Computing the entanglement of formation of a bipartite state is generally difficult, but special symmetries of a state can simplify the problem. For instance, this allows one to determine the entanglement of formation of Werner states and isotropic states. We consider a slightly more general class of states, rotationally symmetric states, also known as SU(2)-invariant states. These states are invariant under global rotations of both subsystems, and one can examine entanglement in cases where the subsystems have different dimensions. We derive an analytic expression for the entanglement of formation of rotationally symmetric states of a spin- $j$ particle and a spin- ${1\over2}$ particle. We also give expressions for the I-concurrence, I-tangle, and convex-roof-extended negativity.Computing the entanglement of formation of a bipartite state is generally difficult, but special symmetries of a state can simplify the problem. For instance, this allows one to determine the entanglement of formation of Werner states and isotropic states. We consider a slightly more general class of states, rotationally symmetric states, also known as SU(2)-invariant states. These states are invariant under global rotations of both subsystems, and one can examine entanglement in cases where the subsystems have different dimensions. We derive an analytic expression for the entanglement of formation of rotationally symmetric states of a spin-$j$ particle and a spin-${1\over2}$ particle. We also give expressions for the I-concurrence, I-tangle, and convex-roof-extended negativity.

arXiv:quant-ph/0506151Jun 17, 20050 citations

Entanglement of formation of rotationally symmetric states

Kiran K. Manne, Carlton M. Caves

Computing the entanglement of formation of a bipartite state is generally difficult, but special symmetries of a state can simplify the problem. For instance, this allows one to determine the entanglement of formation of Werner states and isotropic states. We consider a slightly more general class of states, rotationally symmetric states, also known as SU(2)-invariant states. These states are invariant under global rotations of both subsystems, and one can examine entanglement in cases where the subsystems have different dimensions. We derive an analytic expression for the entanglement of formation of rotationally symmetric states of a spin- $j$ particle and a spin- ${1\\over2}$ particle. We also give expressions for the I-concurrence, I-tangle, and convex-roof-extended negativity.Computing the entanglement of formation of a bipartite state is generally difficult, but special symmetries of a state can simplify the problem. For instance, this allows one to determine the entanglement of formation of Werner states and isotropic states. We consider a slightly more general class of states, rotationally symmetric states, also known as SU(2)-invariant states. These states are invariant under global rotations of both subsystems, and one can examine entanglement in cases where the subsystems have different dimensions. We derive an analytic expression for the entanglement of formation of rotationally symmetric states of a spin-$j$ particle and a spin-${1\\over2}$ particle. We also give expressions for the I-concurrence, I-tangle, and convex-roof-extended negativity.

Quantum Information and ComputationMar 1, 20086 citations

Analytic computation of the entanglement of formation for bipartite isotropic states using their symmetry

Y. Akbari-Kourbolagh, E. Shahhoseini

Abstract It is well-known that computing the entanglement of formation of a general bipartite state is a difficult task. We use the local symmetry of bipartite isotropic states in arbitrary dimensions to compute their entanglement of formation analytically. Our computation is an explicit and detailed version of the computation performed by Terhal et al [B. M. Terhal and K. G. H. Vollbrecht, Phys. Rev. Lett. 85, 2625 (2000)]. However, it is not just a review but we also have our contribution and novelty. For instance, we provide a rigorous proof for the entanglement of formation of two qutrit isotropic states and notably we give a proof supporting the conjecture proposed by Terhal et al about the entanglement of formation of bipartite isotropic states for the last range of allowed parameter values.Abstract It is well-known that computing the entanglement of formation of a general bipartite state is a difficult task. We use the local symmetry of bipartite isotropic states in arbitrary dimensions to compute their entanglement of formation analytically. Our computation is an explicit and detailed version of the computation performed by Terhal et al [B. M. Terhal and K. G. H. Vollbrecht, Phys. Rev. Lett. 85, 2625 (2000)]. However, it is not just a review but we also have our contribution and novelty. For instance, we provide a rigorous proof for the entanglement of formation of two qutrit isotropic states and notably we give a proof supporting the conjecture proposed by Terhal et al about the entanglement of formation of bipartite isotropic states for the last range of allowed parameter values.

Physica ScriptaJun 10, 20240 citations

Entanglement in SU(2)-invariant quantum systems: The PPT criterion and others

John Schliemann

We study entanglement in mixed bipartite quantum states which are invariant under simultaneous SU(2) transformations in both subsystems. Previous results on the behavior of such states under partial transposition are substantially extended. The spectrum of the partial transpose of a given SU(2)-invariant density matrix $\\rho$ is entirely determined by the diagonal elements of $\\rho$ in a basis of tensor-product states of both spins with respect to a common quantization axis. We construct a set of operators which act as entanglement witnesses on SU(2)-invariant states. A sufficient criterion for $\\rho$ having a negative partial transpose is derived in terms of a simple spin correlator. The same condition is a necessary criterion for the partial transpose to have the maximum number of negative eigenvalues. Moreover, we derive a series of sum rules which uniquely determine the eigenvalues of the partial transpose in terms of a system of linear equations. Finally we compare our findings with other entanglement criteria including the reduction criterion, the majorization criterion, and the recently proposed local uncertainty relations.We study entanglement in mixed bipartite quantum states which are invariant under simultaneous SU(2) transformations in both subsystems. Previous results on the behavior of such states under partial transposition are substantially extended. The spectrum of the partial transpose of a given SU(2)-invariant density matrix $\\rho$ is entirely determined by the diagonal elements of $\\rho$ in a basis of tensor-product states of both spins with respect to a common quantization axis. We construct a set of operators which act as entanglement witnesses on SU(2)-invariant states. A sufficient criterion for $\\rho$ having a negative partial transpose is derived in terms of a simple spin correlator. The same condition is a necessary criterion for the partial transpose to have the maximum number of negative eigenvalues. Moreover, we derive a series of sum rules which uniquely determine the eigenvalues of the partial transpose in terms of a system of linear equations. Finally we compare our findings with other entanglement criteria including the reduction criterion, the majorization criterion, and the recently proposed local uncertainty relations.

Mar 14, 20050 citations

Bipartite entanglement of quantum states in a pair basis

M. Roncaglia, Arianna Montorsi, Marco Genovese

The unambiguous detection and quantification of entanglement is a hot topic of scientific research, though it is limited to low dimensions or specific classes of states. Here we identify an additional class of quantum states, for which bipartite entanglement measures can be efficiently computed, providing rigorous results. Such states are written in arbitrary $d\ifmmode\times\else\texttimes\fi{}d$ dimensions, where each basis state in the subsystem A is paired with only one state in B. This class, to which we refer as pair basis states, is remarkably relevant in many physical situations, including quantum optics. We find that negativity is a necessary and sufficient measure of entanglement for mixtures of states written in the same pair basis. We also provide analytical expressions for a tight lower-bound estimation of the entanglement of formation, a central quantity in quantum information.The unambiguous detection and quantification of entanglement is a hot topic of scientific research, though it is limited to low dimensions or specific classes of states. Here we identify an additional class of quantum states, for which bipartite entanglement measures can be efficiently computed, providing rigorous results. Such states are written in arbitrary $d\ifmmode\times\else\texttimes\fi{}d$ dimensions, where each basis state in the subsystem A is paired with only one state in B. This class, to which we refer as pair basis states, is remarkably relevant in many physical situations, including quantum optics. We find that negativity is a necessary and sufficient measure of entanglement for mixtures of states written in the same pair basis. We also provide analytical expressions for a tight lower-bound estimation of the entanglement of formation, a central quantity in quantum information.

arXiv:1308.6783Physical Review ADec 1, 201440 citations

Symmetry and Classification of Multipartite Entangled States

Adam Burchardt

One of the key manifestations of quantum mechanics is the phenomenon of quantum entanglement. While the entanglement of bipartite systems is already well understood, our knowledge of entanglement in multipartite systems is still limited. This dissertation covers various aspects of the quantification of entanglement in multipartite states and the role of symmetry in such systems. Firstly, we establish a connection between the classification of multipartite entanglement and knot theory and investigate the family of states that are resistant to particle loss. Furthermore, we construct several examples of such states using the Majorana representation as well as some combinatorial methods. Secondly, we introduce classes of highly-symmetric but not fully-symmetric states and investigate their entanglement properties. Thirdly, we study the well-established class of Absolutely Maximally Entangled (AME) quantum states. On one hand, we provide construction of new states belonging to this family, for instance, an AME state of 4 subsystems with six levels each, on the other, we tackle the problem of equivalence of such states. Finally, we present a novel approach for the general problem of verification of the equivalence between any pair of arbitrary quantum states based on a single polynomial entanglement measure.One of the key manifestations of quantum mechanics is the phenomenon of quantum entanglement. While the entanglement of bipartite systems is already well understood, our knowledge of entanglement in multipartite systems is still limited. This dissertation covers various aspects of the quantification of entanglement in multipartite states and the role of symmetry in such systems. Firstly, we establish a connection between the classification of multipartite entanglement and knot theory and investigate the family of states that are resistant to particle loss. Furthermore, we construct several examples of such states using the Majorana representation as well as some combinatorial methods. Secondly, we introduce classes of highly-symmetric but not fully-symmetric states and investigate their entanglement properties. Thirdly, we study the well-established class of Absolutely Maximally Entangled (AME) quantum states. On one hand, we provide construction of new states belonging to this family, for instance, an AME state of 4 subsystems with six levels each, on the other, we tackle the problem of equivalence of such states. Finally, we present a novel approach for the general problem of verification of the equivalence between any pair of arbitrary quantum states based on a single polynomial entanglement measure.

arXiv:2204.13441Apr 28, 20221 citation

j

particle and a spin-

{1\over2}

particle. We also give expressions for the I-concurrence, I-tangle, and convex-roof-extended negativity.

j

particle and a spin-

{1\over2}

particle. We also give expressions for the I-concurrence, I-tangle, and convex-roof-extended negativity.

j

particle and a spin-

{1\\over2}

particle. We also give expressions for the I-concurrence, I-tangle, and convex-roof-extended negativity.

\\rho

is entirely determined by the diagonal elements of

\\rho

in a basis of tensor-product states of both spins with respect to a common quantization axis. We construct a set of operators which act as entanglement witnesses on SU(2)-invariant states. A sufficient criterion for

\\rho

having a negative partial transpose is derived in terms of a simple spin correlator. The same condition is a necessary criterion for the partial transpose to have the maximum number of negative eigenvalues. Moreover, we derive a series of sum rules which uniquely determine the eigenvalues of the partial transpose in terms of a system of linear equations. Finally we compare our findings with other entanglement criteria including the reduction criterion, the majorization criterion, and the recently proposed local uncertainty relations.