Construction methods of nonlocal sets of orthogonal product states on multipartite quantum systems

Abstract Nonlocal set of orthogonal product states (OPSs) can improve the confidentiality of information when it is used to design quantum cryptographic protocols. It is a difficult question how to construct a nonlocal set of OPSs on general multipartite and high dimensional quantum systems. Different from the previous works, we first present a novel method for constructing a nonlocal product set with 3 d – 2 members on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>⊗</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>⊗</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> quantum system for d ≥ 3. Then, we extend this construction method to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mrow> <mml:msub> <mml:mi>d</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:msup> <mml:mo>⊗</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mrow> <mml:msub> <mml:mi>d</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:msup> <mml:mo>⊗</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mrow> <mml:msub> <mml:mi>d</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> quantum system and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mo>⊗</mml:mo> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>n</mml:mi> </mml:msubsup> <mml:msup> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mrow> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> quantum system respectively, where 3 ≤ d 1 ≤ d 2 ≤ d 3 ≤ … ≤ d n and n ≥ 3. The nonlocal set of OPSs constructed by our method contains fewer elements than those constructed by the existing methods, except for one special case. More importantly, the set of states constructed by our method has a completely different structure from those constructed by the existing methods since our nonlocal set does not contain a “stopper” state. Our result is helpful to further understand the different structures of nonlocal sets on multipartite systems.