Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Strong antidistinguishability does not imply maximal contextuality

HyunHo Cha

Published Jun 11, 2026• 0 citations
Publisher

Abstract

Srikumar et al. introduced a notion of contextuality for finite sets of pure quantum states and proved that maximal contextuality implies strong antidistinguishability. They conjectured the converse, namely that every strongly antidistinguishable non-orthogonal set of pure states is maximally contextual. We give a counterexample in the four-dimensional Hilbert space.

DOI
10.5281/zenodo.20645144

Related papers

How Contextuality and Antidistinguishability Are Related

Maiyuren Srikumar, Stephen D. Bartlett, Angela Karanjai

Contextuality is a key characteristic that separates quantum from classical phenomena and an important tool in understanding the potential advantage of quantum computation. However, when assessing the quantum resources available for quantum information processing, there is no formalism to determine whether a set of states can exhibit contextuality and whether such proofs of contextuality indicate anything about the resourcefulness of that set. Introducing a well-motivated notion of what it means for a set of states to be contextual, we establish a relationship between contextuality and antidistinguishability of sets of states. We go beyond the traditional notions of contextuality and antidistinguishability and treat both properties as resources, demonstrating that the degree of contextuality within a set of states has a direct connection to its level of antidistinguishability. If a set of states is contextual, then it must be weakly antidistinguishable and vice versa. However, critical contextuality emerges as a stronger property than traditional antidistinguishability.

arXiv:2411.09919•Physical Review Letters•Feb 2, 2026•1 citation

Inner products of pure states and their antidistinguishability

Vincent Russo, Jamie Sikora

A set of d quantum states is said to be antidistinguishable if there exists a d-outcome POVM that can perfectly identify which state was not measured. A conjecture by Havlíček and Barrett states that if a set of d pure states has small pair-wise inner products, then the set must be antidistinguishable. In this note we provide a certificate of antidistinguishability via semidefinite programming duality and use it to provide a counterexample to this conjecture when d = 4.

arXiv:2206.08313•Physical Review A•Mar 27, 2023•8 citations

Necessary and Sufficient Condition for State-Independent Contextual Measurement Scenarios

Ravishankar Ramanathan, Paweł Horodecki

The problem of identifying measurement scenarios capable of revealing state-independent contextuality in a given Hilbert space dimension is considered. We begin by showing that for any given dimension d and any measurement scenario consisting of projective measurements, (i) the measure of contextuality of a quantum state is entirely determined by its spectrum, so that pure and maximally mixed states represent the two extremes of contextual behavior, and that (ii) state-independent contextuality is equivalent to the contextuality of the maximally mixed state up to a global unitary transformation. We then derive a necessary and sufficient condition for a measurement scenario represented by an orthogonality graph to reveal state-independent contextuality. This condition is given in terms of the fractional chromatic number of the graph χf(G) and is shown to identify all state-independent contextual measurement scenarios including those that go beyond the original Kochen-Specker paradigm.

arXiv:1212.5933•Physical Review Letters•Jan 29, 2014•36 citations

Antidistinguishability of pure quantum states

Teiko Heinosaari, Oskari Kerppo

Abstract The Pusey–Barrett–Rudolph theorem has recently provoked a lot of discussion regarding the reality of the quantum state. In this article we focus on a property called antidistinguishability, which is a main component in constructing the proof for the PBR theorem. In particular we study algebraic conditions for a set of pure quantum states to be antidistinguishable, and a novel sufficient condition is presented. We also discuss a more general criterion which can be used to show that the sufficient condition is not necessary. Lastly, we consider how many quantum states needs to be added into a set of pure quantum states in order to make the set antidistinguishable. It is shown that in the case of qubit states the answer is one, while in the general but finite dimensional case the answer is at most n , where n is the size of the original set.

arXiv:1804.10457•Journal of Physics A Mathematical and Theoretical•Jul 9, 2018•29 citations

Single-shot antidistinguishability of unitary operations

Satyaki Manna, Anandamay Das Bhowmik

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.

arXiv:2510.14609•Physical Review A•Jan 29, 2026•1 citation

State-independent quantum contextuality and maximum nonlocality

Adán Cabello

Recently, Yu and Oh [Phys. Rev. Lett. 108, 030402 (2012)] have conjectured that the simplest set of vectors needed to prove state-independent contextuality on a qutrit requires 13 vectors. Here we first prove that a necessary and sufficient condition for a set of vectors in any dimension $d \ge 3$ to prove contextuality for a system prepared in a maximally mixed state is that the graph $G_C$ in which vertices represent vectors and edges link orthogonal ones has chromatic number $χ(G_G)$ larger than $d$. Then, we prove Yu and Oh's conjecture. Finally, we prove that any set satisfying $χ(G_G)>d$ assisted with $d$-party maximum entanglement generates nonlocality which cannot be improved without violating the no-signalling principle. This shows that any set satisfying $χ(G_G)>d$ is a valuable resource for quantum information processing.

arXiv:1112.5149•Dec 21, 2011•10 citations
Srikumar et al. introduced a notion of contextuality for finite sets of pure quantum states and proved that maximal contextuality implies strong antidistinguishability. They conjectured the converse, namely that every strongly antidistinguishable non-orthogonal set of pure states is maximally contextual. We give a counterexample in the four-dimensional Hilbert space.
Contextuality is a key characteristic that separates quantum from classical phenomena and an important tool in understanding the potential advantage of quantum computation. However, when assessing the quantum resources available for quantum information processing, there is no formalism to determine whether a set of states can exhibit contextuality and whether such proofs of contextuality indicate anything about the resourcefulness of that set. Introducing a well-motivated notion of what it means for a set of states to be contextual, we establish a relationship between contextuality and antidistinguishability of sets of states. We go beyond the traditional notions of contextuality and antidistinguishability and treat both properties as resources, demonstrating that the degree of contextuality within a set of states has a direct connection to its level of antidistinguishability. If a set of states is contextual, then it must be weakly antidistinguishable and vice versa. However, critical contextuality emerges as a stronger property than traditional antidistinguishability.
The problem of identifying measurement scenarios capable of revealing state-independent contextuality in a given Hilbert space dimension is considered. We begin by showing that for any given dimension d and any measurement scenario consisting of projective measurements, (i) the measure of contextuality of a quantum state is entirely determined by its spectrum, so that pure and maximally mixed states represent the two extremes of contextual behavior, and that (ii) state-independent contextuality is equivalent to the contextuality of the maximally mixed state up to a global unitary transformation. We then derive a necessary and sufficient condition for a measurement scenario represented by an orthogonality graph to reveal state-independent contextuality. This condition is given in terms of the fractional chromatic number of the graph χf(G) and is shown to identify all state-independent contextual measurement scenarios including those that go beyond the original Kochen-Specker paradigm.
Abstract The Pusey–Barrett–Rudolph theorem has recently provoked a lot of discussion regarding the reality of the quantum state. In this article we focus on a property called antidistinguishability, which is a main component in constructing the proof for the PBR theorem. In particular we study algebraic conditions for a set of pure quantum states to be antidistinguishable, and a novel sufficient condition is presented. We also discuss a more general criterion which can be used to show that the sufficient condition is not necessary. Lastly, we consider how many quantum states needs to be added into a set of pure quantum states in order to make the set antidistinguishable. It is shown that in the case of qubit states the answer is one, while in the general but finite dimensional case the answer is at most n , where n is the size of the original set.
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.
Recently, Yu and Oh [Phys. Rev. Lett. 108, 030402 (2012)] have conjectured that the simplest set of vectors needed to prove state-independent contextuality on a qutrit requires 13 vectors. Here we first prove that a necessary and sufficient condition for a set of vectors in any dimension d≥3d \ge 3d≥3 to prove contextuality for a system prepared in a maximally mixed state is that the graph GCG_CGC​ in which vertices represent vectors and edges link orthogonal ones has chromatic number χ(GG)χ(G_G)χ(GG​) larger than ddd. Then, we prove Yu and Oh's conjecture. Finally, we prove that any set satisfying χ(GG)>dχ(G_G)>dχ(GG​)>d assisted with ddd-party maximum entanglement generates nonlocality which cannot be improved without violating the no-signalling principle. This shows that any set satisfying χ(GG)>dχ(G_G)>dχ(GG​)>d is a valuable resource for quantum information processing.