## Tight bounds for antidistinguishability and circulant sets of pure quantum states

**Abstract:**

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 n pure states is equivalent to a property of its Gram matrix called (n−1)-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 sqrt((n−2)/(2n−2)) then the set must be antidistinguishable, and we show that this bound is tight when n ≤ 4. We also give a simpler proof that if all the inner products are strictly larger than (n−2)/(n−1), then the set cannot be antidistinguishable, and we show that this bound is tight for all n.

**Authors:**

- Nathaniel Johnston
- Vincent Russo
- Jamie Sikora

** ****Download:**

- Preprint from arXiv:2311.17047 [quant-ph]
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** ****Cite as:**

- N. Johnston, V. Russo, and J. Sikora.
*Tight bounds for antidistinguishability and circulant sets of pure quantum states*. E-print: arXiv:2311.17047 [quant-ph], 2023.