an expression for its absolute moments. These results allow us to quantify the typical asymptotic distance between the two random mixed states using various distance measures; in particular, we obtain the almost sure asymptotic behavior of the operator norm distance and the trace distance.
We analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions N and M. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary N≤M, a general relation between the n-point densities and the cross moments of the eigenvalues of the reduced density matrix, i.e., the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level densities, and also an exact expression for the variance of von Neumann entropy at finite N,M. Then, we focus on the moments E{K^{a}} of the Schmidt number K, the reciprocal of the purity. This is a random variable supported on [1,N], which quantifies the number of degrees of freedom effectively contributing to the entanglement. We derive a wealth of analytical results for E{K^{a}} for N=2 and 3 and arbitrary M, and also for square N=M systems by spotting for the latter a connection with the probability P(x_{min}^{GUE}≥sqrt[2N]ξ) that the smallest eigenvalue x_{min}^{GUE} of an N×N matrix belonging to the Gaussian unitary ensemble is larger than sqrt[2N]ξ. As a by-product, we present an exact asymptotic expansion for P(x_{min}^{GUE}≥sqrt[2N]ξ) for finite N as ξ→∞. Our results are corroborated by numerical simulations whenever possible, with excellent agreement.