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 $n\le 4$. 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 $n$.

21 pages, 3 figures. Comments welcome

References in corpus (14)

• Quantum state discrimination and its applications
• Is the quantum state real? An extended review of $\psi$-ontology theorems
• Qudit versions of the qubit "pi-over-eight" gate
• One-Shot Operational Quantum Resource Theory
• On the Optimality of Square Root Measurements in Quantum State Discrimination
• Exact Identification of a Quantum Change Point
• Imperfect 1-out-of-2 quantum oblivious transfer: bounds, a protocol, and its experimental implementation
• Quantum Advantage in Information Retrieval
• Noncontextuality Inequalities from Antidistinguishability
• On the optimal error exponents for classical and quantum antidistinguishability
• Deterministic transformations of multilevel coherent states under incoherence-preserving operations
• Necessary and sufficient optimality conditions for classical simulations of quantum communication processes
• A note on the inner products of pure states and their antidistinguishability
• Absolutely $k$-Incoherent Quantum States and Spectral Inequalities for Factor Width of a Matrix

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• Single-shot antidistinguishability of unitary operations