arXiv:1803.06689 · Journal of Mathematical Physics

Controllability of symmetric spin networks

Francesca Albertini, Domenico D’Alessandro

Published May 1, 2018• 25 citations

Abstract

We consider a network of n spin 12 systems which are pairwise interacting via Ising interaction and are controlled by the same electro-magnetic control field. Such a system presents symmetries since the Hamiltonian is unchanged if we permute two spins. This prevents full (operator) controllability, in that not every unitary evolution can be obtained. We prove however that controllability is verified if we restrict ourselves to unitary evolutions which preserve the above permutation invariance. For low dimensional cases, n = 2 and n = 3, we provide an analysis of the Lie group of available evolutions and give explicit control laws to transfer between two arbitrary permutation invariant states. This class of states includes highly entangled states such as Greenberger-Horne-Zeilinger (GHZ) states and W states, which are of interest in quantum information.

DOI: 10.1063/1.5004652

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arXiv:1803.06689 · Journal of Mathematical Physics

Controllability of symmetric spin networks

Francesca Albertini, Domenico D’Alessandro

Published May 1, 2018• 25 citations

arXiv HTML PDF

Abstract

DOI: 10.1063/1.5004652

Related papers

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Jiahui Chen, Hui Zhou, Chang‐Kui Duan +1 more

Entanglement, a unique quantum resource with no classical counterpart, remains at the heart of quantum information. The Greenberger-Horne-Zeilinger (GHZ) and $W$ states are two inequivalent classes of multipartite entangled states which cannot be transformed into each other by means of local operations and classic communication. In this paper, we present the methods to prepare the GHZ and $W$ states via global controls on a long-range Ising spin model. For the GHZ state, general solutions are analytically obtained for an arbitrary-size spin system, while for the $W$ state, we find a standard way to prepare the $W$ state that is analytically illustrated in three- and four-spin systems and numerically demonstrated for larger-size systems. The number of parameters required in the numerical search increases only linearly with the size of the system.Entanglement, a unique quantum resource with no classical counterpart, remains at the heart of quantum information. The Greenberger-Horne-Zeilinger (GHZ) and $W$ states are two inequivalent classes of multipartite entangled states which cannot be transformed into each other by means of local operations and classic communication. In this paper, we present the methods to prepare the GHZ and $W$ states via global controls on a long-range Ising spin model. For the GHZ state, general solutions are analytically obtained for an arbitrary-size spin system, while for the $W$ state, we find a standard way to prepare the $W$ state that is analytically illustrated in three- and four-spin systems and numerically demonstrated for larger-size systems. The number of parameters required in the numerical search increases only linearly with the size of the system.

arXiv:1703.03000Physical Review AMar 31, 201727 citations

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.

We describe a framework for the controllability analysis of networks of

n

quantum systems of an arbitrary dimension

d

, {\it qudits}, with dynamics determined by Hamiltonians that are invariant under the permutation group

S_n

. Because of the symmetry, the underlying Hilbert space,

{\cal H}=(\mathbb{C}^d)^{\otimes n}

, splits into invariant subspaces for the Lie algebra of

S_n

-invariant elements in

u(d^n)

, denoted here by

u^{S_n}(d^n)

. The dynamical Lie algebra

{\cal L}

, which determines the controllability properties of the system, is a Lie subalgebra of such a Lie algebra

u^{S_n}(d^n)

. If

{\cal L}

acts as

su\left( \dim(V) \right)

on each of the invariant subspaces

V

, the system is called {\it subspace controllable}. Our approach is based on recognizing that such a splitting of the Hilbert space

{\cal H}

coincides with the {\it Clebsch-Gordan} splitting of

(\mathbb{C}^d)^{\otimes n}

into {\it irreducible representations} of

su(d)

. In this view,

u^{S_n}(d^n)

, is the direct sum of certain

su(n_j)

for some

n_j

's we shall specify, and its {\it center} which is the Abelian (Lie) algebra generated by the {\it Casimir operators}. Generalizing the situation previously considered in the literature, we consider dynamics with arbitrary local simultaneous control on the qudits and a symmetric two body interaction. Most of the results presented are for general

n

and

d

but we recast previous results on

n

qubits in this new general framework and provide a complete treatment and proof of subspace controllability for the new case of

n=3

d=3

, that is, {\it three qutrits}.

Entanglement, a unique quantum resource with no classical counterpart, remains at the heart of quantum information. The Greenberger-Horne-Zeilinger (GHZ) and

W

states are two inequivalent classes of multipartite entangled states which cannot be transformed into each other by means of local operations and classic communication. In this paper, we present the methods to prepare the GHZ and

W

states via global controls on a long-range Ising spin model. For the GHZ state, general solutions are analytically obtained for an arbitrary-size spin system, while for the

W

state, we find a standard way to prepare the

W

state that is analytically illustrated in three- and four-spin systems and numerically demonstrated for larger-size systems. The number of parameters required in the numerical search increases only linearly with the size of the system.