Tight bounds for antidistinguishability and circulant sets of pure quantum states

A set of pure quantum states is said to be antidistinguishable if upon sampling one at random, there exists a measurement to perfectly determine some state that was not sampled. We show that antidistinguishability of a set of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math> pure states is equivalent to a property of its Gram matrix called <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>-incoherence, thus establishing a connection with quantum resource theories that lets us apply a wide variety of new tools to antidistinguishability. As a particular application of our result, we present an explicit formula (not involving any semidefinite programming) that determines whether or not a set with a circulant Gram matrix is antidistinguishable. We also show that if all inner products are smaller than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msqrt><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:msqrt></mml:math> then the set must be antidistinguishable, and we show that this bound is tight when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi><mml:mo>&amp;#x2264;</mml:mo><mml:mn>4</mml:mn></mml:math>. We also give a simpler proof that if all the inner products are strictly larger than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>, then the set cannot be antidistinguishable, and we show that this bound is tight for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>.