Proof of Page’s conjecture on the average entropy of a subsystem

Abstract

Page conjectured very recently [Phys. Rev. Lett. 71, 1291(1993] that if a quantum system of Hilbert space dimension mn is in a random pure state, the average entropy of a subsystem of dimension m & n is given by S "= P&" &1 -™ . We outline a proof of this elegant formula.PACS numbers: 05.30.Ch, 02.90. +p, 03.65.w, 05.90.+m Very recently Page [1] considered the problem of getting entropy out of a system in a pure quantum state. He took a quantum system AB of Hilbert dimension mn with normalized density matrix, and divided it into two coupled

Entangling power of the quantum baker s map

“…This succinct formula was conjectured by Page [42] and later proved by others [43,44,45] (see also [19,35,46,47]). An expression for the average purity,…”

Negativity of random pure states

“…This is a consequence of the remarkable fact that the uniform distribution of the k-sphere S k is concentrated largely on the equator for large k, and any polar cap smaller than a hemisphere has a relative volume exponentially small in k. Examples in quantum information theory include the entropy of the reduced density matrix, entanglement of formation, distillable common randomness [11]. The entropy of reduced density matrices of typical states has also been conjectured and calculated independently [12][13][14][15], as has been their concurrence, purity and the linear entropy [16]. Not much is however known of one of the most common and computable measures of entanglement, the negativity [17,18], in random Haar distributed pure states.…”

Simple proof of Page’s conjecture on the average entropy of a subsystem

“…The first proof of this conjecture was given by Foong and Kanno [3]. Here we show that a simpler proof can be achieved by noting that the second term in the right-hand side of (1) can be written as a one-dimensional integral in terms of the one-point correlation function of a Laguerre ensemble of complex Hermitian random matrices (see, e.g., Ref.…”