Minimum trace norm of real symmetric and Hermitian matrices with zero diagonal

We establish tight lower bounds for the trace norm <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>‖</m:mo> <m:mo>⋅</m:mo> <m:msub> <m:mrow> <m:mo>‖</m:mo> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> ( · { }₁) of real symmetric and Hermitian matrices with zero diagonal entries in terms of their entrywise <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> {L}¹ -norms <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>‖</m:mo> <m:mo>⋅</m:mo> <m:msub> <m:mrow> <m:mo>‖</m:mo> </m:mrow> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> ( · { }₍₁₎) . For the space of nonzero real symmetric matrices of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> </m:math> n , we prove that the minimum possible ratio <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfrac> <m:mrow> <m:mo>‖</m:mo> <m:mi>A</m:mi> <m:msub> <m:mrow> <m:mo>‖</m:mo> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mrow> <m:mo>‖</m:mo> <m:mi>A</m:mi> <m:msub> <m:mrow> <m:mo>‖</m:mo> </m:mrow> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msub> </m:mrow> </m:mfrac> </m:math> { A{ }₁}{ A{ }₍₁₎} is exactly <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfrac> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:mfrac> </m:math> 2/n . In the Hermitian case, this minimum ratio is given by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>tan</m:mi> <m:mfenced open="(" close=")"> <m:mrow> <m:mfrac> <m:mrow> <m:mi>π</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mi>n</m:mi> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> </m:math> tan ({[-0.75em]{}{0ex}},π /2n) . Through duality arguments, we derive sharp upper bounds for the spectral norm distance between a matrix and the space of diagonal matrices. For instance, any real symmetric matrix with off-diagonal entries bounded by 1 in absolute value lies within a spectral norm distance of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfrac> <m:mrow> <m:mi>n</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:mfrac> </m:math> n/2 from a diagonal matrix, while the corresponding bound for Hermitian matrices is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>cot</m:mi> <m:mfenced open="(" close=")"> <m:mrow> <m:mfrac> <m:mrow> <m:mi>π</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mi>n</m:mi> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> </m:math> ({[-0.75em]{}{0ex}},π /2n) . Applications to graph energy and quantum coherence are discussed, highlighting implications for algebraic graph theory and quantum resource theory.