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Average G-concurrence of random pure states

Valerio Cappellini, Hans-Jürgen Sommers

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

Average entanglement of random pure states of anN×N composite system is analyzed. We compute the average value of the determinantD of the reduced state, which forms an entanglement monotone. Calculating higher moments of the determinant we characterize the probability distributionP(D). Similar results are obtained for the rescaledN th root of the determinant, called G‐concurrence. We show that in the limitN→∞ this quantity becomes concentrated at a single pointG ⋆ = 1/e. The position of the concentration point changes if one consider an arbitraryN×K bipartite system, in the joint limitN,K→∞,K/N fixed.

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Average entanglement of random pure states of anN×N composite system is analyzed. We compute the average value of the determinantD of the reduced state, which forms an entanglement monotone. Calculating higher moments of the determinant we characterize the probability distributionP(D). Similar results are obtained for the rescaledN th root of the determinant, called G‐concurrence. We show that in the limitN→∞ this quantity becomes concentrated at a single pointG ⋆ = 1/e. The position of the concentration point changes if one consider an arbitraryN×K bipartite system, in the joint limitN,K→∞,K/N fixed.
The average entanglement of random pure states of an $N\ifmmode\times\else\texttimes\fi{}N$ composite system is analyzed. We compute the average value of the determinant DDD of the reduced state, which forms an entanglement monotone. Calculating higher moments of the determinant, we characterize the probability distribution P(D)P(D)P(D). Similar results are obtained for the rescaled NthN\text{th}Nth root of the determinant, called the GGG concurrence. We show that in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ this quantity becomes concentrated at a single point ${G}_{\ensuremath{\star}}=1∕e$. The position of the concentration point changes if one consider an arbitrary $N\ifmmode\times\else\texttimes\fi{}K$ bipartite system, in the joint limit $N,K\ensuremath{\rightarrow}\ensuremath{\infty}$, with K∕NK∕NK∕N fixed.
We study the entanglement properties of random pure stabilizer states in spin-1/2 particles. We obtain a compact and exact expression for the probability distribution of the entanglement values across any bipartite cut. This allows for exact derivations of the average entanglement and the degree of concentration of measure around this average. We also give simple bounds on these quantities. We find that for large systems the average entanglement is near maximal and the measure is concentrated around it.
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 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.
Abstract We investigate the fidelity of Haar random bipartite pure states from a fixed reference quantum state and their bipartite entanglement. By plotting the fidelity and entanglement on perpendicular axes, we observe that the resulting plots exhibit non-uniform distributions. The distribution depends on the entanglement of the fixed reference quantum state used to quantify the fidelity of the random pure bipartite states. We find that the average fidelity of typical random pure bipartite qubits within a narrow entanglement range with respect to a randomly chosen fixed bipartite qubit is 1 4 . Extending our study to higher dimensional bipartite qudits, we find that the average fidelity of typical random pure bipartite qudits with respect to a randomly chosen fixed bipartite qudit remains constant within a narrow entanglement range. The values of these constants are 1 d 2 , with d being the dimension of the local Hilbert space of the bipartite qudit system, suggesting a consistent relationship between entanglement and fidelity across different dimensions. The probability distribution functions of fidelity with respect to a product state are analytically studied and used as a reference for the benchmarking of distributed quantum computing devices.