Largest Schmidt eigenvalue of random pure states and conductance distribution in chaotic cavities

Abstract

A strategy to evaluate the distribution of the largest Schmidt eigenvalue for entangled random pure states of bipartite systems is proposed. We point out that the multiple integral defining the sought quantity for a bipartition of sizes N, M is formally identical (upon simple algebraic manipulations) to the one providing the probability density of Landauer conductance in open chaotic cavities supporting N and M electronic channels in the two leads. Known results about the latter can then be straightforwardly employed in the former problem for both systems with broken (β = 2) and preserved (β = 1) time reversal symmetry. The analytical results, yielding a continuous but not everywhere analytic distribution, are in excellent agreement with numerical simulations.

Exact eigenvalue order statistics for the reduced density matrix of a bipartite system

“…m , for which an implicit formula has been derived in Ref. [17]. As pointed out therein, the determination of p…”

“…, the leading large-m behaviour of the average values of the set of eigenvalues [9], the PDFs of the smallest and largest eigenvalues λ (m) min [3,9,11,15] and λ (m) max [17], their mean values and higher moments (λ (m) min ) q and (λ (m) max ) q , the average Tr (ρ (m) a ) q where q is any positive integer [11,17,19] (listed in no particular order). Their joint PDF is then given by…”

Volume of the set of LOCC-convertible quantum states

“…where we used (48) and we have defined Of course, f cont (p) ≥ 0 and´1 0 f cont (p)dp = 1/2. In fact, for 0 ≤ p ≤ 1, f cont (p) is a polynomial in p of degree 2m − 2.…”

Recursion scheme for the largest $\beta$ -Wishart–Laguerre eigenvalue and Landauer conductance in quantum transport

“…It should be noted that in Mathematica list indices start from 1 and not from 0, therefore the indices in the Eqs. (5), (6), (8), (9) and (20) are accordingly shifted when implemented in the code.…”