arXiv:2311.17260

Entanglement of Quantum States which are Zero on the Symmetric Sector

Domenico D’Alessandro

Published Nov 28, 2023• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.2311.17260

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arXiv:2311.17260

Entanglement of Quantum States which are Zero on the Symmetric Sector

Domenico D’Alessandro

Published Nov 28, 2023• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.2311.17260

Related papers

On a class of entanglement witnesses for multipartite quantum systems

Domenico D’Alessandro

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.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.

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The unitary group acting on the Hilbert space

{\cal H}:=(C^2)^{\otimes 3}

of three quantum bits admits a Lie subgroup,

U^{S_3}(8)

, of elements which permute with the symmetric group of permutations of three objects. Under the action of such a Lie subgroup, the Hilbert space

{\cal H}

splits into three invariant subspaces of dimensions

4

2

and

2

respectively, each corresponding to an irreducible representation of

su(2)

. The subspace of dimension

4

is uniquely determined and corresponds to states that are themselves invariant under the action of the symmetric group. This is the so called {\it symmetric sector.} The subspaces of dimension two are not uniquely determined and we parametrize them all. We provide an analysis of pure states that are in the subspaces invariant under

U^{S_3}(8)

. This concerns their entanglement properties, separability criteria and dynamics under the Lie subgroup

U^{S_3}(8)

. As a physical motivation for the states and dynamics we study, we propose a physical set-up which consists of a symmetric network of three spin

\frac{1}{2}

particles under a common driving electro-magnetic field. {For such system, we solve the control theoretic problem of driving a separable state to a state with maximal distributed entanglement.

k

-positive map witness, optimal Schmidt-number witnesses, and entanglement measures.