Abstract The symmetric subspace of multi-qubit systems, that is, the space of states invariant under permutations, is commonly encountered in applications in the context of quantum information and communication theory. It is known that the symmetric subspace can be described in terms of irreducible representations of the group SU(2), whose representation spaces form a basis of symmetric states, the so-called Dicke states. In this work, we present deformations of the symmetric subspace as deformations of this group structure, which are promoted to a quantum group U q ( su ( 2 ) ) . We see that deformations of the symmetric subspace obtained in this manner correspond to local deformations of the inner product of each spin, in such a way that departure from symmetry can be encoded in a position-dependent inner product. The consequences and possible extensions of these results are also discussed.
The unitary group acting on the Hilbert space
H:=(C2)⊗3 of three quantum bits admits a Lie subgroup,
US3(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
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
US3(8). This concerns their entanglement properties, separability criteria and dynamics under the Lie subgroup
US3(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
21 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.
Abstract A natural symmetry group of a multicomponent quantum system is a special combination of a symmetry group acting within a single component (“local group”) and a group that permutes the components (“spatial symmetry group”). This combination is called the wreath product. Unitary representations of wreath products describe quantum evolutions of multipartite systems. It is known that any unitary representation of a finite group is contained in some permutation representation. We describe an algorithm for decomposing permutation representations of wreath products into irreducible components. This decomposition makes it possible to study the quantum behavior (entanglement, non-local correlations, etc.) of multipartite systems in invariant subspaces of the permutation Hilbert space.
A quantum many-body scar system usually contains a special nonthermal subspace (approximately) decoupled from the rest of the Hilbert space. In this work, we propose a general structure called deformed symmetric spaces for the decoupled subspaces hosting quantum many-body scars, which are irreducible sectors of simple Lie groups transformed by matrix-product operators (or projected entangled pair operators), of which the entanglement entropies are proved to obey sub-volume-law scaling and thus violate the eigenstate thermalization hypothesis. A deformed symmetric space, in general, is required to have at least a U(1) sub-Lie-group symmetry to allow coherent periodic dynamics from certain low-entangled initial states. We enumerate several possible deforming transformations based on the subgroup symmetry requirement and recover many exact scar eigenstates in the existing scar models, of which the Hamiltonians can be reconstructed using the standard quantum inverse method. In particular, a two-dimensional scar model is proposed, which hosts a periodic dynamical trajectory on which all states are topologically ordered.
Abstract Symmetries are omnipresent in physics and have been used to reduce the number of degrees of freedom of systems. In this work, we investigate the properties of M SU ( n ) , M -invariant subspaces of the special unitary Lie group SU(n). This group is relevant to quantum computing and quantum systems in general. We demonstrate that for certain choices of M , the subset M SU ( n ) inherits many topological and group properties from SU(n). We then present a combinatorial method for computing the dimension of such subspaces when M is a representation of a permutation group acting on a tensor product of spaces G SU ( n ) , or a Hamiltonian H ( n ) SU ( n ) . The Kronecker product of su ( 2 ) matrices is employed to construct the Lie algebras associated with different permutation-invariant groups GSU(n). Numerical results on the number of dimensions support the developed theory.
The study of entanglement properties of multiqubit states that are invariant under permutations of qubits is motivated by potential applications in quantum computing, quantum communication, and quantum metrology. In this work we generalize the notions of symmetrization, Dicke states, and the Majorana representation to the alternating, cyclic, and dihedral subgroups of the full group of permutations. We use these tools to characterize states that are invariant under these subgroups and analyze their entanglement properties.
Abstract Dicke states are completely symmetric states of multiple qubits (2-level systems), and qudit Dicke states are their d -level generalization. We define here q -deformed qudit Dicke states using the quantum algebra s u q ( d ) . We show that these states can be compactly expressed as a weighted sum over permutations with q -factors involving the so-called inversion number, an important permutation statistic in Combinatorics. We use this result to compute the bipartite entanglement entropy of these states. We also discuss the preparation of these states on a quantum computer, and show that introducing a q -dependence does not change the circuit gate count.