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 $n$ pure states is equivalent to a property of its Gram matrix called $(n-...
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Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2025-02-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2025-02-04-1622/pdf/ |
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| author | Nathaniel Johnston Vincent Russo Jamie Sikora |
| author_facet | Nathaniel Johnston Vincent Russo Jamie Sikora |
| author_sort | Nathaniel Johnston |
| collection | DOAJ |
| description | 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 $n \leq 4$. 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 $n$. |
| format | Article |
| id | doaj-art-abc8ac636c1447359775d011c7e0fe36 |
| institution | Kabale University |
| issn | 2521-327X |
| language | English |
| publishDate | 2025-02-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj-art-abc8ac636c1447359775d011c7e0fe362025-02-04T17:08:26ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2025-02-019162210.22331/q-2025-02-04-162210.22331/q-2025-02-04-1622Tight bounds for antidistinguishability and circulant sets of pure quantum statesNathaniel JohnstonVincent RussoJamie SikoraA 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 $n \leq 4$. 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 $n$.https://quantum-journal.org/papers/q-2025-02-04-1622/pdf/ |
| spellingShingle | Nathaniel Johnston Vincent Russo Jamie Sikora Tight bounds for antidistinguishability and circulant sets of pure quantum states Quantum |
| title | Tight bounds for antidistinguishability and circulant sets of pure quantum states |
| title_full | Tight bounds for antidistinguishability and circulant sets of pure quantum states |
| title_fullStr | Tight bounds for antidistinguishability and circulant sets of pure quantum states |
| title_full_unstemmed | Tight bounds for antidistinguishability and circulant sets of pure quantum states |
| title_short | Tight bounds for antidistinguishability and circulant sets of pure quantum states |
| title_sort | tight bounds for antidistinguishability and circulant sets of pure quantum states |
| url | https://quantum-journal.org/papers/q-2025-02-04-1622/pdf/ |
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