Lower bounds on entanglement entropy without twin copy

We discuss the possibility of estimating experimentally the von Neumann entanglement entropy <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msubsup><a:mi>S</a:mi><a:mrow><a:mi>A</a:mi></a:mrow><a:mrow><a:mi>v</a:mi><a:mi>N</a:mi></a:mrow></a:msubsup></a:math> of a symmetric bipartite quantum system <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>A</b:mi><b:mi>B</b:mi></b:mrow></b:math> by using the basic measurement counts (bitstrings) for a single copy of a prepared state. Using exact diagonalization and analog simulations performed with the publicly available QuEra facilities for chains and ladders of Rydberg atoms, we calculate the Shannon entropy <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msubsup><c:mi>S</c:mi><c:mrow><c:mi>A</c:mi><c:mi>B</c:mi></c:mrow><c:mi>X</c:mi></c:msubsup></c:math> associated with the bitstrings of adiabatically prepared ground states and the reduced entropies <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:msubsup><d:mi>S</d:mi><d:mi>A</d:mi><d:mi>X</d:mi></d:msubsup></d:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:msubsup><e:mi>S</e:mi><e:mi>B</e:mi><e:mi>X</e:mi></e:msubsup></e:math> obtained from the marginal probabilities in <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>A</f:mi></f:math> and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>B</g:mi></g:math>. We then calculate the classical mutual information <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:msubsup><h:mi>I</h:mi><h:mrow><h:mi>A</h:mi><h:mi>B</h:mi></h:mrow><h:mi>X</h:mi></h:msubsup><h:mo>=</h:mo><h:msubsup><h:mi>S</h:mi><h:mi>A</h:mi><h:mi>X</h:mi></h:msubsup><h:mo>+</h:mo><h:msubsup><h:mi>S</h:mi><h:mi>B</h:mi><h:mi>X</h:mi></h:msubsup><h:mo>−</h:mo><h:msubsup><h:mi>S</h:mi><h:mrow><h:mi>A</h:mi><h:mi>B</h:mi></h:mrow><h:mi>X</h:mi></h:msubsup></h:mrow></h:math>, which is a lower bound on <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:msubsup><i:mi>S</i:mi><i:mrow><i:mi>A</i:mi></i:mrow><i:mrow><i:mi>v</i:mi><i:mi>N</i:mi></i:mrow></i:msubsup></i:math>. We show that for a broad range of lattice spacing and detuning, <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:msubsup><j:mi>I</j:mi><j:mrow><j:mi>A</j:mi><j:mi>B</j:mi></j:mrow><j:mi>X</j:mi></j:msubsup></j:math> is typically 20% below <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:msubsup><k:mi>S</k:mi><k:mrow><k:mi>A</k:mi></k:mrow><k:mrow><k:mi>v</k:mi><k:mi>N</k:mi></k:mrow></k:msubsup></k:math> in regions where <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:msubsup><l:mi>S</l:mi><l:mrow><l:mi>A</l:mi></l:mrow><l:mrow><l:mi>v</l:mi><l:mi>N</l:mi></l:mrow></l:msubsup></l:math> is large and a less close bound in regions where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msubsup><m:mi>S</m:mi><m:mrow><m:mi>A</m:mi></m:mrow><m:mrow><m:mi>v</m:mi><m:mi>N</m:mi></m:mrow></m:msubsup></m:math> is low. We argue that this use of the easily available bitstrings provides a robust and efficient way to explore empirically the phase diagram of qubit-based quantum simulators and identify critical regions.