Operational applications of the diamond norm and related measures in quantifying the non-physicality of quantum maps
arXiv:2102.07773 · doi:10.22331/q-2021-08-09-522
Abstract
Although quantum channels underlie the dynamics of quantum states, maps which are not physical channels -- that is, not completely positive -- can often be encountered in settings such as entanglement detection, non-Markovian quantum dynamics, or error mitigation. We introduce an operational approach to the quantitative study of the non-physicality of linear maps based on different ways to approximate a given linear map with quantum channels. Our first measure directly quantifies the cost of simulating a given map using physically implementable quantum channels, shifting the difficulty in simulating unphysical dynamics onto the task of simulating linear combinations of quantum states. Our second measure benchmarks the quantitative advantages that a non-completely-positive map can provide in discrimination-based quantum games. Notably, we show that for any trace-preserving map, the quantities both reduce to a fundamental distance measure: the diamond norm, thus endowing this norm with new operational meanings in the characterisation of linear maps. We discuss applications of our results to structural physical approximations of positive maps, quantification of non-Markovianity, and bounding the cost of error mitigation.
arXiv comment25 pages. v3: published version
References in corpus (50)
- Quantum entanglement
- Separability of Mixed States: Necessary and Sufficient Conditions
- Entanglement detection
- Quantum Teleportation is a Universal Computational Primitive
- On the volume of the set of mixed entangled states
- General Benchmarks for Quantum Repeaters
- Fundamental Limits of Repeaterless Quantum Communications
- Error mitigation for short-depth quantum circuits
- Entanglement and non-Markovianity of quantum evolutions
- Quantum Non-Markovianity: Characterization, Quantification and Detection
- Distance measures to compare real and ideal quantum processes
- Efficient variational quantum simulator incorporating active error minimisation
- Robustness of entanglement
- Practical Quantum Error Mitigation for Near-Future Applications
- On the degree of non-Markovianity of quantum evolution
- Direct detection of quantum entanglement
- Completely Positive Maps and Classical Correlations
- Estimating outcome probabilities of quantum circuits using quasiprobabilities
- General Resource Theories in Quantum Mechanics and Beyond: Operational Characterization via Discrimination Tasks
- Optimal discrimination of quantum operations
- Physical accessibility of non-completely positive maps
- Quantifying Operations with an Application to Coherence
- Quantifying magic for multi-qubit operations
- A resource theory of quantum memories and their faithful verification with minimal assumptions
- Resource Theory of Coherence - Beyond States
- How to quantify a dynamical quantum resource
- Using and reusing coherence to realize quantum processes
- Operational approach to open dynamics and quantifying initial correlations
- Structural approximations to positive maps and entanglement breaking channels
- Entanglement Cost of Quantum Channels
- Application of the Resource Theory of Channels to Communication Scenarios
- Fundamental limitations on distillation of quantum channel resources
- Entanglement of a bipartite channel
- Entanglement cost and quantum channel simulation
- Universal and Operational Benchmarking of Quantum Memories
- Maps for general open quantum systems and a theory of linear quantum error correction
- Amortized entanglement of a quantum channel and approximately teleportation-simulable channels
- Why initial system-environment correlations do not imply the failure of complete positivity: a causal perspective
- One-Shot Manipulation of Dynamical Quantum Resources
- Realizing Physical Approximation of the Partial Transpose
- Almost all quantum channels are equidistant
- Quantification of quantum dynamics with input-output games
- Relationship between costs for quantum error mitigation and non-Markovian measures
- Quantifying non-classical and beyond-quantum correlations in the unified operator formalism
- Designing Quantum Information Processing via Structural Physical Approximation
- Base norms and discrimination of generalized quantum channels
- Sampling Overhead Analysis of Quantum Error Mitigation: Uncoded vs. Coded Systems
- Improving compressed sensing with the diamond norm
- Implementing positive maps with multiple copies of an input state
- Note on the saturation of the norm inequalities between diamond and nuclear norm
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- Quantum Error Mitigation
- Fundamental limits of quantum error mitigation
- Universal Sampling Lower Bounds for Quantum Error Mitigation
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- Virtual quantum broadcasting
- Virtual quantum resource distillation: General framework and applications
- Noise effects on purity and quantum entanglement in terms of physical implementability
- Causal classification of spatiotemporal quantum correlations
- Experimental virtual distillation of entanglement and coherence
- Shadow Simulation of Quantum Processes
- Power of quantum measurement in simulating unphysical operations
- Noisy Probabilistic Error Cancellation and Generalized Physical Implementability
- Agnostic Process Tomography
- Quantum Approximation of Normalized Schatten Norms and Applications to Learning
- Guaranteeing Completely Positive Quantum Evolution
- Virtual Quantum Markov Chains
- Complete characterisation of state conversions by work extraction
- Photonic non-Markovianity identification by quantum process capabilities of non-CP processes
- A Novel Single-Layer Quantum Neural Network for Approximate SRBB-Based Unitary Synthesis
- Virtual phase-covariant quantum broadcasting for qubits