Joint measurability structures realizable with qubit measurements: incompatibility via marginal surgery

Abstract

Measurements in quantum theory exhibit incompatibility, i.e., they can fail to be jointly measurable. An intuitive way to represent the (in)compatibility relations among a set of measurements is via a hypergraph representing their joint measurability structure: its vertices represent measurements and its hyperedges represent (all and only) subsets of compatible measurements. Projective measurements in quantum theory realize (all and only) joint measurability structures that are graphs. On the other hand, general measurements represented by positive operator-valued measures (POVMs) can realize arbitrary joint measurability structures. Here we explore the scope of joint measurability structures realizable with qubit POVMs. We develop a technique that we term marginal surgery to obtain nontrivial joint measurability structures starting from a set of compatible measurements. We show explicit examples of marginal surgery on a special set of qubit POVMs to construct joint measurability structures such as $N$-cycle and $N$-Specker scenarios for any integer $N\ge 3$. We also show the realizability of various joint measurability structures with $N\in \{4,5,6\}$ vertices. In particular, we show that all possible joint measurability structures with $N=4$ vertices are realizable. We conjecture that all joint measurability structures are realizable with qubit POVMs. This contrasts with the unbounded dimension required in R. Kunjwal et al., Phys. Rev. A 89, 052126 (2014). Our results also render this previous construction maximally efficient in terms of the required Hilbert space dimension. We also obtain a sufficient condition for the joint measurability of any set of binary qubit POVMs which powers many of our results and should be of independent interest.

arXiv comment37 pages, 32 figures, 1 table, comments welcome!

References in corpus (34)

• Bell nonlocality
• Contextuality for preparations, transformations, and unsharp measurements
• Joint Measurability, Einstein-Podolsky-Rosen Steering, and Bell Nonlocality
• Specker's Parable of the Over-protective Seer: A Road to Contextuality, Nonlocality and Complementarity (PLUS AN ERRATUM)
• An Invitation to Quantum Incompatibility
• Joint measurability of generalized measurements implies classicality
• Almost quantum correlations
• An experimental test of noncontextuality without unwarranted idealizations
• Measurements incompatible in Quantum Theory cannot be measured jointly in any other local theory
• A Combinatorial Approach to Nonlocality and Contextuality
• Quantifying quantum resources with conic programming
• All sets of incompatible measurements give an advantage in quantum state discrimination
• Quantatitive relations between measurement incompatibility, quantum steering, and nonlocality
• Quantum Incompatibility Witnesses
• Notes on Joint Measurability of Quantum Observables
• From the Kochen-Specker theorem to noncontextuality inequalities without assuming determinism
• A Complete Resource Theory of Quantum Incompatibility as Quantum Programmability
• Operational relevance of resource theories of quantum measurements
• Degree of Complementarity Determines the Nonlocality in Quantum Mechanics
• Verifying the quantumness of a channel with an untrusted device
• Measurement incompatibility does not give rise to Bell violation in general
• The status of determinism in proofs of the impossibility of a noncontextual model of quantum theory
• Quantum realization of arbitrary joint measurability structures
• Steering, incompatibility, and Bell inequality violations in a class of probabilistic theories
• Adaptive strategy for joint measurements
• From statistical proofs of the Kochen-Specker theorem to noise-robust noncontextuality inequalities
• Experimental generalized contextuality with single-photon qubits
• Minimal state-dependent proof of measurement contextuality for a qubit
• A necessary condition for incompatibility of observables in general probabilistic theories
• Approximate joint measurement of qubit observables through an Arthur-Kelly type model
• Beyond the Cabello-Severini-Winter framework: Making sense of contextuality without sharpness of measurements
• Almost Quantum Correlations are Inconsistent with Specker's Principle
• Quantum theory realises all joint measurability graphs
• Hypergraph framework for irreducible noncontextuality inequalities from logical proofs of the Kochen-Specker theorem

Cited by in corpus (6)

• Incompatible measurements in quantum information science
• Contextuality in entanglement-assisted one-shot classical communication
• Order preserving maps on quantum measurements
• Constrained Measurement Incompatibility from Generalised Contextuality of Steered Preparation
• Necessary and Sufficient Condition for Randomness Certification from Incompatibility
• On compatibility of binary qubit measurements