Novel method for one-way local distinguishability of generalized Bell states in arbitrary dimension

Abstract In this paper the local distinguishability of generalized Bell states in arbitrary dimension is investigated. We firstly study the decomposition of a basis which consists of d 2 number of generalized Pauli matrices. We discover that this basis is equal to the union of D number of different sets, where <?CDATA $D=\frac{2}{\phi (d)}{\sum }_{\genfrac{}{}{0pt}{}{t\in {\mathbb{Z}}_{d}}{\mathrm{gcd}(t,d)=1}}{\sum }_{i=2}^{\lfloor \frac{d}{t}\rfloor }\enspace \phi (i)+1$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> </mml:mfrac> <mml:msub> <mml:mrow> <mml:mo>∑</mml:mo> </mml:mrow> <mml:mrow> <mml:mfrac linethickness="0pt"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mrow> <mml:mi>gcd</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:msub> <mml:msubsup> <mml:mrow> <mml:mo>∑</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mrow> <mml:mo>⌊</mml:mo> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:mfrac> </mml:mrow> <mml:mo>⌋</mml:mo> </mml:mrow> </mml:mrow> </mml:msubsup> <mml:mspace width="0.17em" /> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>i</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:math> and ϕ is Euler ϕ -function. Then we define the generator of the matrices in this decomposition, and exhibit an algorithm to calculate generators of a given set of matrices. This algorithm shows that generators of a given set can be calculated simply and efficiently. Secondly, we show that a set <?CDATA $\mathcal{L}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">L</mml:mi> </mml:math> of GBSs can be distinguished by one-way LOCC if the cardinality of <?CDATA ${\mathcal{G}}_{\mathcal{L}}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="script">G</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="script">L</mml:mi> </mml:mrow> </mml:msub> </mml:math> is less than Dϕ ( d ), where <?CDATA ${\mathcal{G}}_{\mathcal{L}}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="script">G</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="script">L</mml:mi> </mml:mrow> </mml:msub> </mml:math> is a set of generators of all the elements in difference set of a set <?CDATA $\mathcal{L}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">L</mml:mi> </mml:math> of GBSs. The previous results in [2004 Phys. Rev. Lett. 92 177905; 2019 Phys. Rev. A 99 022307; 2021 Quantum Inf. Process. 20 52] can be covered by our result. Finally, for the uncovered cases in [2021 Quantum Inf. Process. 20 52], we give a new result to partly solve that problem.