Single-shot antidistinguishability of unitary operations

TL;DR

This work investigates the single-shot antidistinguishability of unitary operations, focusing on sets of three unitaries under two probing schemes: single-system and entangled probes. It reduces antidistinguishability to the problem of distinguishing evolved probe states and proves that maximally entangled probes provide a universal, dimension-dependent resource for the task, fully characterizing the qubit case and revealing breakdowns in higher dimensions. The authors establish closure properties for antidistinguishable sets, provide methods to construct antidistinguishable sets from non-antidistinguishable ones, and supply concrete counterexamples in dimension $d\ge 3$. These results extend antidistinguishability from quantum states to unitary channels and motivate further study of higher-dimensional and multi-shot scenarios. The work deepens our understanding of how entanglement and probe choices influence the negative identification of quantum processes.

Abstract

The notion of antidistinguishability captures the possibility of ruling out certain alternatives in a quantum experiment without identifying the actual outcome. Although extensively studied for quantum states, the antidistinguishability of quantum channels remains largely unexplored. In this work, we investigate the single-shot antidistinguishability of unitary operations. We analyse two scenarios: antidistinguishability with single-system probes and with entangled probes. For sets of three unitaries, we first prove that all maximally entangled states are equivalent in their performance as probe. In the qubit case, we further establish that maximally entangled probes are always sufficient: if a set of three qubit unitaries is antidistinguishable with either a single-system or non-maximally entangled probe, then it is also antidistinguishable with a maximally entangled one. However, in higher dimension, this equivalence fails. In \textit{dimension 3}, there exists a set of unitaries that are antidistinguishable with non-maximally entangled probe or single-system probe but not with maximally entangled probe. We also establish that union of two antidistinguishable sets of three qubit unitaries also forms a set of antidistinguishable unitaries. Lastly, we provide methods to construct antidistinguishable unitaries from non-antidistinguishable ones.

Theorems & Definitions (20)

• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof