Abstract A way to characterize multipartite entanglement in pure states of a spin chain with sites and local dimension is by means of the Cayley hyperdeterminant. The latter quantity is a polynomial constructed with the components of the wave function which is invariant under local unitary transformation. For spin 1/2 chains (i.e. ) with and sites, the hyperdeterminant coincides with the concurrence and the tangle respectively. In this paper we consider spin chains with sites where the hyperdeterminant is a polynomial of degree 24 containing around terms. This huge object can be written in terms of more simple polynomials and of degrees 8 and 12 respectively. Correspondingly we compute , and the hyperdeterminant for eigenstates of the following spin chain Hamiltonians: the transverse Ising model, the XXZ Heisenberg model and the Haldane–Shastry model. Those invariants are also computed for random states, the ground states of random matrix Hamiltonians in the Wigner–Dyson Gaussian ensembles and the quadripartite entangled states defined by Verstraete et al in 2002. Finally, we propose a generalization of the hyperdeterminant to thermal density matrices. We observe how these polynomials are able to capture the phase transitions present in the models studied as well as a subclass of quadripartite entanglement present in the eigenstates.
Abstract A way to characterize multipartite entanglement in pure states of a spin chain with sites and local dimension is by means of the Cayley hyperdeterminant. The latter quantity is a polynomial constructed with the components of the wave function which is invariant under local unitary transformation. For spin 1/2 chains (i.e. ) with and sites, the hyperdeterminant coincides with the concurrence and the tangle respectively. In this paper we consider spin chains with sites where the hyperdeterminant is a polynomial of degree 24 containing around terms. This huge object can be written in terms of more simple polynomials and of degrees 8 and 12 respectively. Correspondingly we compute , and the hyperdeterminant for eigenstates of the following spin chain Hamiltonians: the transverse Ising model, the XXZ Heisenberg model and the Haldane–Shastry model. Those invariants are also computed for random states, the ground states of random matrix Hamiltonians in the Wigner–Dyson Gaussian ensembles and the quadripartite entangled states defined by Verstraete et al in 2002. Finally, we propose a generalization of the hyperdeterminant to thermal density matrices. We observe how these polynomials are able to capture the phase transitions present in the models studied as well as a subclass of quadripartite entanglement present in the eigenstates.
Simple physical interactions between spin-1∕2 particles may result in quantum states that exhibit exotic correlations that are difficult to find if one simply explores state spaces of multipartite systems. In particular, we present a detailed investigation of the well-known Ising model of a chain (ring) of spin-1∕2 particles (qubits) in a transverse magnetic field. We present explicit expressions for eigenstates of the model Hamiltonian for arbitrary number of spin-1∕2 particles in the chain in the standard (computer) basis, and we investigate quantum entanglement between individual qubits. We analyze bipartite as well as multipartite entanglement in the ground state of the model. In particular, we show that bipartite entanglement between pairs of qubits of the Ising chain (measured in terms of a concurrence) as a function of the parameter $\ensuremath{\lambda}$ has a maximum around the point $\ensuremath{\lambda}=1$, and it monotonically decreases for large values of $\ensuremath{\lambda}$. We prove that in the limit $\ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty}$ this state is locally unitary equivalent to an N-partite Greenberger-Horn-Zeilinger state. We also analyze a very specific eigenstate of the Ising Hamiltonian with a zero eigenenergy (we denote this eigenstate as the X-state). This X-state exhibits the ``extreme'' entanglement in a sense that an arbitrary subset A of $k\ensuremath{\leqslant}n$ qubits in the Ising chain composed of N=2n+1 qubits is maximally entangled with the remaining qubits (set B) in the chain. In addition, we prove that by performing a local operation just on the subset B, one can transform the X-state into a direct product of k singlets shared by the parties A and B. This property of the X-state can be utilized for new secure multipartite communication protocols.
We classify multipartite entanglement in a unified manner, focusing on a duality between the set of separable states and that of entangled states. Hyperdeterminants, derived from the duality, are natural generalizations of entanglement measures, the concurrence, 3-tangle for 2, 3 qubits respectively. Our approach reveals how inequivalent multipartite entangled classes of pure states constitute a partially ordered structure under local actions, significantly different from a totally ordered one in the bipartite case. Moreover, the generic entangled class of the maximal dimension, given by the nonzero hyperdeterminant, does not include the maximally entangled states in Bell's inequalities in general (e.g., in the n≥4 qubits), contrary to the widely known bipartite or 3-qubit cases. It suggests that not only are they never locally interconvertible with the majority of multipartite entangled states, but they would have no grounds for the canonical n-partite entangled states. Our classification is also useful for that of mixed states.
Organising the space of entanglement structures of a multipartite quantum system is a much more challenging task than its bipartite version: while the local unitary (LU) orbit of a bipartite pure state can be conveniently characterized by its entanglement spectrum, invariants of multipartite entanglement structures are comparatively difficult to define and work with. The root cause of this difference is that the bipartite problem can be reduced to the analysis of matrix invariants, while its multipartite version is governed by a much richer space of tensor invariants. The present work explores the latter through the lens of so-called trace-invariants, which are in one-to-one correspondence with combinatorial objects known as colored graphs. We first explain why trace-invariant evaluations can serve as labels of LU-orbits of multipartite pure states, how this strategy extends to random states, and how the effect of local operations (LO) can be analyzed through such data. We then focus on entanglement classification within an (infinite-dimensional) subspace of reference states, whose basic building blocks are GHZ states of various dimensions. We show that relatively simple subclasses of trace-invariants are sufficient to separate the LU-orbits of reference states, and enable a complete (resp. an incomplete) characterization of their relations in the LO (resp. LOCC) resource theory of entanglement. Finally, we investigate how a (still infinite) subclass of reference states of local dimension N can be efficiently distinguished at leading and subleading orders in an asymptotic large-N expansion (among themselves, or from Haar-random states). This analysis relies crucially on combinatorial quantities associated to colored graphs, some of which have already played instrumental roles in the recent literature on random tensors. Results of broader relevance are reported along the way.
We find that multidimensional determinants ``hyperdeterminants,'' related to entanglement measures (the so-called concurrence, or 3-tangle for two or three qubits, respectively), are derived from a duality between entangled states and separable states. By means of the hyperdeterminant and its singularities, the single copy of multipartite pure entangled states is classified into an onion structure of every closed subset, similar to that by the local rank in the bipartite case. This reveals how inequivalent multipartite entangled classes are partially ordered under local actions. In particular, the generic entangled class of the maximal dimension, distinguished as the nonzero hyperdeterminant, does not include the maximally entangled states in Bell's inequalities in general (e.g., in the n>4 qubits), contrary to the widely known bipartite or three-qubit cases. It suggests that not only are they never locally interconvertible with the majority of multipartite entangled states, but they would have no grounds for the canonical n-partite entangled states. Our classification is also useful for the mixed states.
The spectral statistics and entanglement within the eigenstates of generic spin chain Hamiltonians are analysed. A class of random matrix ensembles is defined which include the most general nearest-neighbour qubit chain Hamiltonians. For these ensembles, and their generalisations, it is seen that the long chain limiting spectral density is a Gaussian and that this convergence holds on the level of individual Hamiltonians. The rate of this convergence is numerically seen to be slow. Higher eigenvalue correlation statistics are also considered, the canonical nearest-neighbour level spacing statistics being numerically observed and linked with ensemble symmetries. A heuristic argument is given for a conjectured form of the full joint probability density function for the eigenvalues of a wide class of such ensembles. This is numerically verified in a particular case. For many translationally-invariant nearest-neighbour qubit Hamiltonians it is shown that there exists a complete orthonormal set of eigenstates for which the entanglement present in a generic member, between a fixed length block of qubits and the rest of the chain, approaches its maximal value as the chain length increases. Many such Hamiltonians are seen to exhibit a simple spectrum so that their eigenstates are unique up to phase. The entanglement within the eigenstates contrasts the spectral density for such Hamiltonians, which is that seen for a non-interacting chain of qubits. For such non-interacting chains, their always exists a basis of eigenstates for which there is no entanglement present.
We find that the hyperdeterminant DetA, related to an entanglement measure (the concurrence, 3-tangle for the 2,3-qubit respectively), is derived from a duality between the entangled states and separable states. In terms of DetA and its singularities, the single copy of multipartite pure entangled states is classified into an onion structure of every closed subset, similarly to the local rank in the bipartite case. This reveals that many inequivalent multipartite entangled states are partially ordered such that entanglement measures like DetA as well as local ranks are needed to distinguish them. In particular, the nonzero DetA distinguishes generic entangled states of the maximal dimension (the outermost class of the onion structure). It suggests that the majority of multipartite entangled states never locally converts to the maximally entangled states in Bell's inequalities even probabilistically in general (e.g., in the ngeq4-qubit), contrary to the widely known bipartite or 3-qubit cases. The classification is also useful for that of mixed states.