Bounds on the entanglement entropy by the number entropy in non-interacting fermionic systems
Maximilian Kiefer-Emmanouilidis, Razmik Unanyan, Jesko Sirker, Michael Fleischhauer
Abstract
Entanglement in a pure state of a many-body system can be characterized by the Rényi entropies S^{(\alpha)}=\ln\textrm{tr}(\rho^\alpha)/(1-\alpha) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mo>ln</mml:mo> <mml:mtext mathvariant="normal">tr</mml:mtext> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:msup> <mml:mi>ρ</mml:mi> <mml:mi>α</mml:mi> </mml:msup> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mi>/</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> of the reduced density matrix \rho <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>ρ</mml:mi> </mml:math> of a subsystem. These entropies are, however, difficult to access experimentally and can typically be determined for small systems only. Here we show that for free fermionic systems in a Gaussian state and with particle number conservation, S^{(2)} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:msup> </mml:math> can be tightly bound—from above and below—by the much easier accessible Rényi number entropy S^{(2)}_N=-\ln \sum_n p^2(n) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>N</mml:mi> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mo>ln</mml:mo> <mml:msub> <mml:mo>∑</mml:mo> <mml:mi>n</mml:mi> </mml:msub> <mml:msup> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> which is a function of the probability distribution p(n) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> of the total particle number in the considered subsystem only. A dynamical growth in entanglement, in particular, is therefore always accompanied by a growth—albeit logarithmically slower—of the number entropy. We illustrate this relation by presenting numerical results for quenches in non-interacting one-dimensional lattice models including disorder-free, Anderson-localized, and critical systems with off-diagonal (bond) disorder.