Constructions of locally distinguishable sets of maximally entangled states which require two-way LOCC

Abstract Most of known locally distinguishable sets of maximally entangled states (MESs) can be perfectly distinguished with one-way local operations and classical communication (1-LOCC), and there are not many known locally distinguishable sets which require two-way LOCC (2-LOCC). Recently some such sets of MESs have been built, that is, they can be locally distinguished only with 2-LOCC. In this work, we show that there exist a lot of locally distinguishable MESs which require 2-LOCC (denote such sets by 2-LOCC sets). Firstly generalized Bell states are employed to construct lots of 1-LOCC indistinguishable sets of MESs in the quantum systems <?CDATA ${\mathbb{C}}^{a+\left(a+1\right)r}\otimes {\mathbb{C}}^{a+\left(a+1\right)r}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> </mml:msup> <mml:mo>⊗</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> </mml:msup> </mml:math> and <?CDATA ${\mathbb{C}}^{a+\left(a+2\right)r}\otimes {\mathbb{C}}^{a+\left(a+2\right)r}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> </mml:msup> <mml:mo>⊗</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> </mml:msup> </mml:math> . Then we build a lot of 2-LOCC sets by using Weyl commutation relation and two constructive methods. Especially, it is shown that there are at least 8 and 105 2-LOCC sets in quantum systems <?CDATA ${\mathbb{C}}^{10}\otimes {\mathbb{C}}^{10}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>10</mml:mn> </mml:mrow> </mml:msup> <mml:mo>⊗</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>10</mml:mn> </mml:mrow> </mml:msup> </mml:math> and <?CDATA ${\mathbb{C}}^{19}\otimes {\mathbb{C}}^{19}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>19</mml:mn> </mml:mrow> </mml:msup> <mml:mo>⊗</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>19</mml:mn> </mml:mrow> </mml:msup> </mml:math> respectively.