Entanglement Detection Beyond Measuring Fidelities

Abstract

One of the most widespread methods to determine if a quantum state is entangled, or to quantify its entanglement dimensionality, is by measuring its fidelity with respect to a pure state. In this Letter we find a large class of states whose entanglement cannot be detected in this manner; we call them unfaithful. We find that unfaithful states are ubiquitous in information theory. For small dimensions, we check numerically that most bipartite states are both entangled and unfaithful. Similarly, numerical searches in low dimensions show that most pure entangled states remain entangled but become unfaithful when a certain amount of white noise is added. We also find that faithfulness can be self-activated, i.e., there exist instances of unfaithful states whose tensor powers are faithful. To explore how the fidelity approach limits the quantification of entanglement dimensionality, we generalize the notion of an unfaithful state to that of a D-unfaithful state, one that cannot be certified as D-dimensionally entangled by measuring its fidelity with respect to a pure state. For describing such states, we additionally introduce a hierarchy of semidefinite programming relaxations that fully characterizes the set of states of Schmidt rank at most D.

arXiv comment5+3 pages, 2 figures, 1 table. Supplemented with GitHub repository. v2: improved presentation (including new appendices that give additional technical details). v3: corrected errors in Table I

References in corpus (33)

• Separability Criterion for Density Matrices
• Entanglement detection
• Efficient classical simulation of slightly entangled quantum computations
• Measurement-based quantum computation with cluster states
• Security of quantum key distribution using d-level systems
• Generation of Hyperentangled Photons Pairs
• On the role of entanglement in quantum computational speed-up
• Witnessing multipartite entanglement
• Entanglement and Extreme Spin Squeezing
• Fully device independent quantum key distribution
• Direct Fidelity Estimation from Few Pauli Measurements
• Optimal eavesdropping in cryptography with three-dimensional quantum states
• A complete family of separability criteria
• A Schmidt number for density matrices
• Entanglement Certification $-$ From Theory to Experiment
• Distinguishing separable and entangled states
• Measurements in two bases are sufficient for certifying high-dimensional entanglement
• Separability of n-particle mixed states: necessary and sufficient conditions in terms of linear maps
• Distribution of high-dimensional entanglement via an intra-city free-space link
• Experimental GHZ Entanglement beyond Qubits
• Entanglement's Benefit Survives an Entanglement-Breaking Channel
• Average fidelity between random quantum states
• Detecting multipartite entanglement
• Universal quantum computation with little entanglement
• Evaluation of convex roof entanglement measures
• Hilbert--Schmidt volume of the set of mixed quantum states
• Weak randomness in device independent quantum key distribution and the advantage of using high dimensional entanglement
• Complete hierarchies of efficient approximations to problems in entanglement theory
• Quantum lithography, entanglement and Heisenberg-limited parameter estimation
• The power of symmetric extensions for entanglement detection
• Random Bures mixed states and the distribution of their purity
• Class of PPT bound entangled states associated to almost any set of pure entangled states
• High-Dimensional Entanglement in States with Positive Partial Transposition

Cited by in corpus (33)

• Predicting Many Properties of a Quantum System from Very Few Measurements
• Efficient generation of high-dimensional entanglement through multi-path downconversion
• Optimal Entanglement Certification from Moments of the Partial Transpose
• Statistical Methods for Quantum State Verification and Fidelity Estimation
• Semidefinite programming relaxations for quantum correlations
• Geometry of faithful entanglement
• Characterizing entanglement dimensionality from randomized measurements
• Resource-efficient high-dimensional entanglement detection via symmetric projections
• Fundamental Limitation on the Detectability of Entanglement
• Bounding entanglement dimensionality from the covariance matrix
• Quantum-Inspired Hierarchy for Rank-Constrained Optimization
• Complete hierarchy for high-dimensional steering certification
• Timelessness strictly inside the Quantum Realm
• Optimized detection of high-dimensional entanglement
• Analyzing quantum entanglement with the Schmidt decomposition in operator space
• Detecting the dimensionality of genuine multi-particle entanglement
• Semidefinite relaxations for high-dimensional entanglement in the steering scenario
• Exploring the relationship between the faithfulness and entanglement of two qubits
• A generic framework for genuine multipartite entanglement detection
• Quantum LOSR networks cannot generate graph states with high fidelity
• Quantum Preparation Games
• Genuine multipartite entanglement verification with convolutional neural networks
• Faithful coherent states
• Characterizing generalized axisymmetric quantum states in $d\times d$ systems
• Estimating the Schmidt numbers of quantum states via symmetric measurements
• Simulating quantum instruments with projective measurements and quantum post-processing
• Uncertainty relations between quantum Fisher information and entanglement monotones
• Positive maps from the walled Brauer algebra
• On fully entangled fraction and quantum conditional entropies for states with maximally mixed marginals
• Certifying entanglement dimensionality by $k$-reduction moments
• Quantum Advantage via Efficient Post-processing on Qudit Classical Shadow tomography
• Semi-device-independent certification of high-dimensional quantum channels
• Universal entanglement detection with geometrically optimal witnesses