Constructing a ball of separable and absolutely separable states for

Constructing a ball of separable and absolutely separable states for $2\otimes d$ quantum systemConstructing a ball of separable and absolutely separable states for $$2\otimes d$$ quantum system

Satyabrata Adhikari

arXiv:2007.00891The European Physical Journal DMar 1, 20211 citation

Characterizing the boundary of the set of absolutely separable states and their generation via noisy environments

Saronath Halder, Shiladitya Mal, Aditi Sen

We characterize the boundary of the convex compact set of absolutely separable states, referred as AS, that cannot be transformed to entangled states by global unitary operators, in $2\ensuremath{\bigotimes}d$ Hilbert space. However, we show that the absolutely separable states of rank $(2d\ensuremath{-}1)$ are extreme points of such sets. We then discuss conditions to examine if a given full-rank absolutely separable state is an interior point or a boundary point of AS. Moreover, we construct two-qubit absolutely separable states which are boundary points but not extreme points of AS and prove the existence of full-rank extreme points of AS. Properties of certain interior points are also explored. We further show that by examining the boundary of the above set, it is possible to develop an algorithm to generate the absolutely separable states which stay outside the maximal ball. By considering paradigmatic noise models, we find the amount of local noise which the input entangled states can sustain, so that the output states do not become absolutely separable. Interestingly, we report that with the decrease of entanglement of the pure input state, critical depolarizing noise value, transferring an entangled state to an absolutely separable one, increases, thereby showing advantages of sharing nonmaximally entangled states. Furthermore, when the input two-qubit states are Haar uniformly generated, we report a hierarchy among quantum channels according to the generation of absolutely separable states.We characterize the boundary of the convex compact set of absolutely separable states, referred as AS, that cannot be transformed to entangled states by global unitary operators, in $2\ensuremath{\bigotimes}d$ Hilbert space. However, we show that the absolutely separable states of rank $(2d\ensuremath{-}1)$ are extreme points of such sets. We then discuss conditions to examine if a given full-rank absolutely separable state is an interior point or a boundary point of AS. Moreover, we construct two-qubit absolutely separable states which are boundary points but not extreme points of AS and prove the existence of full-rank extreme points of AS. Properties of certain interior points are also explored. We further show that by examining the boundary of the above set, it is possible to develop an algorithm to generate the absolutely separable states which stay outside the maximal ball. By considering paradigmatic noise models, we find the amount of local noise which the input entangled states can sustain, so that the output states do not become absolutely separable. Interestingly, we report that with the decrease of entanglement of the pure input state, critical depolarizing noise value, transferring an entangled state to an absolutely separable one, increases, thereby showing advantages of sharing nonmaximally entangled states. Furthermore, when the input two-qubit states are Haar uniformly generated, we report a hierarchy among quantum channels according to the generation of absolutely separable states.

arXiv:1911.13145Physical Review AMay 26, 202117 citations

Integral representations of separable states

Bronisław Jakubczyk, Gabriel Pietrzkowski

arXiv:0802.0291Reports on Mathematical PhysicsFeb 1, 20092 citations

Maximum entanglement of mixed symmetric states under unitary transformations

Eduardo Serrano-Ensástiga, John Martin

We study the maximum entanglement that can be produced by a global unitary transformation for systems of two and three qubits constrained to the fully symmetric states. This restriction to the symmetric subspace appears naturally in the context of bosonic or collective spin systems. We also study the symmetric states that remain separable after any global unitary transformation, called symmetric absolutely separable (SAS) states, or absolutely classical for spin states. The results for the two-qubit system are deduced analytically. In particular, we determine the maximal radius of a ball of SAS states around the maximally mixed state in the symmetric sector, and the minimal radius of a ball that contains the set of SAS states. As an application of our results, we also analyse the temperature dependence of the maximum entanglement that can be obtained from the thermal state of a spin-1 system with a spin-squeezing Hamiltonian. For the symmetric three-qubit case, our results are mostly numerical, and we conjecture a 3-parameter family of states that achieves the maximum negativity in the unitary orbit of any mixed state. In addition, we derive upper bounds, apparently tight, on the radii of balls containing only/all SAS states.We study the maximum entanglement that can be produced by a global unitary transformation for systems of two and three qubits constrained to the fully symmetric states. This restriction to the symmetric subspace appears naturally in the context of bosonic or collective spin systems. We also study the symmetric states that remain separable after any global unitary transformation, called symmetric absolutely separable (SAS) states, or absolutely classical for spin states. The results for the two-qubit system are deduced analytically. In particular, we determine the maximal radius of a ball of SAS states around the maximally mixed state in the symmetric sector, and the minimal radius of a ball that contains the set of SAS states. As an application of our results, we also analyse the temperature dependence of the maximum entanglement that can be obtained from the thermal state of a spin-1 system with a spin-squeezing Hamiltonian. For the symmetric three-qubit case, our results are mostly numerical, and we conjecture a 3-parameter family of states that achieves the maximum negativity in the unitary orbit of any mixed state. In addition, we derive upper bounds, apparently tight, on the radii of balls containing only/all SAS states.

arXiv:2112.05102SciPost PhysicsSep 28, 20236 citations

Constructing a ball of separable and absolutely separable states for

2\otimes d

quantum system