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International Journal of Quantum Information

On a class of entanglement witnesses for multipartite quantum systems

Domenico D’Alessandro

Published Jul 10, 2025• 0 citations
Publisher

Abstract

The Hilbert space for [Formula: see text], [Formula: see text]-dimensional, quantum systems, [Formula: see text], contains the subspace of states that are invariant under any permutation, the symmetric sector, [Formula: see text]. Separable states must have a nonzero component on the symmetric sector. More in detail, if [Formula: see text] denotes the orthogonal projection onto [Formula: see text], [Formula: see text], for a bound [Formula: see text] and for any separable state [Formula: see text]. Therefore, if one knows a (lower bound for) [Formula: see text], the operator [Formula: see text] is an entanglement witness for the system. In this paper we calculate sharp bounds [Formula: see text] analytically for the case [Formula: see text] and general [Formula: see text], and numerically for the case of [Formula: see text] qubits ([Formula: see text]). We show with an example that the resulting entanglement witnesses are able to detect entanglement of PPT states.

DOI
10.1142/s0219749925500169

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The Hilbert space for [Formula: see text], [Formula: see text]-dimensional, quantum systems, [Formula: see text], contains the subspace of states that are invariant under any permutation, the symmetric sector, [Formula: see text]. Separable states must have a nonzero component on the symmetric sector. More in detail, if [Formula: see text] denotes the orthogonal projection onto [Formula: see text], [Formula: see text], for a bound [Formula: see text] and for any separable state [Formula: see text]. Therefore, if one knows a (lower bound for) [Formula: see text], the operator [Formula: see text] is an entanglement witness for the system. In this paper we calculate sharp bounds [Formula: see text] analytically for the case [Formula: see text] and general [Formula: see text], and numerically for the case of [Formula: see text] qubits ([Formula: see text]). We show with an example that the resulting entanglement witnesses are able to detect entanglement of PPT states.
We consider a quantum system of n qudits and the Clebsch-Gordan decomposition of the associated Hilbert space. In this decomposition, one of the subspaces is the so-called symmetric subspace or symmetric sector, that is, the subspace of all states that are invariant under the action of the symmetric group. We prove that any separable state must have a nonzero component on the symmetric sector, or, equivalently, any state which has zero component on the symmetric sector must be entangled. For the cases of n=2,3 particles, and in arbitrary dimension d, this result can be refined by providing sharp lower bounds on the size of the component of separable states on the symmetric sector. This leads us to identify a class of entanglement witnesses for these systems. We provide an example showing that in the multipartite case, this class of witnesses detects PPT entangled states.
We present a method to detect entanglement partitions of multipartite quantum systems, by exploiting their inherent symmetries. Structures like genuinely multipartite entanglement, mmm-separability and entanglement depth are detected as very special cases. This formulation enables us to characterize all the entanglement partitions of all three- and four- partite states and witnesses with unitary and permutation symmetry. In particular, we find and parametrize a complete set of bound entangled states therein. For larger systems, we provide a large family of analytical witnesses detecting many-body states of arbitrary size where none of the parties is separable from the rest. This method relies on weak Schur sampling with projective measurements, and thus can be implemented in a quantum computer. Beyond physics, our results extend to the mathematical literature: we establish new inequalities between matrix immanants, and characterize the set of such inequalities for matrices of size three and four.
Abstract Entanglement witnesses (EWs) are a versatile tool to detect entangled states and characterize related properties of entanglement in quantum information theory. The verification of entangled states via EWs relies on the fact that separable states form a convex set; this also means that the framework presented by EWs generally applies to other quantum resources where free resources contain the convexity. A witness W corresponds to an observable satisfying Tr [ W σ sep ] ⩾ 0 for all separable states σ sep ; entangled states are detected once the inequality is violated. Recently, mirrored EWs have been introduced by showing that there exist non-trivial upper bounds to EWs, u W ⩾ Tr [ W σ sep ] ⩾ 0. An upper bound to a witness W signifies the existence of the other one M , called a mirrored EW, such that W + M = u W I ⊗ I . The framework of mirrored EWs shows that a single EW can be even more useful, as it can detect a larger set of entangled states by lower and upper bounds. In this work, we develop and investigate mirrored EWs for multipartite qubit states and also for high-dimensional systems, to find the efficiency and effectiveness of mirrored EWs in detecting entangled states. We provide mirrored EWs for n -partite GHZ states, graph states such as two-colorable states and tripartite bound entangled states. We also show that optimal EWs can be reflected with each other. For bipartite systems, we present mirrored EWs for existing optimal EWs and also construct a mirrored pair of optimal EWs in dimension three. Finally, we generalize mirrored EWs such that a pair of EWs can be connected by another EW, i.e. W + M = K is also an EW. Our results enhance the capability of EWs to detect a larger set of entangled states in multipartite and high-dimensional quantum systems.
We present general numerical methods to construct witness operators for entanglement detection and estimation of the fidelity. Our methods are applied to detecting entanglement in the vicinity of a six-qubit Dicke state with three excitations and also to further entangled symmetric states. All our witnesses are designed to keep the measurement effort small. We also present general results on the efficient local decomposition of permutationally invariant operators, which makes it possible to measure projectors to symmetric states efficiently.
We present general numerical methods to construct witness operators for entanglement detection and estimation of the fidelity. Our methods are applied to detecting entanglement in the vicinity of a six-qubit Dicke state with three excitations and also to further entangled symmetric states. All our witnesses are designed to keep the measurement effort small. We also present general results on the efficient local decomposition of permutationally invariant operators, which makes it possible to measure projectors to symmetric states efficiently.
Abstract We study the entanglement properties of quantum hypergraph states of n qubits, focusing on multipartite entanglement . We compute multipartite entanglement for hypergraph states with a single hyperedge of maximum cardinality, for hypergraph states endowed with all possible hyperedges of cardinality equal to n − 1 and for those hypergraph states with all possible hyperedges of cardinality greater than or equal to n − 1 . We then find a lower bound to the multipartite entanglement of a generic quantum hypergraph state. We finally apply the multipartite entanglement results to the construction of entanglement witness operators , able to detect genuine multipartite entanglement in the neighbourhood of a given hypergraph state. We first build entanglement witnesses of the projective type, then propose a class of witnesses based on the stabilizer formalism, hence called stabilizer witnesses , able to reduce the experimental effort from an exponential to a linear growth in the number of local measurement settings with the number of qubits.