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- W2023707592 abstract "A collection of pairwise mutually unbiased bases (in short: MUB) in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d greater-than 1> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>d>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> dimensions may consist of at most <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d plus 1> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>d+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bases. Such “complete” collections are known to exist in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper C Superscript d> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>C</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {C}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding=application/x-tex>d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a power of a prime. However, in general, little is known about the maximum number <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N left-parenthesis d right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>N(d)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of bases that a collection of MUB in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper C Superscript d> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>C</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {C}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can have. In this work it is proved that a collection of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding=application/x-tex>d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> MUB in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper C Superscript d> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>C</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {C}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can always be completed. Hence <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N left-parenthesis d right-parenthesis not-equals d> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:mi>d</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>N(d)neq d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d greater-than 1> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>d>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we have a dichotomy: <italic>either</italic> <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N left-parenthesis d right-parenthesis equals d plus 1> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>N(d)=d+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (so that there exists a complete collection of MUB) <italic>or</italic> <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N left-parenthesis d right-parenthesis less-than-or-equal-to d minus 1> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>d</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>N(d)leq d-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the course of the proof an interesting new characterization is given for a linear subspace of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M Subscript d Baseline left-parenthesis double-struck upper C right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>C</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>M_d(mathbb {C})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to be a subalgebra." @default.
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- W2023707592 date "2013-01-23" @default.
- W2023707592 modified "2023-10-17" @default.
- W2023707592 title "A gap for the maximum number of mutually unbiased bases" @default.
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