An equality between entanglement and uncertainty

Abstract

Heisenberg's uncertainty principle implies that if one party (Alice) prepares a system and randomly measures one of two incompatible observables, then another party (Bob) cannot perfectly predict the measurement outcomes. This implication assumes that Bob does not possess an additional system that is entangled to the measured one; indeed the seminal paper of Einstein, Podolsky and Rosen (EPR) showed that maximal entanglement allows Bob to perfectly win this guessing game. Although not in contradiction, the observations made by EPR and Heisenberg illustrate two extreme cases of the interplay between entanglement and uncertainty. On the one hand, no entanglement means that Bob's predictions must display some uncertainty. Yet on the other hand, maximal entanglement means that there is no more uncertainty at all. Here we follow an operational approach and give an exact relation - an equality - between the amount of uncertainty as measured by the guessing probability, and the amount of entanglement as measured by the recoverable entanglement fidelity. From this equality we deduce a simple criterion for witnessing bipartite entanglement and a novel entanglement monogamy equality.

v2: published as "Entanglement-assisted guessing of complementary measurement outcomes", 11 pages, 1 figure

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• Quantum Information Processing with Finite Resources -- Mathematical Foundations
• Entanglement sampling and applications
• Quantum-memory-assisted entropic uncertainty relations
• Relating different quantum generalizations of the conditional Renyi entropy
• Quantum uncertainty relation using coherence
• Reformulation of the No-Free-Lunch Theorem for Entangled Data Sets
• Quantum Achievability Proof via Collision Relative Entropy
• Tightening the tripartite quantum memory assisted entropic uncertainty relation
• Renyi relative entropies of quantum Gaussian states
• The Conditional Uncertainty Principle
• Separability criteria via sets of mutually unbiased measurements
• Uncertainty, joint uncertainty, and the quantum uncertainty principle
• Pretty good measures in quantum information theory
• Certainty relations, mutual entanglement and non-displacable manifolds
• Entropic uncertainty relations with quantum memory in a multipartite scenario
• Quantum Side Information: Uncertainty Relations, Extractors, Channel Simulations
• Uncertainty Relations in the Presence of Quantum Memory for Mutually Unbiased Measurements
• Multipartite entanglement detection based on generalized state-dependent entropic uncertainty relation for multiple measurements
• Experimental investigation of uncertainty relations for non-Hermitian operators
• Quantum preparation uncertainty and lack of information
• Tripartite quantum-memory-assisted entropic uncertainty relations for multiple measurements
• Characterization of measurement uncertainties using the correlations between local outcomes obtained from maximally entangled pairs
• Better bounds on optimal measurement and entanglement recovery, with applications to uncertainty and monogamy relations
• Experimental investigation of quantum uncertainty relations with classical shadows
• Enhanced quantum channel uncertainty relations by skew information
• Tightening the entropic uncertainty relations with quantum memory in a multipartite scenario
• Average of uncertainty-product for bounded observables
• Loss of Information in Quantum Guessing Game
• General framework of quantum complementarity from a measurement-based perspective
• Tight bounds from multiple-observable entropic uncertainty relations
• Biphoton phase-space correlations from Gouy-phase measurements using double slits
• Constrained dynamics of maximally entangled bipartite system
• Multiparticle Entanglement Resolution Analyzer based on Quantum-Control-Assisted Uncertainty Relation
• Stronger sum uncertainty relations for non-Hermitian operators
• The quantum uncertainty relations of quantum channels
• Experimental investigation of majorization uncertainty relations in the high-dimensional systems
• Uncertainty relation and the constrained quadratic programming
• Tighter entropic uncertainty relations in the presence of quantum memories for complete sets of mutually unbiased bases
• The uncertainty of quantum states with respect to the projective measurement