Separability Criterion for Density Matrices

1996/04/30 by Asher Peres · 5,274 citations
Computer Science · Mathematics · Physics and Astronomy · #Combinatorics #Density matrix #Eigenvalues and eigenvectors #Materials science #Mathematical analysis #Mathematical physics #Mathematics #Matrix (chemical analysis) #Physics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum mechanics #Separable space #quant-ph

paper · pdf · doi:10.1103/physrevlett.77.1413

published in Physical Review Letters 77(8), 1413-1415 (American Physical Society) · 6 pages LaTeX, contains a simplified derivation and two new examples

arxiv created 1996/06/17 · openalex publication_date 1996/08/19 · arxiv updated 2011/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

A quantum system consisting of two subsystems is separable if its density matrix can be written as \ensuremathρ\phantom\rule0ex0ex=\phantom\rule0ex0ex\ensuremathΣAw_A\ensuremathρ_A^\ensuremath'\ensuremath\bigotimes\ensuremathρ_A^\ensuremath'\ensuremath', where \ensuremathρ_A^\ensuremath' and \ensuremathρ_A^\ensuremath'\ensuremath' are density matrices for the two subsystems, and the positive weights w_A satisfy \ensuremathΣw_A\phantom\rule0ex0ex=\phantom\rule0ex0ex1. In this Letter, it is proved that a necessary condition for separability is that a matrix, obtained by partial transposition of \ensuremathρ, has only non-negative eigenvalues. Some examples show that this criterion is more sensitive than Bell's inequality for detecting quantum inseparability.

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• Assessing the significance of fidelity as a figure of merit in quantum state reconstruction of discrete and continuous-variable systems
• Exact solvability, non-integrability, and genuine multipartite entanglement dynamics of the Dicke model
• Optomechanical entanglement in the presence of laser phase noise
• Entanglement negativity in the harmonic chain out of equilibrium
• Self-trapping triggered by losses in cavity QED
• Monogamy and entanglement in tripartite quantum states
• Viable entanglement detection of unknown mixed states in low dimensions
• Provable entanglement and information cost for qubit-based quantum key-distribution protocols
• Multipartite pure-state entanglement
• The Jamiołkowski Isomorphism and a Simplified Proof for the Correspondence Between Vectors Having Schmidt Number k and k-Positive Maps
• Evaluating Convex Roof Entanglement Measures
• Nonequilibrium dissipation-driven steady many-body entanglement
• On maximal entanglement between two pairs in four-qubit pure states
• Constrained bounds on measures of entanglement
• Entanglement oscillations in non-Markovian quantum channels
• Global Geometric Difference Between Separable and Positive Partial Transpose States
• Mixed-state entanglement and quantum teleportation through noisy channels
• Entanglement redistribution in the Schwarzschild spacetime
• Systematic construction of genuine-multipartite-entanglement criteria in continuous-variable systems using uncertainty relations
• Complete conditions for legitimate Wigner distributions
• Cloning entangled photons to scales one can see
• Spin correlation of hyperon-antihyperon systems in e⁺e^- annihilation
• QUBIT4MATLAB V3.0: A program package for quantum information science and quantum optics for MATLAB
• Extended studies of separability functions and probabilities and the relevance of Dyson indices
• The geometry of absolute separability and other convex matrix properties from spectrum
• Thermal equilibrium of two quantum Brownian particles
• Disentanglement in bipartite continuous-variable systems
• Who's afraid of not completely positive maps?
• Entanglement Structure: Entanglement Partitioning in Multipartite Systems and Its Experimental Detection Using Optimizable Witnesses
• Separability criteria with angular and Hilbert space averages
• Relativistic Quantum Information of Anyons
• Asymmetry of bipartite quantum discord
• Entanglement of tripartite states with decoherence in non-inertial frames
• Conical Designs and Categorical Jordan Algebraic Post-Quantum Theories
• Indistinguishable near-infrared single photons from an individual organic molecule
• Single-photon quantum key distribution in the presence of loss
• Lower bounds on concurrence and separability conditions
• Entanglement generation and protection by detuning modulation
• Manipulating multiqudit entanglement witnesses by using linear programming
• Full separability criterion for tripartite quantum systems
• A contour for the entanglement negativity of bosonic Gaussian states
• Entanglement of mixed macroscopic superpositions: An entangling-power study
• Heuristic for estimation of multiqubit genuine multipartite entanglement
• Relations between entanglement witnesses and Bell inequalities
• Mode transformations and entanglement relativity in bipartite Gaussian states
• Entanglement generated between a single atom and a laser pulse
• Qubit-qutrit separability-probability ratios
• Entanglement verification for quantum-key-distribution systems with an underlying bipartite qubit-mode structure
• Small quantum networks in the qudit stabilizer formalism
• Interplay between computable measures of entanglement and other quantum correlations
• Finite temperature entanglement negativity in conformal field theory
• Experimental Detection of Multipartite Entanglement using Witness Operators
• Schmidt number for density matrices
• Transfer of quantum-enhanced information through a many-body system
• Maximal entanglement versus entropy for mixed quantum states
• Entanglement swapping with quantum-memory-compatible photons
• Multipartite entanglement and quantum state exchange
• Local description of quantum inseparability
• Disentanglement in the macroscopic limit
• Gravity-induced entanglement in optomechanical systems
• Experimental characterization of continuous-variable entanglement
• Family of Bell-like Inequalities as Device-Independent Witnesses for Entanglement Depth
• Entanglement verification with detection-efficiency mismatch
• Effect of an uniaxial single-ion anisotropy on the quantum and thermal entanglement of a mixed spin-(1/2, S) Heisenberg dimer
• Classification of mixed high-dimensional multiparticle systems
• Distillation Protocols: Output Entanglement and Local Mutual Information
• Minimal tomography with entanglement witnesses
• Study on Entanglement and its Utility in Information Processing
• Entropic and information inequality for nonlinearly transformed two-qubit X-states
• Class of bound entangled states ofN+Nqubits revealed by nondecomposable maps
• Bell-correlated activable bound entanglement in multiqubit systems
• Dynamical entanglement transfer for quantum-information networks
• Non-Gaussian entanglement in gravitating masses: The role of cumulants
• Stability of encoded macroscopic quantum superpositions
• Dynamical Entanglement
• Quantum energy teleportation between spin particles in a Gibbs state
• Bell's inequality violation with spins in silicon
• Negativity of random pure states
• Cones of positive maps and their duality relations
• Entanglement amplification via local weak measurements
• Entanglement between distant qubits in cyclicXXchains
• Entanglement and communication-reducing properties of noisyN-qubit states
• Triplet-like correlation symmetry of continuous variable entangled states
• Holographic Entanglement negativity in flat space generalized minimal massive gravity
• Maximal coherence and the resource theory of purity
• The power of noisy fermionic quantum computation
• Entanglement, Bell Inequalities and Decoherence in Particle Physics
• Entanglement generation in atoms immersed in a thermal bath of external quantum scalar fields with a boundary
• Ancilla-assisted sequential approximation of nonlocal unitary operations
• Triangle inequalities in coherence measures and entanglement concurrence
• Bipartite and tripartite entanglement in a Bose-Einstein acoustic black hole
• Detection and typicality of bound entangled states
• Structural physical approximations and entanglement witnesses
• Non-equilibrium entangled steady state of two independent two-level systems
• A family of norms with applications in quantum information theory
• Experimental Investigation of Criteria for Continuous Variable Entanglement
• Can We Detect the Quantum Nature of Weak Gravitational Fields?
• On the `fake' inferred entanglement associated with the maximum entropy inference of quantum states
• Conservation relation of nonclassicality and entanglement for Gaussian states in a beam splitter
• Entanglement splitting of pure bipartite quantum states
• An inverse mass expansion for the mutual information in free scalar QFT at finite temperature
• Decoherence of entangled kaons and its connection to entanglement measures
• Quantum channels in random spin chains
• Population-only decay map forn-qubitn-partite inseparability detection
• Entanglement detection with scrambled data
• Retrodiction of measurement outcomes on a single quantum system reveals entanglement with its environment
• Entanglement and the truncated moment problem
• Residual effect on the robustness of multiqubit entanglement
• Separability and entanglement of identical bosonic systems
• Separability and entanglement of four-mode Gaussian states
• Entanglement spectrum degeneracy and the Cardy formula in 1+1 dimensional conformal field theories
• Entangled quantum currents in distant mesoscopic Josephson junctions
• Characteristics and benchmarks of entanglement of mixed states: The two-qubit case
• Measurable concurrence of mixed states
• Measurement-base-independent test for genuine multipartite entanglement
• Entanglement in Finitely Correlated Spin States
• Entanglement entropy of theQ<mml:mspace width="0.16em"/>≥<mml:mspace width="0.16em"/>4quantum Potts chain
• Simulating Quantum Dynamics with Entanglement Mean Field Theory
• Using non-positive maps to characterize entanglement witnesses
• Sudden death and long-lived entanglement of two trapped ions
• Determining quantum correlations in bipartite systems - from qubit to qutrit and beyond
• Variational characterizations of separability and entanglement of formation
• Environment Induced Entanglement in Markovian Dissipative Dynamics
• Entangling two atoms via spontaneous emission
• On quantum operations as quantum states
• A parametrization of bipartite systems based onSU(4) Euler angles
• Decoherence and entropy of primordial fluctuations. I. Formalism and interpretation
• Verifying multipartite mode entanglement of W states
• Measuring quantum discord using the most distinguishable steered states
• Resilience of multiphoton entanglement under losses
• Spin entanglement via scanning tunneling microscope current
• The optimal approximation of qubit states with limited quantum states
• Amortized entanglement of a quantum channel and approximately teleportation-simulable channels
• Family of multipartite separability criteria based on a correlation tensor
• A class of positive atomic maps
• Detecting EPR steering via two classes of local measurements
• Purity as a witness for initial system-environment correlations in open-system dynamics
• Remote control of quantum correlations in a two-qubit receiver by a three-qubit sender
• Multi-qubit stabilizer and cluster entanglement witnesses
• Quantum information and relativity theory
• A class of bound entangled states
• Hermitian tensor product approximation of complex matrices and separability
• Detecting genuine non-Gaussian entanglement
• The Exclusivity Principle and the Set of Quantum Correlations
• Quantum measurement incompatibility in subspaces
• A Few Steps More Towards NPT Bound Entanglement
• Gaussian entanglement of symmetric two-mode Gaussian states
• Sudden change in dynamics of genuine multipartite entanglement of cavity-reservoir qubits
• Fine-grained dynamics of entanglement in non-integrable quenches far across the Ising quantum critical point
• Separability of completely symmetric states in a multipartite system
• General solution of the time evolution of two interacting harmonic oscillators
• Hawking radiation, entanglement, and teleportation in the background of an asymptotically flat static black hole
• Quantum coherences,K-way negativities and multipartite entanglement
• Entanglement Spectrum in General Free Fermionic Systems
• Harvesting Large Scale Entanglement in de Sitter Space with Multiple Detectors
• Tripartite entanglement in parametric down-conversion with spatially structured pump
• Asymmetric broadcasting of quantum correlations
• Quantum discord for two-qubitXstates: Analytical formula with very small worst-case error
• CLASSICALITY WITNESS FOR TWO-QUBIT STATES
• Entanglement and phase properties of noisy NOON states
• THERMAL DISCORD AND NEGATIVITY IN A TWO-SPIN-QUTRIT SYSTEM UNDER DIFFERENT MAGNETIC FIELDS
• Speed of disentanglement in multiqubit systems under a depolarizing channel
• Gaussian intrinsic entanglement for states with partial minimum uncertainty
• Additivity of entanglement of formation via an entanglement-breaking space
• The bell inequality: A measure of entanglementa?
• Local uncertainty relations serving as measures of entanglement in a bipartite two-level system
• The ℏ→ 0 Limit of the Entanglement Entropy
• Absence of Entanglement Growth in Dicke Superradiance
• Gaussian continuous-variable isotropic state
• Construction of quantum states with bound entanglement
• Disentanglement and inseparability correlation in a two-qubit system
• Tripartite separability conditions exponentially violated by Gaussian states
• Perturbative expansion of entanglement negativity using patterned matrix calculus
• On the Geometric Distance Between Quantum States with Positive Partial Transposition and Private States
• Thermal entanglement in ferrimagnetic chains
• Entanglement dynamics of qubits in a common environment
• Does the uncertainty relation determine the quantum state?
• Quantum entanglement of anharmonic oscillators
• State estimation from a pair of conjugate qudits
• Steering and Correlations for a Single Qudit State on the Example of Spin j =3/2
• Entanglement dynamics of two-bipartite system under the influence of dissipative environments
• Bell’s inequalities and distillability inN-quantum-bit systems
• Uncertainty Relation and Inseparability Criterion
• Dyson indices and Hilbert–Schmidt separability functions and probabilities
• Detecting entanglement of continuous variables with three mutually unbiased bases
• Qudit hypergraph states
• On the comparison of volumes of quantum states
• Extremal states of positive partial transpose in a system of three qubits
• Role of correlations in the two-body-marginal problem
• Separability conditions and limit temperatures for entanglement detection in two-qubit HeisenbergXYZmodels
• Distilling entanglement from random cascades with partial “which path” ambiguity
• On the complementary quantum capacity of the depolarizing channel
• Singularity in entanglement negativity across finite-temperature phase transitions
• Degradation of entanglement in Markovian noise
• Local two-qubit entanglement-annihilating channels
• Iterations of nonlinear entanglement witnesses
• Ideal pairing of the Stokes and anti-Stokes photons in the Raman process
• Characterization of Equivariant Maps and Application to Entanglement Detection
• Maximization of thermal entanglement of arbitrarily interacting two qubits
• Witness for edge states and its characteristics
• Experimental Detection of Entanglement with Optimal-Witness Families
• Separability of n-particle mixed states: necessary and sufficient conditions in terms of linear maps
• A necessary and sufficient criterion for multipartite separable states
• Optimal manipulations with qubits: Universal quantum entanglers
• Quantum correlations and distinguishability of quantum states
• Entanglement witnesses: construction, analysis and classification
• Entanglement negativity in the critical Ising chain
• Multipartite Bound Entangled States that Violate Bell’s Inequality
• Entanglement generation between a charge qubit and its bosonic environment during pure dephasing: Dependence on the environment size
• Bound on distillable entanglement
• Structural physical approximations of unphysical maps and generalized quantum measurements
• Colloquium: Multiparticle quantum superpositions and the quantum-to-classical transition
• Quantum separability and entanglement detection via entanglement-witness search and global optimization
• Displaced photon-number entanglement tests
• Einstein-Podolsky-Rosen uncertainty limits for bipartite multimode states
• Entanglement and nonclassicality: A mutual impression
• How to construct indecomposable entanglement witnesses
• Implementing positive maps with multiple copies of an input state
• Entanglement, Nonlinear Dynamics, and the Heisenberg Limit
• Invariance of bipartite separability and PPT-probabilities over Casimir invariants of reduced states
• Construction of exposed indecomposable positive linear maps between matrix algebras
• Quantum resource theories
• Gaussian entanglement revisited
• Realignment separability criterion assisted with filtration for detecting continuous-variable entanglement
• Symmetry-resolved entanglement detection using partial transpose moments
• Searching for extremal PPT entangled states
• Passing quantum correlations to qubits using any two-mode state
• Approximating the set of separable states using the positive partial transpose test
• Quantum Communication with Zero-Capacity Channels
• Squeezed vacuum reservoir effect for entanglement decay in the nonlinear quantum scissor system
• Postselection-induced entanglement swapping from a vacuum-excitation entangled state to separate quantum systems
• Entanglement generation and transfer between remote atomic qubits interacting with squeezed field
• Statistical mixtures of states can be more quantum than their superpositions: Comparison of nonclassicality measures for single-qubit states
• Entanglement Wedge Cross Section from the Dual Density Matrix
• Factorization and entanglement in generalXYZspin arrays in nonuniform transverse fields
• The generalized partial transposition criterion for separability of multipartite quantum states
• Generation of phase-covariant quantum cloning
• Partial transposition as a direct link between concurrence and negativity
• Quantum Information
• Entanglement properties of positive operators with ranges in completely entangled subspaces
• Correlations in interference and diffraction
• Observable estimation of entanglement for arbitrary finite-dimensional mixed states
• Collective versus local measurements on two parallel or antiparallel spins
• Teleportation using coupled oscillator states
• Field-Space Entanglement Dynamics Between Tunnel-Coupled Luttinger Liquids
• Amendable Gaussian channels: Restoring entanglement via a unitary filter
• Dynamical creation of entanglement versus disentanglement in a system of three-level atoms with vacuum-induced coherences
• Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases
• Characterizing High Schmidt Number Witnesses in Arbitrary Dimensions System
• Multipartite entanglement based on realignment moments
• Quantum Entanglement between gauge boson pairs at a Muon Collider
• Practical Advantage of Classical Communication in Entanglement Detection
• Computable entanglement conversion witness that is better than the negativity
• Entanglement of systems of dipolar coupled nuclear spins at the adiabatic demagnetization
• Quantum memory in spontaneous emission processes
• Ratios of maximal concurrence-parameterized separability functions, and generalized Peres–Horodecki conditions
• Quantum neural networks facilitating quantum state classification
• Separability and Entanglement of Composite Quantum Systems
• Quantum copying: A network
• Complete positivity of nonlinear evolution: A case study
• Non-positive-partial-transpose subspaces can be as large as any entangled subspace
• Experimental Quantification of the Entanglement of Noisy Twin Beams
• Experimental violation of local causality in a quantum network
• Markov Gap and Bound Entanglement in Haar Random State
• Biochemistry and theory of proton-coupled electron transfer. [europepmc]
• Performances and robustness of quantum teleportation with identical particles. [europepmc]
• Entanglement classifier in chemical reactions. [europepmc]
• Quantum process capability. [europepmc]
• Nonclassicality by Local Gaussian Unitary Operations for Gaussian States. [europepmc]
• Uncertainty Relations for Coarse-Grained Measurements: An Overview. [europepmc]
• Geometry and Entanglement of Two-Qubit States in the Quantum Probabilistic Representation. [europepmc]
• Efficient Quantum Teleportation of Unknown Qubit Based on DV-CV Interaction Mechanism. [europepmc]
• Generation of Werner-like states via a two-qubit system plunged in a thermal reservoir and their application in solving binary classification problems. [europepmc]
• A complete hierarchy for the pure state marginal problem in quantum mechanics. [europepmc]
• Measuring dimensionality and purity of high-dimensional entangled states. [europepmc]
• Free versus bound entanglement, a NP-hard problem tackled by machine learning. [europepmc]
• Unconventional Thermal and Magnetic-Field-Driven Changes of a Bipartite Entanglement of a Mixed Spin-(1/2, S ) Heisenberg Dimer with an Uniaxial Single-Ion Anisotropy. [europepmc]
• Entangled Qubit States and Linear Entropy in the Probability Representation of Quantum Mechanics. [europepmc]
• Almost complete solution for the NP-hard separability problem of Bell diagonal qutrits. [europepmc]
• Bayesian tomography of high-dimensional on-chip biphoton frequency combs with randomized measurements. [europepmc]
• Entanglement of orbital angular momentum in non-sequential double ionization. [europepmc]
• Entanglement detection with artificial neural networks. [europepmc]
• Comparing bound entanglement of bell diagonal pairs of qutrits and ququarts. [europepmc]
• Numerically Exact Solution for a Real Polaritonic System under Vibrational Strong Coupling in Thermodynamic Equilibrium: Loss of Light-Matter Entanglement and Enhanced Fluctuations. [europepmc]
• Enhancing detection of topological order by local error correction. [europepmc]
• Observation of quantum entanglement with top quarks at the ATLAS detector. [europepmc]
• The Quantum Information Science Challenge for Chemistry. [europepmc]
• Large-scale cluster quantum microcombs. [europepmc]
• Quantum spin-engineering in on-surface molecular ferrimagnets. [europepmc]

Related

• Quantum nonlocality and inseparability
• Separability of Density Matrices and Conditional Information Transmission
• Strong and weak separability conditions for two-qubits density matrices
• Convexity and the Separability Problem of Quantum Mechanical Density Matrices
• Separability and entanglement in 2xN composite quantum systems