The polynomial invariants of four qubits

Abstract

We describe explicitly the algebra of polynomial functions on the Hilbert space of four qubit states which are invariant under the SLOCC group $SL(2,\mathbb{C}{)}^4$. From this description, we obtain a closed formula for the hyperdeterminant in terms of low degree invariants.

5 pages, revtex4; some typos corrected

Cited by in corpus (123)

• Quantum entanglement
• Absolute Maximal Entanglement and Quantum Secret Sharing
• Quantifying entanglement resources
• Absolutely Maximally Entangled states, combinatorial designs and multi-unitary matrices
• SLOCC classification of n qubits invoking the proportional relationships for spectrums and for standard Jordan normal forms
• All Maximally Entangled Four Qubits States
• Geometric phases in quantum information
• Geometry of quantum systems: density states and entanglement
• Classification of multipartite entanglement in all dimensions
• The black-hole/qubit correspondence: an up-to-date review
• On polynomial invariants of several qubits
• Entanglement in the symmetric sector of $n$ qubits
• Measuring Polynomial Invariants of Multi-Party Quantum States
• Polynomial invariants for discrimination and classification of four-qubit entanglement
• The Hyperdeterminant and Triangulations of the 4-Cube
• Multipartite entanglement in 2 x 2 x n quantum systems
• On the geometry of four qubit invariants
• Algebraic invariants of five qubits
• Multipartite Entanglement Measure and Complete Monogamy Relation
• The moduli space of three qutrit states
• Evolution and Symmetry of Multipartite Entanglement
• Genuine multipartite entanglement measure
• Multipartite-entanglement monotones and polynomial invariants
• Classification of Entanglement in Symmetric States
• STU Black Holes as Four Qubit Systems
• On the geometry of a class of N-qubit entanglement monotones
• Special entangled quantum systems and the Freudenthal construction
• k-stretchability of entanglement, and the duality of k-separability and k-producibility
• Topological phases and multiqubit entanglement
• The invariant-comb approach and its relation to the balancedness of multipartite entangled states
• Quantification of entanglement via uncertainties
• Entanglement of four qubit systems: a geometric atlas with polynomial compass I (the finite world)
• Possibility of generalized monogamy relations for multipartite entanglement beyond three qubits
• Method for classifying multiqubit states via the rank of the coefficient matrix and its application to four-qubit states
• Geometric descriptions of entangled states by auxiliaries varieties
• Normal Forms and Tensor Ranks of Pure States of Four Qubits
• Stochastic Local Operations with Classical Communication of Absolutely Maximally Entangled States
• Qubits from extra dimensions
• Qubits and chirotopes
• SLOCC Invariants for Multipartite Mixed States
• Embedding qubits into fermionic Fock space, peculiarities of the four-qubit case
• Entanglement of four-qubit systems: a geometric atlas with polynomial compass II (the tame world)
• Quantum Entanglement involved in Grover's and Shor's algorithms: the four-qubit case
• A bipartite class of entanglement monotones for N-qubit pure states
• Invariant-based entanglement monotones as expectation values and their experimental detection
• Minimal Renyi-Ingarden-Urbanik entropy of multipartite quantum states
• An algebraic classification of entangled states
• Classification scheme of pure multipartite states based on topological phases
• Local Unitary Invariants for N-qubit Pure State
• Four-tangle for pure states
• Classification of qubit entanglement: SL(2,C) versus SU(2) invariance
• Dynamic symmetry approach to entanglement
• Two-Center Black Holes, Qubits and Elliptic Curves
• Hyperdeterminants as integrable discrete systems
• A brief introduction to multipartite entanglement
• The attractor mechanism as a distillation procedure
• Singularity of type $D_4$ arising from four qubit systems
• Proof of the Gour-Wallach conjecture
• Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals
• Permutation-invariant monotones for multipartite entanglement characterization
• Dimension of Tensor Network varieties
• The mixed two qutrit system: local unitary invariants, entanglement monotones, and the SLOCC group SL(3,C)
• Three-Tangle of a General Three-Qubit State in the Representation of Majorana Stars
• Stochastic local operations and classical communication (SLOCC) and local unitary operations (LU) classifications of n qubits via ranks and singular values of the spin-flipping matrices
• Polynomial invariants of degree 4 for even-$n$ qubits and their applications in entanglement classification
• Unitary Invariants and Classification of Four-Qubit States via Negativity Fonts
• Complete monogamy of the multipartite quantum mutual information
• Classification of Bipartite and Tripartite Qutrit Entanglement under SLOCC
• Flux Attractors and Generating Functions
• Multipartite entanglement in spin chains and the Hyperdeterminant
• Mermin Polynomials for Entanglement Evaluation in Grover's algorithm and Quantum Fourier Transform
• Nilpotent polynomials approach to four-qubit entanglement
• Learning Algebraic Models of Quantum Entanglement
• Local invariants of braiding quantum gates -- associated link polynomials and entangling power
• Cayley's hyperdeterminant: a combinatorial approach via representation theory
• SLOCC determinant invariants of order 2^{n/2} for even n qubits
• Classification of four qubit states and their stabilisers under SLOCC operations
• A Complete Set of Invariants for LU-Equivalence of Density Operators
• A link between symmetries of critical states and the structure of SLOCC classes in multipartite systems
• Quantum entanglement in finite-dimensional Hilbert spaces
• Complete Genuine Multipartite Entanglement Monotone
• SL-invariant entanglement measures in higher dimensions: the case of spin $1$ and $3/2$
• Quantum circuits of CNOT gates
• New invariants for entangled states
• On the algebra of local unitary invariants of pure and mixed quantum states
• Unitary invariants of qubit systems
• Negativity Fonts, multiqubit invariants and Four qubit Maximally Entangled States
• Stochastic local operations and classical communication equations and classification of even $n$ qubits
• Maximal entanglement of two delocalized spin-$\frac{1}{2}$ particles
• Constructing entanglement measures for fermions
• The co-Pieri rule for Kronecker coefficients
• Sequential generation of Polynomial Invariants and N-body non-local correlations
• Local unitary symmetries and entanglement invariants
• Induction for secant varieties of Segre varieties
• Entanglement and its facets in condensed matter systems
• Invariant and polynomial identities for higher rank matrices
• Coarse-grained entanglement classification through orthogonal arrays
• A spectral theory for transverse tensor operators
• Entanglement Classification via Single Entanglement Measure
• On Monogamy of four qubit entanglement
• Candidate entanglement invariants for two Dirac spinors
• On the Entropy Region of Gaussian Random Variables
• Stability of Flip and Exchange Symmetric Entangled State Classes under Invertible Local Operations
• A Note on State Decomposition Independent Local Invariants
• Polynomial identities for hypermatrices
• Poincare series for local unitary invariants of mixed states of the qubit-qutrit system
• Simple forms of SLOCC equivalent four-qubit \c{hi} state
• Genuinely multipartite entangled states in higher dimensions: a generalization of balancedness
• Canonical Forms of $2\times 2\times 2$ and $2\times 2\times 2\times 2$ Symmetric Tensors over Prime Fields
• Tensors Masquerading as Matchgates: Relaxing Planarity Restrictions on Pfaffian Circuits
• A Complete Set of Local Invariants for a Family of Multipartite Mixed States
• Low degree Lorentz invariant polynomials as potential entanglement invariants for multiple Dirac spinors
• Maximal Entanglement: Applications in Quantum Information and Particle Physics
• On two maximally entangled couples
• The necessary and sufficient condition for perfect teleportation and superdense coding and all the suitable states for teleportation and superdense coding
• Angular momentum decomposition of the three-dimensional Wigner harmonic oscillator
• On the ring of local polynomial invariants for a pair of entangled qubits
• Canonical forms of Order-$k$ ($k=2,3,4$) Symmetric Tensors of Format $3\times \cdots \times 3$ Over Prime Fields
• Entanglement classification via integer partitions
• A study of two-qubit density matrices with fermionic purifications
• An algebraically independent generating set of the algebra of local unitary invariants
• Monogamy constraints on entanglement of four-qubit pure states
• Rank-based SLOCC classification for odd n qubits