Mixed-state quantum anomaly and multipartite entanglement

Abstract

Quantum entanglement measures of many-body states have been increasingly useful to characterize phases of matter. Here we explore a surprising connection between mixed state entanglement and 't Hooft anomaly. More specifically, we consider lattice systems in $d$ space dimensions with anomalous symmetry $G$ where the anomaly is characterized by an invariant in the group cohomology $H^{d+2}(G,U(1))$. We show that any mixed state $\rho$ that is strongly symmetric under $G$, in the sense that $G\rho \propto \rho$, is necessarily $(d+2)$-nonseparable, i.e. is not the mixture of tensor products of $d+2$ states in the Hilbert space. Furthermore, such states cannot be prepared from any $(d+2)$-separable states using finite-depth local quantum channels, so the nonseparability is long-ranged in nature. We provide proof of these results in $d\le 1$, and plausibility arguments in $d>1$. The anomaly-nonseparability connection thus allows us to generate simple examples of mixed states with nontrivial long-ranged multipartite entanglement. In particular, in $d=1$ we found an example of intrinsically mixed quantum phase, in the sense that states in this phase cannot be two-way connected to any pure state through finite-depth local quantum channels. We also analyze mixed anomaly involving both strong and weak symmetries, including systems constrained by the Lieb-Schultz-Mattis type of anomaly. We find that, while strong-weak mixed anomaly in general does not constrain quantum entanglement, it does constrain long-range correlations of mixed states in nontrivial ways. Namely, such states are not symmetrically invertible and not gapped Markovian, generalizing familiar properties of anomalous pure states.

27 pages, 9 figures; New results on strong-weak mixed anomaly and other revisions

References in corpus (58)

• Quantum entanglement
• Three qubits can be entangled in two inequivalent ways
• Entanglement detection
• Persistent entanglement in arrays of interacting particles
• Entanglement spectrum of a topological phase in one dimension
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
• Symmetry protection of topological order in one-dimensional quantum spin systems
• Unextendible Product Bases and Bound Entanglement
• Lieb-Schultz-Mattis in Higher Dimensions
• Four qubits can be entangled in nine different ways
• Information-theoretic aspects of quantum inseparability of mixed states
• Braiding statistics approach to symmetry-protected topological phases
• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• Symmetries and conserved quantities in Lindblad master equations
• 2D symmetry protected topological orders and their protected gapless edge excitations
• Lieb-Robinson Bounds and the Exponential Clustering Theorem
• Commensurability, excitation gap and topology in quantum many-particle systems on a periodic lattice
• A note on symmetry reductions of the Lindblad equation: transport in constrained open spin chains
• Symmetry-protected topological orders for interacting fermions -- Fermionic topological nonlinear $\sigma$ models and a special group supercohomology theory
• Quantum conditional mutual information and approximate Markov chains
• Detection of high-dimensional genuine multi-partite entanglement of mixed states
• Translational symmetry and microscopic constraints on symmetry-enriched topological phases: a view from the surface
• Valence Bonds in Random Quantum Magnets: Theory and Application to YbMgGaO4
• Reflection positivity and invertible topological phases
• Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge
• Non-Abelian Topological Order and Anyons on a Trapped-Ion Processor
• Topological Order at Non-zero Temperature
• Quantum criticality under decoherence or weak measurement
• Topological phases with generalized global symmetries
• Measurement as a shortcut to long-range entangled quantum matter
• Multipartite entanglement measures
• Lieb-Schultz-Mattis, Luttinger, and 't Hooft -- anomaly matching in lattice systems
• Symmetry breaking and error correction in open quantum systems
• Average Symmetry-Protected Topological Phases
• The entropy vector formalism and the structure of multidimensional entanglement in multipartite systems
• Lattice Homotopy Constraints on Phases of Quantum Magnets
• Non-Fermi liquids as ersatz Fermi liquids: general constraints on compressible metals
• Strong-to-Weak Spontaneous Symmetry Breaking in Mixed Quantum States
• Separable Multipartite Mixed States - Operational Asymptotically Necessary and Sufficient Conditions
• Intrinsic Mixed-state Topological Order
• Topological characterization of Lieb-Schultz-Mattis constraints and applications to symmetry-enriched quantum criticality
• Electric polarization as a nonquantized topological response and boundary Luttinger theorem
• Symmetry protected topological phases under decoherence
• Classification of phases for mixed states via fast dissipative evolution
• Separability criteria for several classes of $n$-partite quantum states
• A family of multipartite separability criteria based on correlation tensor
• Stability of mixed-state quantum phases via finite Markov length
• Symmetry-enforced many-body separability transitions
• Detection of $k$-partite entanglement and $k$-nonseparability of multipartite quantum states
• Separability criteria via sets of mutually unbiased measurements
• Building models of topological quantum criticality from pivot Hamiltonians
• Parameterized multipartite entanglement measures
• A UV perspective on mixed anomalies at critical points between bosonic symmetry-protected phases
• Matrix Product Operator Algebras II: Phases of Matter for 1D Mixed States
• Quantum interactive proofs and the complexity of separability testing
• Reviving the Lieb-Schultz-Mattis Theorem in Open Quantum Systems
• Detecting Quantum Anomalies in Open Systems

Cited by in corpus (19)

• Locally Purified Density Operators for Symmetry-Protected Topological Phases in Mixed States
• Tracking the variation of entanglement Rényi negativity: a quantum Monte Carlo study
• Holographic View of Mixed-State Symmetry-Protected Topological Phases in Open Quantum Systems
• Spontaneous symmetry breaking in open quantum systems: strong, weak, and strong-to-weak
• Decoherence and wavefunction deformation of $D_4$ non-Abelian topological order
• Gauge theory and mixed state criticality
• A New Framework for Quantum Phases in Open Systems: Steady State of Imaginary-Time Lindbladian Evolution
• Stabilizing Non-Abelian Topological Order against Heralded Noise via Local Lindbladian Dynamics
• Mixed-State Topological Order under Coherent Noise
• Many-body quantum catalysts for transforming between phases of matter
• Exact, Average, and Broken Symmetries in a Simple Adaptive Monitored Circuit
• Instability of steady-state mixed-state symmetry-protected topological order to strong-to-weak spontaneous symmetry breaking
• Parent Lindbladians for Matrix Product Density Operators
• Revealing effects of local dimension on variable-range interacting model by connecting Lieb-Robinson bounds and multipartite entanglement
• Entanglement Spectrum Resolved by Loop Symmetries
• Universal Decay of Mutual Information and Conditional Mutual Information in Gapped Pure- and Mixed-State Quantum Matter
• Spacetime Markov length: a diagnostic for fault tolerance via mixed-state phases
• Symmetry-enforced minimal entanglement and correlation in quantum spin chains
• Holographic duality between bulk topological order and boundary mixed-state order