Lower entropy bounds and particle number fluctuations in a Fermi sea

In this letter we demonstrate, in an elementary manner, that given a partition of the single particle Hilbert space into orthogonal subspaces, a Fermi sea may be factored into pairs of entangled modes, similar to a BCS state. We derive expressions for the entropy and for the particle number fluctuations of a subspace of a Fermi sea, at zero and finite temperatures, and relate these by a lower bound on the entropy. As an application we investigate analytically and numerically these quantities for electrons in the lowest Landau level of a quantum Hall sample. PACS numbers: 03.67.Mn, 05.70.−a, 05.30.Fk The study of quantum many particle states, when measurements are only applied to a given subsystem, are at the heart of many questions in physics. Examples where the entropy of such subsystems is interesting range from the quantum mechanical origins of black hole entropy, where the existence of an event horizon thermalizes the field density matrix inside the black hole [1, 2] to entanglement structure of spin systems [3]. In this work we address the relation of entanglement entropy of fermions with the fluctuations in the number of fermions. A general treatment of a BCS-like factorization of a Gaussian state on a bi-partite system was carried out in [4]. Here we concentrate on a particular case and show in an elementary way how a given subspace of the single particle Hilbert space, a Fermi sea, may be factorized into pairs of entangled modes in and out of the subspace, thereby writing the state as a BCS state. We then use this construction to calculate various properties of the fermions in one of the subsystems. While upper bounds on entropy were the subject of numerous investigations, especially since Bekenstein’s bound [5], lower bounds on entropy are less known. We show that given a ‘Fermi sea’, the entropy of the ground state, restricted to a particular subspace A of the single particle space, relates to the particle number fluctuations in the 0305-4470/06/040085+07$30.00 © 2006 IOP Publishing Ltd Printed in the UK L85 L86 Letter to the Editor subspace via the inequality: SA (4 log 2) N A −(8 log 2) 〈〈 N A 〉〉 , (1) where SA is the entropy associated with the subspace A, N2 A are the fluctuations in the particle number in A and 〈〈 N4 A 〉〉 is the fourth cummulant of particle number1. The importance of this result lies in the fact that particle fluctuations are, in principle, measurable, and are fundamentally related to the quantum noise in various systems. On the technical side, note that the right-hand side has the advantage of being easy to calculate analytically in a wider class of problems. We start by examining the ground state of non-interacting fermions, in arbitrary external potential, when measurements are applied to a given part of the space. The basic example is a Fermi sea or a Dirac sea where we are interested in the relative entropy of a given region of space, and in fluctuations in the number of particles there, but one may also consider entanglement in Fermion traps (Fermi degeneracy of potassium atoms (40K) has been observed by DeMarco and Jin [6]). The discussion is also relevant for systems which behave like a non-interacting Fermi gas, as in problems of transport at the zero temperature limit2, in ideal metals, where transport may be approximated well within a non-interacting theory, due to good screening. In an ideal single channel conductor the analogy is done by mapping excitations that travel at the Fermi velocity to a time–energy coordinate representation in discussion of quantum pumps [7–9]. The analogy is especially manifest in the problem of switching noise [10]. The ground state of a noninteracting Fermi gas, containing N particles, is obtained by occupying the allowed statesφi ∈ H (H is the single particle Hilbert space) up to energyEf , i.e. |gs〉 = ∏ E(φi) 0, otherwise they are already of the required form. Letter to the Editor L87 where the factor dl is the lth eigenvalue of M, and serves to normalize the Al with the inner product on H. Similarly we take Bl = ∑ k U † lkP ⊥ A φk √ 1−dl , which are orthonormal since M(A⊥) = I −M(A) is diagonalized by U as well. Obviously Ai ∈ A and Bi ∈ A⊥. Since U is unitary we write E = span{U φi}. Using U † ilφl = √ diAi + √ 1 − diBi , the ground state may be written as (up to an overall phase)