Uncertainty relations from graph theory

Abstract

Quantum measurements are inherently probabilistic and quantum theory often forbids to precisely predict the outcomes of simultaneous measurements. This phenomenon is captured and quantified through uncertainty relations. Although studied since the inception of quantum theory, the problem of determining the possible expectation values of a collection of quantum measurements remains, in general, unsolved. By constructing a close connection between observables and graph theory, we derive uncertainty relations valid for any set of dichotomic observables. These relations are, in many cases, tight, and related to the size of the maximum clique of the associated graph. As applications, our results can be straightforwardly used to formulate entropic uncertainty relations, separability criteria and entanglement witnesses.

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• Triply efficient shadow tomography
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• Uncertainty relations from state polynomial optimization
• Bounds on the ground state energy of quantum $p$-spin Hamiltonians
• Imprecision plateaus in quantum steering
• An Effective Way to Determine the Separability of Quantum State
• Bound entanglement-assisted prepare-and-measure scenarios based on four-dimensional quantum messages
• Optimizing Sparse SYK
• Positivity of state, trace, and moment polynomials, and applications in quantum information
• Multiqubit monogamy relations beyond shadow inequalities
• Frustration graph formalism for qudit observables