Convex geometry of quantum resource quantification

Abstract

We introduce a framework unifying the mathematical characterisation of different measures of general quantum resources and allowing for a systematic way to define a variety of faithful quantifiers for any given convex quantum resource theory. The approach allows us to describe many commonly used measures such as matrix norm-based quantifiers, robustness measures, convex roof-based measures, and witness-based quantifiers together in a common formalism based on the convex geometry of the underlying sets of resource-free states. We establish easily verifiable criteria for a measure to possess desirable properties such as faithfulness and strong monotonicity under relevant free operations, and show that many quantifiers obtained in this framework indeed satisfy them for any considered quantum resource. We derive various bounds and relations between the measures, generalising and providing significantly simplified proofs of results found in the resource theories of quantum entanglement and coherence. We also prove that the quantification of resources in this framework simplifies for pure states, allowing us to obtain more easily computable forms of the considered measures and show that several of them are equal on pure states. Further, we investigate the dual formulation of resource quantifiers, characterising sets of resource witnesses. We present an explicit application of the results to the resource theories of multi-level coherence, entanglement of Schmidt number k, multipartite entanglement, as well as magic states, providing insight into the quantification of the resources and introducing new quantifiers, such as a measure of entanglement of Schmidt number k which generalises the convex roof-extended negativity, a measure of k-coherence which generalises the L1 norm of coherence, and a hierarchy of norm-based quantifiers of k-partite entanglement generalising the greatest cross norm.

45 pages, 2 figures. v6: section 4.1 rewritten to simplify proofs and correct inconsistencies; theorem numbering consistent with published version

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• Quantum Resource Theories
• Simulation of quantum circuits by low-rank stabilizer decompositions
• General Resource Theories in Quantum Mechanics and Beyond: Operational Characterization via Discrimination Tasks
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• Quantifying quantum resources with conic programming
• Operational Advantage of Quantum Resources in Subchannel Discrimination
• Lower bounds on the non-Clifford resources for quantum computations
• One-shot coherence distillation
• Quantifying quantum speedups: improved classical simulation from tighter magic monotones
• Stabilizer entropies and nonstabilizerness monotones
• Quantifying magic for multi-qubit operations
• Gaussian quantum resource theories
• One-Shot Operational Quantum Resource Theory
• Conformal field theories are magical
• Probabilistic distillation of quantum coherence
• Operational relevance of resource theories of quantum measurements
• Application of the Resource Theory of Channels to Communication Scenarios
• Experimental progress on quantum coherence: detection, quantification, and manipulation
• Benchmarking one-shot distillation in general quantum resource theories
• All quantum resources provide an advantage in exclusion tasks
• Optimal resource cost for error mitigation
• Dynamical Entanglement
• Entanglement of a bipartite channel
• Skew informations from an operational view via resource theory of asymmetry
• Quantifying the resource content of quantum channels: An operational approach
• Correlation in Catalysts Enables Arbitrary Manipulation of Quantum Coherence
• Converting multilevel nonclassicality into genuine multipartite entanglement
• Quantum coherence as a signature of chaos
• Operational quantification of continuous-variable quantum resources
• Lower bound for the T count via unitary stabilizer nullity
• Framework for resource quantification in infinite-dimensional general probabilistic theories
• Fisher information universally identifies quantum resources
• Random Matrix Theory of the Isospectral twirling
• Partial coherence and quantum correlation with fidelity and affinity distances
• Stabilizer extent is not multiplicative
• One-Shot Manipulation of Dynamical Quantum Resources
• Resource Preservability
• One-shot entanglement distillation beyond local operations and classical communication
• Tight constraints on probabilistic convertibility of quantum states
• General Quantum Resource Theories: Distillation, Formation and Consistent Resource Measures
• $\alpha$-Logarithmic negativity
• Universal limitations on implementing resourceful unitary evolutions
• Communication, Dynamical Resource Theory, and Thermodynamics
• Inequalities witnessing coherence, nonlocality, and contextuality
• One-Shot Yield-Cost Relations in General Quantum Resource Theories
• Monotones in General Resource Theories
• Mana and thermalization: probing the feasibility of near-Clifford Hamiltonian simulation
• Some notes on the robustness of k-coherence and k-entanglement
• Bounding and simulating contextual correlations in quantum theory
• Every quantum helps: Operational advantage of quantum resources beyond convexity
• Arbitrary Amplification of Quantum Coherence in Asymptotic and Catalytic Transformation
• Epsilon-smooth measure of coherence
• Bridging magic and non-Gaussian resources via Gottesman-Kitaev-Preskill encoding
• Robustness- and weight-based resource measures without convexity restriction: Multicopy witness and operational advantage in static and dynamical quantum resource theories
• One-shot manipulation of entanglement for quantum channels
• Asymptotically Consistent Measures of General Quantum Resources: Discord, Non-Markovianity, and Non-Gaussianity
• Simple Bounds for One-shot Pure-State Distillation in General Resource Theories
• Mixed-state additivity properties of magic monotones based on quantum relative entropies for single-qubit states and beyond
• Anomalies of weight-based coherence measure and mixed maximally coherent states
• Resource Marginal Problems
• Distance-based resource quantification for sets of quantum measurements
• Accessible bounds for general quantum resources
• Hierarchy of correlation quantifiers comparable to negativity
• Gibbs-preserving operations requiring infinite amount of quantum coherence
• A Compendious Review of Majorization-Based Resource Theories: Quantum Information and Quantum Thermodynamics
• Unified Framework for Calculating Convex Roof Resource Measures
• Robustness of contextuality under different types of noise as quantifiers for parity-oblivious multiplexing tasks
• Quantum network-entanglement measures
• One-Shot Static Entanglement Cost of Bipartite Quantum Channels
• Projective robustness for quantum channels and measurements and their operational significance
• Birkhoff-James Orthogonality in the Trace Norm, with Applications to Quantum Resource Theories
• Quantum process discrimination with restricted strategies
• Noisy Probabilistic Error Cancellation and Generalized Physical Implementability
• Improving quantum channel discrimination with resourceful states
• Harvesting stabilizer entropy and non-locality from a quantum field
• Completely positive completely positive maps (and a resource theory for non-negativity of quantum amplitudes)
• Stochastic approximate state conversion for entanglement and general quantum resource theories
• Weight-based measure of quantum memory as a universal and operational benchmark
• Entropic and operational characterizations of dynamic quantum resources