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.

References in corpus (30)

• Entanglement detection
• Entropic Uncertainty Relations and their Applications
• The Bloch Vector for N-Level Systems
• Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement
• Violation of local uncertainty relations as a signature of entanglement
• Security in Quantum Cryptography
• Entropic uncertainty relations - A survey
• Graph-Theoretic Approach to Quantum Correlations
• Stronger uncertainty relations for the sum of variances
• Entanglement Detection in the Stabilizer Formalism
• Characterization of the Positivity of the Density Matrix in Terms of the Coherence Vector Representation
• Quantum root-mean-square error and measurement uncertainty relations
• Entropic uncertainty relations and entanglement
• Heisenberg uncertainty for qubit measurements
• Entanglement criteria based on local uncertainty relations are strictly stronger than the computable cross norm criterion
• Steering criteria from general entropic uncertainty relations
• Correlation complementarity yields Bell monogamy relations
• State-independent Uncertainty Relations and Entanglement Detection in Noisy Systems
• Entropic uncertainty from effective anti-commutators
• Higher entropic uncertainty relations for anti-commuting observables
• Restricted numerical range: a versatile tool in the theory of quantum information
• Tight state-independent uncertainty relations for qubits
• Local commutativity versus Bell inequality violation for entangled states and versus non-violation for separable states
• Symmetries in quantum networks lead to no-go theorems for entanglement distribution and to verification techniques
• Classification of joint numerical ranges of three hermitian matrices of size three
• Entanglement detection with imprecise measurements
• The Significance of the $C$-Numerical Range and the Local $C$-Numerical Range in Quantum Control and Quantum Information
• Calibration robust entanglement detection beyond Bell inequalities
• State independent uncertainty relations from eigenvalue minimization
• Entropic uncertainty relations and the stabilizer formalism

Cited by in corpus (12)

• Semi-definite programming and quantum information
• Triply efficient shadow tomography
• Bounding the joint numerical range of Pauli strings by graph parameters
• 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