Finding out all locally indistinguishable sets of generalized Bell states

In general, for a bipartite quantum system <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>d</mml:mi></mml:mrow></mml:msup><mml:mo>&amp;#x2297;</mml:mo><mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:math> and an integer <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math> such that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>4</mml:mn><mml:mo>&amp;#x2264;</mml:mo><mml:mi>k</mml:mi><mml:mo>&amp;#x2264;</mml:mo><mml:mi>d</mml:mi></mml:math>,there are few necessary and sufficient conditions for local discrimination of sets of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math> generalized Bell states (GBSs) and it is difficult to locally distinguish <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math>-GBS sets.The purpose of this paper is to completely solve the problem of local discrimination of GBS sets in some bipartite quantum systems.Firstly three practical and effective sufficient conditions are given,Fan<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi /><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>,</mml:mo></mml:mrow></mml:msup></mml:math>s and Wang et al.<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi /><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>,</mml:mo></mml:mrow></mml:msup></mml:math>s results [Phys Rev Lett 92, 177905 (2004); Phys Rev A 99, 022307 (2019)] can be deduced as special cases of these conditions.Secondly in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mo>&amp;#x2297;</mml:mo><mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:math>, a necessary and sufficient condition for local discrimination of GBS sets is provided, and a list of all locally indistinguishable 4-GBS sets is provided,and then the problem of local discrimination of GBS sets is completely<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>5</mml:mn></mml:mrow></mml:msup><mml:mo>&amp;#x2297;</mml:mo><mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>5</mml:mn></mml:mrow></mml:msup></mml:math>, a concise necessary and sufficient condition for one-way local discrimination of GBS sets is obtained,which gives an affirmative answer to the case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>5</mml:mn></mml:math> of the problem proposed by Wang et al.