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Abstract We present two results on multiqubit Werner states, defined to be those states that are invariant under the collective action of any given single-qubit unitary that acts simultaneously on all the qubits. Motivated by the desire to characterize entanglement properties of Werner states, we construct a basis for the real linear vector space of Werner invariant Hermitian operators on the Hilbert space of pure states; it follows that any mixed Werner state can be written as a mixture of these basis operators with unique coefficients. Continuing a study of ‘polygon diagram’ Werner states constructed in earlier work, with a goal to connect diagrams to entanglement properties, we consider a family of multiqubit states that generalize the singlet, and show that their 2-qubit reduced density matrices are separable.
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Symmetric states and dynamics of three quantum bits
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.
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- Award ID(s):
- 1710558
- PAR ID:
- 10337045
- Date Published:
- Journal Name:
- Quantum Information and Computation
- Volume:
- 22
- Issue:
- 7&8
- ISSN:
- 1533-7146
- Page Range / eLocation ID:
- 541 to 568
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.26421/QIC22.7-8-1
